5.7: De Morgan’s Laws
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Learning Objectives
After completing this section, you should be able to:
- Use De Morgan’s Laws to negate conjunctions and disjunctions.
- Construct the negation of a conditional statement.
- Use truth tables to evaluate De Morgan’s Laws.
The contributions to logic made by Augustus De Morgan and George Boole during the 19th century acted as a bridge to the development of computers, which may be the greatest invention of the 20th century. Boolean logic is the basis for computer science and digital electronics, and without it the technological revolution of the late 20th and early 21st centuries—including the creation of computer chips, microprocessors, and the Internet—would not have been possible. Every modern computer language uses Boolean logic statements, which are translated into commands understood by the underlying electronic circuits enabling computers to operate. But how did this logic get its name?
People in Mathematics: George Boole
George Boole was born in Lincolnshire, England in 1815. He was the son of a cobbler who provided him some initial education, but Boole was mostly self-taught. He began teaching at 16 years of age, and opened his own school at the age of 20. In 1849, at the age of 34, he was appointed Professor of Mathematics at Queens College in Cork, Ireland. In 1853, he published the paper, An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities , which is the treatise that the field of Boolean algebra and digital circuitry was built on.
Reference: Posamentier, Alfred and Spreitzer Christian, “Chapter 35 George Boole: English (1815-1864)” pp. 279-283, Math Makers: The Lives and Works of 50 Famous Mathematicians , Prometheus Books, 2019.
Negation of Conjunctions and Disjunctions
In Chapter 1, Example 1.37 used a Venn diagram to prove De Morgan’s Law for set complement over union. Because the complement of a set is analogous to negation and union is analogous to an or statement, there are equivalent versions of De Morgan’s Laws for logic.
FORMULA
De Morgan’s Law for negation of a conjunction:
De Morgan’s Law for the negation of a disjunction:
Negation of a conditional:
Writing conditional as a disjunction:
Checkpoint
Recall that the symbol for logical equivalence is:
De Morgan’s Laws allow us to write the negation of conjunctions and disjunctions without using the phrase, “It is not the case that …” to indicate the parentheses. Avoiding this phrase often results in a written or verbal statement that is clearer or easier to understand.
Exercise \(\PageIndex{1}\): Applying De Morgan’s Law for Negation of Conjunctions and Disjunctions
Write the negation of each statement in words without using the phrase, “It is not the case that.”
- Kristin is a biomedical engineer and Thomas is a chemical engineer.
- A person had cake or they had ice cream.
- Answer
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1. Kristin is a biomedical engineer and Thomas is a chemical engineer has the form " \(p \wedge q\)," where \(p\) is the statement, "Kristin is a biomedical engineer," and \(q\) is the statement, "Thomas is a chemical engineer." By De Morgan's Law, the negation of a conjunction, \(\sim(p \wedge q)\), is logically equivalent to \(\sim p \vee \sim q . \sim p\) is "Kristen is not a biomedical engineer," and \(\sim q\) is "Thomas is not a chemical engineer." By De Morgan's Law, the solution has the form " \(\sim p \vee \sim q\)," so the answer is: "Kristin is not a biomedical engineer or Tom is not a chemical engineer."
2. A person had cake or they had ice cream has the form " \(p \vee q\)," where \(p\) is the statement, "A person had cake," and \(q\) is the statement, "A person had ice cream." By De Morgan's Law for the negations of a disjunction, \(\sim(p \vee q) \equiv \sim p \wedge \sim q\). The solution is the statement: "A person did not have cake and they did not have ice cream."
is not a chemical engineer.”
Negation of a Conditional Statement
The negation of any statement has the opposite truth values of the original statement. The negation of a conditional , , is the conjunction of and not , Consider the truth table below for the negation of the conditional.
| T | T | T | F |
| T | F | F | T |
| F | T | T | F |
| F | F | T | F |
The only time the negation of the conditional statement is true is when is true, and is false. This means that is logically equivalent to as the following truth table demonstrates.
| T | T | T | F | F | F | T |
| T | F | F | T | T | T | T |
| F | T | T | F | F | F | T |
| F | F | T | F | T | F | T |
Exercise \(\PageIndex{2}\): Constructing the Negation of a Conditional Statement
Write the negation of each conditional statement.
- If Adele won a Grammy, then she is a singer.
- If Henrik Lundqvist played professional hockey, then he did not win the Stanley Cup.
- Answer
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1. The negation of the conditional statement, \(p \rightarrow q\), is the statement, \(p \wedge \sim q\). The hypothesis of the conditional statement is \(p\) : "Adele won a Grammy," and conclusion of the conditional statement is \(q\) : "Adele is a singer." The negation of the conclusion, \(\sim q\), is the statement: "She is not a singer." Therefore, the answer is \(p \wedge \sim q\) : "Adele won a Grammy, and she is not a singer."
2. The hypothesis is \(p\) : "Henrik Lundqvist played professional hockey," and the conclusion of the conditional statement is \(q\) : "He did not win the Stanley Cup." The negation of \(q\) is the statement: "He won the Stanley Cup." The negation of the conditional statement is equal to \(p \wedge \sim q\) : "Henrick Lundqvist played professional hockey, and he won the Stanley Cup."
Exercise \(\PageIndex{3}\): Constructing the Negation of a Conditional Statement with Quantifiers
Write the negation of each conditional statement.
- If all cats purr, then my partner’s cat purrs.
- If a penguin is a bird, then some birds do not fly.
- Answer
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1. The negation of the conditional statement \(p \rightarrow q\) is the statement \(p \wedge \sim q\). The hypothesis of the conditional statement is \(p\) : "All cats purr," and the conclusion of the conditional statement is \(q\) : "My partner's cat purrs." The negation of the conclusion, \(\sim q\), is the statement: "My partner's cat does not purr." Therefore, the answer is \(p \wedge \sim q\) : "All cats purr, but my partner's cat does not purr."
2. The hypothesis is \(p\) : "A penguin is a bird," and the conclusion of the conditional statement is \(q\) : "Some birds do not fly." The negation of \(q\) is the statement: "All birds fly." Therefore, the negation of the conditional statement is equal to \(p \wedge \sim q\) : "A penguin is a bird, and all birds fly."
Many of the properties that hold true for number systems and sets also hold true for logical statements. The following table summarizes some of the most useful properties for analyzing and constructing logical arguments. These properties can be verified using a truth table.
| Property | Conjunction (AND) | Disjunction (OR) |
|---|---|---|
| Commutative | ||
| Associative | ||
| Distributive | ||
| De Morgan’s | ||
| Conditional |
Exercise \(\PageIndex{4}\)
Negating a Conditional Statement with a Conjunction or Disjunction
Write the negation of each conditional statement applying De Morgan’s Law.
- If mom needs to buy chips, then Mike had friends over and Bob was hungry.
- If Juan had pizza or Chris had wings, then dad watched the game.
- Answer
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1. The conditional has the form "If \(p\) then \(q\) or \(r\)," where \(p\) is "Mom needs to buy chips," \(q\) is "Mike had friends over," and \(r\) is "Bob was hungry." The negation of \(p \rightarrow(q \wedge r)\) is \(p \wedge \sim(q \wedge r)\). Applying De Morgan's Law to the statement \(\sim(q \wedge r)\) the result is \(\sim q \vee \sim r\), so our conditional statement becomes \(p \wedge(\sim q \vee \sim r)\). By the distributive property for conjunction over disjunction, this statement is equivalent to \((p \wedge \sim q) \vee(p \wedge \sim r)\). Translating the statement \((p \wedge \sim q) \vee(p \wedge \sim r)\) into words, the solution is: "Mom needs to buy chips and Mike did not have friends over, or Mom needs to buy chips and Bob was not hungry."
2. The conditional has the form "If \(p\) or \(q\), then \(r\)," where \(p\) is "Juan had pizza," \(q\) is "Chris had wings," and \(r\) is "Dad watched the game." The negation of \((p \vee q) \rightarrow r\) is \((p \vee q) \wedge \sim r\). By the distributive property for disjunction over conjuction, the statement is equivalent to \((p \vee \sim r) \wedge(q \vee \sim r)\). Translating the statement \((p \vee \sim r) \wedge(q \vee \sim r)\) into words, the solution is: "Juan had pizza or dad did not watch the game, and Chris had wings or dad did not watch the game."
Evaluating De Morgan’s Laws with Truth Tables
In Chapter 1, you learned that you could prove the validity of De Morgan’s Laws using Venn diagrams. Truth tables can also be used to prove that two statements are logically equivalent. If two statements are logically equivalent, you can use the form of the statement that is clearer or more persuasive when constructing a logical argument.
The next example will prove the validity of one of De Morgan’s Laws using a truth table. The same procedure can be applied to any two logical statement that you believe are equivalent. If the last column of the truth table is a tautology, then the two statements are logically equivalent.
Exercise \(\PageIndex{5}\): Verifying De Morgan’s Law for Negation of a Conjunction
Construct a truth table to verify De Morgan’s Law for the negation of a conjunction, , is valid.
- Answer
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Step 1 : To verify any logical equivalence, you must first replace the logical equivalence symbol, \(\equiv\), with the biconditional symbol, \(\leftrightarrow\). The statement \(\sim(p \wedge q) \equiv \sim p \vee \sim q\) becomes \(\sim(p \wedge q) \leftrightarrow \sim p \vee \sim q\)
Step 2: Next, you create a truth table for the statement. Because we have two basic statements, \(p\), and \(q\), the truth table will have four rows to account for all the possible outcomes.  In this instance, the last column is all true. Therefore, the statement is valid and De Morgan's Law for the negation of a conjunction is verified.
Your Turn \(\PageIndex{5}\)
1. Construct a truth table to verify De Morgan's Law for the negation of a disjunction, \(\sim(p \vee q) \equiv \sim p \wedge \sim q\), is valid.
Check Your Understanding
1. De Morgan’s Law for the negation of a conjunction states that \(\(\text{~}\left( {p \wedge q} \right)\)is logically equivalent to ___________________.
2. De Morgan’s Law for the negation of a disjunction states that \(\text{~}\left( {p \vee q} \right)\) is logically equivalent to __________________.
3. The negation of the conditional statement, \(\text{~}\left( {p \to q} \right)\), is logically equivalent to _____________.
\(\text{~}\left( {\text{~}\left( {p \to q} \right)} \right) \equiv p \to q\), which means the conditional statement is logically equivalent to \(\text{~}\left( {p \wedge \text{~}q} \right).\) Apply _______________________ to the statement \(\text{~}\left( {p \wedge \text{~}q} \right)\)to show that the conditional statement \(p \to q \equiv \text{~}p \vee q.\)


6.2: Symmetric Groups
Definition: Symmetric Group
When \(A=\{1,2,\ldots, n\}\) (\(n\in \mathbb{Z}^+\)), we call \(S_A\) the symmetric group on \(n\) letters and denote it by \(S_n\text{.}\)
(We can, in fact, define \(S_0\text{,}\) the set of all permutations on the empty set. One can show, using the fact that a function is a relation on a Cartesian product of sets, that \(S_0\) is the trivial group. However, this will not be relevant in this text.)
Remark
Throughout this course, if we are discussing a group \(S_n\text{,}\) you should assume \(n\in \mathbb{Z}^+\text{.}\)
It is important for us to be able to easily describe specific elements of \(S_n\text{.}\) It would be cumbersome to describe, for instance, an element of \(S_3\) by saying it swaps \(1\) and \(2\) and fixes \(3\text{;}\) imagine how much more cumbersome it could be to describe an element of, say, \(S_{100}\text{!}\) One can somewhat concisely describe a permutation \(\sigma\) of \(S_n\) by listing out the elements \(1,2,\ldots,n\) and writing the element \(\sigma(i)\) below each \(i\) for \(i=1,2,\ldots, n\text{.}\) For instance, if \(\sigma\) sends \(1\) to \(2\text{,}\) we'd write the number \(2\) below the number \(1\text{.}\) The convention is to enclose these two rows of numbers in a single set of parentheses, as in the following example.
Example \(\PageIndex{1}\)
We can denote the element \(\sigma\) of \(S_3\) that swaps \(1\) and \(2\) and fixes \(3\) by
\begin{equation*} \sigma = \begin{pmatrix}1& 2& 3\\ 2& 1& 3 \end{pmatrix}, \end{equation*}
and the element \(\tau\) of \(S_3\) that sends \(1\) to \(3\text{,}\) \(2\) to \(1\text{,}\) and \(3\) to \(2\) by
\begin{equation*} \tau =\begin{pmatrix}1& 2&3\\ 3& 1& 2 \end{pmatrix}. \end{equation*}
Then
\begin{equation*} \sigma\tau = \begin{pmatrix}1& 2& 3\\ 3& 2&1 \end{pmatrix} \end{equation*}
and
\begin{equation*} \tau\sigma = \begin{pmatrix}1& 2& 3\\ 1& 3& 2 \end{pmatrix}. \end{equation*}
But even this notation is unnecessarily cumbersome. Instead, we use cycle notation .
Definition: Cycle, Cycle Notation, and Transposition
A permutation \(\sigma\) in \(S_n\) is called a \(k\) -cycle or a cycle of length \(k\) (or, less specifically, a cycle ) if there exist distinct elements \(a_1, a_2,\ldots, a_k\) in \(\{1,2,\ldots,n\}\) such that
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\(\sigma(a_i)=a_{i+1}\) for each \(i=1,2,\ldots, k-1\text{;}\)
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\(\sigma(a_k)=a_1\text{;}\)
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\(\sigma(x)=x\) for every other element of \(\{1,2,\ldots, n\}\text{.}\)
We use the cycle notation \(\sigma = (a_1 a_2 \cdots a_k)\) to describe such a cycle. A \(2\)-cycle is often called a transposition .
Example \(\PageIndex{2}\)
The permutation \(\tau\) in \(S_3\) that sends \(1\) to \(3\text{,}\) \(2\) to \(1\text{,}\) and \(3\) to \(2\) is a cycle. It can be denoted by \(\tau =(132)\text{.}\) Similarly, the cycle \(\rho\) in \(S_3\) swapping \(1\) and \(3\) can be denoted by \(\rho=(13)\text{.}\) On the other hand, the permutation in \(S_4\) that swaps \(1\) with \(2\) and \(3\) with \(4\) is not a cycle.
Remark
Given a \(k\)-cycle \(\sigma=(a_1 a_2\cdots a_k)\text{,}\) there are \(k\) different expressions for \(\sigma\text{.}\) Indeed, we have
\begin{equation*} \sigma=(a_1 a_2\cdots a_k)=(a_2 a_3 \cdots a_k a_1)=(a_3 a_4 \cdots a_k a_1 a_2)=\cdots = (a_k a_1 \cdots a_{k-1}). \end{equation*}
Example \(\PageIndex{3}\)
The permutation \(\tau\) described in Example \(6.2.2\) can also be written as \((321)\) and as \((213)\text{.}\)
However, by convention, we usually “start” a cycle \(\sigma\) with the smallest of the numbers that \(\sigma\) doesn't fix: e.g., we'd write \(\sigma=(213)\) as \((132)\text{.}\)
Definition: Disjoint and Mutual Disjointness
Two cycles \(\sigma=(a_1 a_2 \cdots a_k)\) and \(\tau=(b_1 b_2 \cdots b_m)\) are said to be disjoint if \(a_i \neq b_j\) for all \(i\) and \(j\text{.}\) Cycles \(\sigma_1\text{,}\) \(\sigma_2\text{,}\) \(\ldots\text{,}\) \(\sigma_m\) are disjoint if \(\sigma_i\) and \(\sigma_j\) are disjoint for each \(i \neq j\text{.}\) (Notice: this version of disjointness is what we usually refer to as mutual disjointness .)
Remark
Note that if cycles \(\sigma\) and \(\tau\) are disjoint, then \(\sigma\) and \(\tau\) commute; that is, \(\sigma \tau=\tau \sigma\text{.}\)
Note
If cycles \(\sigma\) and \(\tau\) are not disjoint then they may not commute. For instance, see Example 6.2.3, where \(\sigma\tau \neq \tau \sigma\text{.}\)
Note that any permutation of \(S_n\) is a product of disjoint cycles (where by “product” we mean the permutation resulting from permutation multiplication).
Definition: (Disjoint) Cycle Notation
Writing a permutation in (disjoint) cycle notation means writing it as a product of disjoint cycles, where each cycle is written in cycle notation.
Remark
Note that if \(\sigma\) in \(S_n\) is written in cycle notation and the number \(a\in \{1,2,\ldots, n\}\) appears nowhere in \(\sigma\)'s representation, this means that \(\sigma\) fixes \(a\text{.}\) The only permutation that we cannot really write in cycle notation is the identity element of \(S_n\text{,}\) which we henceforth denote by \(e\text{.}\)
Example \(\PageIndex{4}\)
The permutation
\begin{equation*} \sigma =\begin{pmatrix}1& 2& 3& 4& 5& 6\\ 3& 1& 2&6& 5& 4 \end{pmatrix} \end{equation*}
is the product of disjoint cycles \((132)\) and \((46)\text{,}\) so in cycle notation we have
\begin{equation*} \sigma=(132)(46). \end{equation*}
Note that we could also write \(\sigma\) as \((321)(46)\text{,}\) \((213)(64)\text{,}\) \((64)(132)\text{,}\) etc.
While it is true that we also have \(\sigma=(13)(23)(46)\text{,}\) this is not a disjoint cycle representation of \(\sigma\) since both \((13)\) and \((23)\) “move” the element \(3\text{.}\)
Example \(\PageIndex{5}\)
In \(S_4\text{,}\) let \(\sigma=(243)\) and \(\tau=(13)(24)\text{.}\) Then \(\sigma \tau=(123)\) and \(\tau \sigma = (134).\)
Example \(\PageIndex{6}\)
In \(S_9\text{,}\) let \(\sigma=(134)\text{,}\) \(\tau=(26)(17)\text{,}\) and \(\rho=(358)(12)\text{.}\) Find the following, writing your answers using disjoint cycle notation.
- \(\sigma^{-1}\)
- \(\sigma^{-1}\tau\sigma\)
- \(\sigma^2\)
- \(\sigma^3\)
- \(\rho^2\)
- \(\rho^{-2}\)
- \(\sigma \tau\)
- \(\sigma \rho\)
Example \(\PageIndex{7}\)
Explicitly express all the elements of \(S_4\) in disjoint cycle notation.
Theorem \(\PageIndex{1}\)
Any \(k\)-cycle has order \(k\) in \(S_n\text{.}\) More generally, if permutation \(\sigma\) can be written in disjoint cycle notation as \(\sigma=\sigma_1 \sigma_2 \cdots \sigma_m\text{,}\) then
\begin{align*} o(\sigma)& =\text{lcm}(o(\sigma_1), o(\sigma_2),\ldots, o(\sigma_m))\\ & =\text{lcm}(\mathrm{length}(\sigma_1),\mathrm{length}(\sigma_2),\ldots,\mathrm{length}(\sigma_m)), \end{align*}
where \(\text{lcm}\) denotes the least common multiple.
Note
Permutation \(\sigma\) must be in disjoint cycle notation for the above formula to hold. For instance, let \(\sigma=(12)(23)\) in \(S_3\text{.}\) The transpositions \((12)\) and \((23)\) both have order \(2\text{,}\) but \(o(\sigma)\neq \text{lcm}(2,2)=2\text{.}\) Rather, \(o(\sigma)=3\text{,}\) since in disjoint cycle notation \(\sigma\) can be written as \(\sigma=(123)\text{.}\) You must write a permutation using disjoint cycle notation before attempting to use this method to compute its order!
Example \(\PageIndex{8}\)
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Find the orders of each of the elements in Example \(6.2.6\), including \(\sigma\text{,}\) \(\tau\text{,}\) and \(\rho\) themselves.
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Explicitly list the elements of \(\langle \sigma\rangle\text{,}\) \(\langle \tau\rangle\text{,}\) and \(\langle \rho\rangle\text{.}\)


4.5: Conformations of Cyclohexane
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Objectives
After completing this section, you should be able to
- explain why cyclohexane rings are free of angular strain.
- draw the structure of a cyclohexane ring in the chair conformation.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- chair conformation
- twist-boat conformation
We will find that cyclohexanes tend to have the least angle strain and consequently are the most common cycloalkanes found in nature. A wide variety of compounds including, hormones, pharmaceuticals, and flavoring agents have substituted cyclohexane rings.
testosterone, which contains three cyclohexane rings and one cyclopentane ring
Rings larger than cyclopentane would have angle strain if they were planar. However, this strain, together with the eclipsing strain inherent in a planar structure, can be relieved by puckering the ring. Cyclohexane is a good example of a carbocyclic system that virtually eliminates eclipsing and angle strain by adopting non-planar conformations. Cycloheptane and cyclooctane have greater strain than cyclohexane, in large part due to transannular crowding (steric hindrance by groups on opposite sides of the ring). Cyclohexane has the possibility of forming multiple conformations each of which have structural differences which lead to different amounts of ring strain.
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Conformations of Cyclohexane
A planar structure for cyclohexane is clearly improbable. The bond angles would necessarily be 120º, 10.5º larger than the ideal tetrahedral angle. Also, every carbon-hydrogen bond in such a structure would be eclipsed. The resulting angle and eclipsing strains would severely destabilize this structure. The ring strain of planar cyclohexane is in excess of 84 kJ/mol so it rarely discussed other than in theory.
Cyclohexane in the strained planar configuration showing how the hydrogens become eclipsed.
Chair Conformation of Cyclohexane
The flexibility of cyclohexane allows for a conformation which is almost free of ring strain. If two carbon atoms on opposite sides of the six-membered ring are bent out of the plane of the ring, a shape is formed that resembles a reclining beach chair. This chair conformation is the lowest energy conformation for cyclohexane with an overall ring strain of 0 kJ/mol. In this conformation, the carbon-carbon ring bonds are able to assume bonding angles of ~111 o which is very near the optimal tetrahedral 109.5 o so angle strain has been eliminated.
Also, the C-H ring bonds are staggered so torsional strain has also been eliminated. This is clearly seen when looking at a Newman projection of chair cyclohexane sighted down the two central C-C bonds.
Newman projection of cyclohexane
Interactive Element
The 3D Structure of Chair Cyclohexane
How to Draw the Chair Conformation
Boat Conformation of Cyclohexane
The Boat Conformation of cyclohexane is created when two carbon atoms on opposite sides of the six-membered ring are both lifted up out of the plane of the ring creating a shape which slightly resembles a boat. The boat conformation is less stable than the chair form for two major reasons. The boat conformation has unfavorable steric interactions between a pair of 1,4 hydrogens (the so-called "flagpole" hydrogens) that are forced to be very close together (1.83Å). This steric hindrance creates a repulsion energy of about 12 kJ/mol. An additional cause of the higher energy of the boat conformation is that adjacent hydrogen atoms on the 'bottom of the boat' are forced into eclipsed positions. For these reasons, the boat conformation about 30 kJ/mol less stable than the chair conformation.
A boat structure of cyclohexane (the interfering "flagpole" hydrogens are shown in red)
Twist-Boat Conformation of Cyclohexane
The boat form is quite flexible and by twisting it at the bottom created the twist-boat conformer . This conformation reduces the strain which characterized the boat conformer. The flagpole hydrogens move farther apart (the carbons they are attached to are shifted in opposite directions, one forward and one back) and the eight hydrogens along the sides become largely but not completely staggered. Though more stable than the boat conformation, the twist-boat (sometimes skew-boat) conformation is roughly 23 kJ/mol less stable than the chair conformation.
A twist-boat structure of cyclohexane
Half Chair Conformation of Cyclohexane
Cyclohexane can obtain a partially plane conformation called "half chair" but with only with excessive amounts of ring strain. The half chair conformation is formed by taking planar cyclohexane and lifting one carbon out of the plane of the ring. The half chair conformation has much of the same strain effects predicted by the fully planar cyclohexane. In the planar portion of half chair cyclohexane the C-C bond angles are forced to 120 o which creates significant amounts of angle strain. Also, the corresponding C-H bonds are fully eclipsed which create torsional strain. The out-of-plane carbon allows for some of the ring's bond angles to reach 109.5 o and for some of C-H bonds to not be fully eclipsed. Overall, the half chair conformation is roughly 45 kJ/mol less stable than the chair conformation.
Conformation Changes in Cyclohexane - "Ring Flips"
Cyclohexane is rapidly rotating between the two most stable conformations known as the chair conformations in what is called the "ring flip" shown below. The importance of the ring flip will be discussed in the next section.
"Ring flip" describes the rapid equilibrium of cyclohexane rings between the two chair conformations
It is important to note that one chair does not immediately become the other chair, rather the ring must travel through the higher energy conformations as transitions. At room temperature the energy barrier created by the half chair conformation is easily overcome allowing for equilibration between the two chair conformation on the order of 80,000 times per second. Although cyclohexane is continually converting between these different conformations, the stability of the chair conformation causes it to comprises more than 99.9% of the equilibrium mixture at room temperature.
Exercises
1) Consider the conformations of cyclohexane: half chair, chair, boat, twist boat. Order them in increasing ring strain in the molecule.
Solutions
1) Chair < Twist Boat < Boat < half chair (most ring strain)


10.5: Repeated Eigenvalues with One Eigenvector
Example: Find the general solution of \(\dot{x}_{1}=x_{1}-x_{2}, \dot{x}_{2}=x_{1}+3 x_{2}\) .
The equations in matrix form are
\[\frac{d}{d t}\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right)=\left(\begin{array}{rr} 1 & -1 \\ 1 & 3 \end{array}\right)\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right) \nonumber \]
The ansatz \(\mathrm{x}=\mathrm{v} e^{\lambda t}\) leads to the characteristic equation
\[\begin{aligned} 0 &=\operatorname{det}(\mathbf{A}-\lambda \mathbf{I}) \\ &=\lambda^{2}-4 \lambda+4 \\ &=(\lambda-2)^{2} . \end{aligned} \nonumber \]
Therefore, \(\lambda=2\) is a repeated eigenvalue. The associated eigenvector is found from \(-v_{1}-v_{2}=0\) , or \(v_{2}=-v_{1} ;\) and normalizing with \(v_{1}=1\) , we have
\[\lambda=2, \quad \mathrm{v}=\left(\begin{array}{r} 1 \\ -1 \end{array}\right) \nonumber \]
We have thus found a single solution to the ode, given by
\[\mathrm{x}_{1}(t)=c_{1}\left(\begin{array}{r} 1 \\ -1 \end{array}\right) e^{2 t} \nonumber \]
and we need to find the missing second solution to be able to satisfy the initial conditions. An ansatz of \(t\) times the first solution is tempting, but will fail. Here, we will cheat and find the missing second solution by solving the equivalent secondorder, homogeneous, constant-coefficient differential equation.
If \(A\) has only a single linearly independent eigenvector \(v\) , then Equation \ref{10.13} can be solved for \(w\) (otherwise, it cannot). Using \(A, \lambda\) and \(v\) of our present example, Equation \ref{10.13} is the system of equations given by
\[\left(\begin{array}{rr} -1 & -1 \\ 1 & 1 \end{array}\right)\left(\begin{array}{l} w_{1} \\ w_{2} \end{array}\right)=\left(\begin{array}{r} 1 \\ -1 \end{array}\right) \nonumber \]
The first and second equation are the same, so that \(w_{2}=-\left(w_{1}+1\right)\) . Therefore,
\[\begin{aligned} w &=\left(\begin{array}{c} w_{1} \\ -\left(w_{1}+1\right) \end{array}\right) \\ &=w_{1}\left(\begin{array}{r} 1 \\ -1 \end{array}\right)+\left(\begin{array}{r} 0 \\ -1 \end{array}\right) \end{aligned} \nonumber \]
Notice that the first term repeats the first found solution, i.e., a constant times the eigenvector, and the second term is new. We therefore take \(w_{1}=0\) and obtain
\[\mathrm{w}=\left(\begin{array}{r} 0 \\ -1 \end{array}\right) \nonumber \]
as before.
The phase portrait for this ode is shown in Fig. 10.3. The dark line is the single eigenvector \(\mathbf{v}\) of the matrix \(\mathbf{A}\) . When there is only a single eigenvector, the origin is called an improper node .


11.5: Activities
Develop an inductive definition of the set of words \(\Sigma^{\ast}\) from the alphabet \(\Sigma = \{ \text{a},\text{b},\text{c} \}\text{.}\)
Then verify that the word \(\text{ccababb}\) is in the set by tracing it back to the base clause.
- Hint.
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Steps:
- Think of a simple way to form new words from old (inductive clause).
- Then think about the basic words you need to get the process started (base clause).
- Finally, decide whether you are certain you can form every possible word in a finite number steps starting at some base word.
Let \(\Sigma = \{\text{a},\text{z}\}\text{.}\) Write an inductive definition for the set of words in \(\Sigma^{\ast}\) that have the same number of \(\text{a}\) letters as \(\text{z}\) letters.


10.4: Sampling methods
As we mentioned in a previous section, the first thing we should do before conducting a survey is to identify the population that we want to study. Suppose we are hired by a politician to determine the amount of support he has among the electorate should he decide to run for another term. What population should we study? Every person in the district? Not every person is eligible to vote, and regardless of how strongly someone likes or dislikes the candidate, they don't have much to do with him being re-elected if they are not able to vote.
What about eligible voters in the district? That might be better, but if someone is eligible to vote but does not register by the deadline, they won't have any say in the election either. What about registered voters? Many people are registered but choose not to vote. What about "likely voters?"
This is the criteria used in much political polling, but it is sometimes difficult to define a "likely voter." Is it someone who voted in the last election? In the last general election? In the last presidential election? Should we consider someone who just turned 18 a "likely voter?" They weren't eligible to vote in the past, so how do we judge the likelihood that they will vote in the next election?
In November 1998, former professional wrestler Jesse "The Body" Ventura was elected governor of Minnesota. Up until right before the election, most polls showed he had little chance of winning. There were several contributing factors to the polls not reflecting the actual intent of the electorate:
- Ventura was running on a third-party ticket and most polling methods are better suited to a two-candidate race.
- Many respondents to polls may have been embarrassed to tell pollsters that they were planning to vote for a professional wrestler.
- The mere fact that the polls showed Ventura had little chance of winning might have prompted some people to vote for him in protest to send a message to the major-party candidates.
But one of the major contributing factors was that Ventura recruited a substantial amount of support from young people, particularly college students, who had never voted before and who registered specifically to vote in the gubernatorial election. The polls did not deem these young people likely voters (since in most cases young people have a lower rate of voter registration and a turnout rate for elections) and so the polling samples were subject to sampling bias : they omitted a portion of the electorate that was weighted in favor of the winning candidate.
A sampling method is biased if every member of the population doesn’t have equal likelihood of being in the sample.
So even identifying the population can be a difficult job, but once we have identified the population, how do we choose an appropriate sample? Remember, although we would prefer to survey all members of the population, this is usually impractical unless the population is very small, so we choose a sample. There are many ways to sample a population, but there is one goal we need to keep in mind: we would like the sample to be representative of the population .
Returning to our hypothetical job as a political pollster, we would not anticipate very accurate results if we drew all of our samples from among the customers at a Starbucks, nor would we expect that a sample drawn entirely from the membership list of the local Elks club would provide a useful picture of district-wide support for our candidate.
One way to ensure that the sample has a reasonable chance of mirroring the population is to employ randomness . The most basic random method is simple random sampling.
A random sample is one in which each member of the population has an equal probability of being chosen. A simple random sample is one in which every member of the population and any group of members has an equal probability of being chosen.
If we could somehow identify all likely voters in the state, put each of their names on a piece of paper, toss the slips into a (very large) hat and draw 1000 slips out of the hat, we would have a simple random sample.
In practice, computers are better suited for this sort of endeavor than millions of slips of paper and extremely large headgear.
It is always possible, however, that even a random sample might end up not being totally representative of the population. If we repeatedly take samples of 1000 people from among the population of likely voters in the state of Washington, some of these samples might tend to have a slightly higher percentage of Democrats (or Republicans) than does the general population; some samples might include more older people and some samples might include more younger people; etc. In most cases, this sampling variability is not significant.
The natural variation of samples is called sampling variability .
This is unavoidable and expected in random sampling, and in most cases is not an issue.
To help account for variability, pollsters might instead use a stratified sample .
In stratified sampling , a population is divided into a number of subgroups (or strata). Random samples are then taken from each subgroup with sample sizes proportional to the size of the subgroup in the population.
Suppose in a particular state that previous data indicated that the electorate was comprised of 39% Democrats, 37% Republicans and 24% independents. In a sample of 1000 people, they would then expect to get about 390 Democrats, 370 Republicans and 240 independents. To accomplish this, they could randomly select 390 people from among those voters known to be Democrats, 370 from those known to be Republicans, and 240 from those with no party affiliation.
Stratified sampling can also be used to select a sample with people in desired age groups, a specified mix ratio of males and females, etc. A variation on this technique is called quota sampling .
Quota sampling is a variation on stratified sampling, wherein samples are collected in each subgroup until the desired quota is met.
Suppose the pollsters call people at random, but once they have met their quota of 390 Democrats, they only gather people who do not identify themselves as a Democrat.
You may have had the experience of being called by a telephone pollster who started by asking you your age, income, etc. and then thanked you for your time and hung up before asking any "real" questions. Most likely, they already had contacted enough people in your demographic group and were looking for people who were older or younger, richer or poorer, etc. Quota sampling is usually a bit easier than stratified sampling, but also does not ensure the same level of randomness.
Another sampling method is cluster sampling , in which the population is divided into groups, and one or more groups are randomly selected to be in the sample.
In cluster sampling , the population is divided into subgroups (clusters), and a set of subgroups are selected to be in the sample
If the college wanted to survey students, since students are already divided into classes, they could randomly select 10 classes and give the survey to all the students in those classes. This would be cluster sampling.
Other sampling methods include systematic sampling .
In systematic sampling , every \(n^{t h}\) member of the population is selected to be in the sample.
To select a sample using systematic sampling, a pollster calls every \(100^{th}\) name in the phone book.
Solution
Systematic sampling is not as random as a simple random sample (if your name is Albert Aardvark and your sister Alexis Aardvark is right after you in the phone book, there is no way you could both end up in the sample) but it can yield acceptable samples.
Perhaps the worst types of sampling methods are convenience samples and voluntary response samples .
Convenience sampling is samples chosen by selecting whoever is convenient.
Voluntary response sampling is allowing the sample to volunteer.
A pollster stands on a street corner and interviews the first 100 people who agree to speak to him. This is a convenience sample.
A website has a survey asking readers to give their opinion on a tax proposal. This is a self-selected sample, or voluntary response sample, in which respondents volunteer to participate.
Usually voluntary response samples are skewed towards people who have a particularly strong opinion about the subject of the survey or who just have way too much time on their hands and enjoy taking surveys.
In each case, indicate what sampling method was used
- Every 4th person in the class was selected
- A sample was selected to contain 25 men and 35 women
- Viewers of a new show are asked to vote on the show’s website
- A website randomly selects 50 of their customers to send a satisfaction survey to
- To survey voters in a town, a polling company randomly selects 10 city blocks, and interviews everyone who lives on those blocks.
- Answer
-
- Systematic
- Stratified or Quota
- Voluntary response
- Simple random
- Cluster


4.1.4: Ethnocentrism and Cultural Relativism
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CULTURAL RELATIVISM
The guiding philosophy of modern anthropology is cultural relativism —the idea that we should seek to understand another person’s beliefs and behaviors from the perspective of their culture rather than our own. Anthropologists do not judge other cultures based on their values nor view other cultural ways of doing things as inferior. Instead, anthropologists seek to understand people’s beliefs within the system they have for explaining things.
Definition: cultural relativism
The idea that we should seek to understand another person’s beliefs and behaviors from the perspective of their culture rather than our own.
Cultural relativism is an important methodological consideration when conducting research. In the field, anthropologists must temporarily suspend their own value, moral, and esthetic judgments and seek to understand and respect the values, morals, and esthetics of the other culture on their terms. This can be a challenging task, particularly when a culture is significantly different from the one in which they were raised.
A Young Anthropologist's Experience
Katie Nelson, Inver Hills Community College
During my first field experience in Brazil, I learned firsthand how challenging cultural relativism could be. Preferences for physical proximity and comfort talking about one’s body are among the first differences likely to be noticed by U.S. visitors to Brazil. Compared to Americans, Brazilians generally are much more comfortable standing close, touching, holding hands, and even smelling one another and often discuss each other’s bodies. Children and adults commonly refer to each other using playful nicknames that refer to their body size, body shape, or skin color. Neighbors and even strangers frequently stopped me on the street to comment on the color of my skin (It concerned some as being overly pale or pink—Was I ill? Was I sunburned?), the texture of my hair (How did I get it so smooth? Did I straighten my hair?), and my body size and shape (“You have a nice bust, but if you lost a little weight around the middle you would be even more attractive!”).
During my first few months in Brazil, I had to remind myself constantly that these comments were not rude, disrespectful, or inappropriate as I would have perceived them to be in the United States. On the contrary, it was one of the ways that people showed affection toward me. From a culturally relativistic perspective, the comments demonstrated that they cared about me, were concerned with my well-being, and wanted me to be part of the community. Had I not taken a culturally relativistic view at the outset and instead judged the actions based on my cultural perspective, I would have been continually frustrated and likely would have confused and offended people in the community. And offending your informants and the rest of the community certainly is not conducive to completing high-quality ethnography! Had I not fully understood the importance of body contact and physical proximity in communication in Brazil, I would have missed an important component of the culture.
ETHNOCENTRISM
Another perspective that has been rejected by anthropologists is ethnocentrism —the tendency to view one’s own culture as most important and correct and as a stick by which to measure all other cultures. People who are ethnocentric view their own cultures as central and normal and reject all other cultures as inferior and morally suspect. As it turns out, many people and cultures are ethnocentric to some degree; ethnocentrism is a common human experience. Why do we respond the way we do? Why do we behave the way we do? Why do we believe what we believe? Most people find these kinds of questions difficult to answer. Often the answer is simply “because that is how it is done.” They believe what they believe because that is what one normally believes and doing things any other way seems wrong.
Definition: ethnocentrism
The tendency to view one’s own culture as most important and correct and as a stick by which to measure all other cultures.
Ethnocentrism is not a useful perspective in contexts in which people from different cultural backgrounds come into close contact with one another, as is the case in many cities and communities throughout the world. People increasingly find that they must adopt culturally relativistic perspectives in governing communities and as a guide for their interactions with members of the community. For anthropologists in the field, cultural relativism is especially important. We must set aside our innate ethnocentrisms and let cultural relativism guide our inquiries and interactions with others so that our observations are not biased. Cultural relativism is at the core of the discipline of anthropology.
Cultural Relativism and Fieldwork
Objectivity and Activist Anthropology
Despite the importance of cultural relativism, it is not always possible and at times is inappropriate to maintain complete objectivity in the field. Researchers may encounter cultural practices that are an affront to strongly held moral values or that violate the human rights of a segment of a population. In other cases, they may be conducting research in part to advocate for a particular issue or for the rights of a marginalized group.
Take, for example, the practice of female genital cutting (FGC), also known as female genital mutilation (FGM), a practice that is common in various regions of the world, especially in parts of Africa and the Middle East. Such practices involving modification of female genitals for non-medical and cultural reasons range from clitoridectomy (partial or full removal of the clitoris) to infibulation, which involves removal of the clitoris and the inner and outer labia and suturing to narrow the vaginal opening, leaving only a small hole for the passage of urine and menstrual fluid Anthropologists working in regions where such practices are common often understandably have a strong negative opinion, viewing the practice as unnecessary medically and posing a risk of serious infection, infertility, and complications from childbirth. They may also be opposed to it because they feel that it violates the right of women to experience sexual pleasure, something they likely view as a fundamental human right. Should the anthropologist intervene to prevent girls and women from being subjected to this practice?
Anthropologist Janice Boddy studied FGC/FGM in rural northern Sudan and sought to explain it from a culturally relativistic perspective. She found that the practice persists, in part, because it is believed to preserve a woman’s chastity and curb her sexual desire, making her less likely to have affairs once she is married. Boddy’s research showed how the practice makes sense in the context of a culture in which a woman’s sexual conduct is a symbol of her family’s honor, which is important culturally. [5]
Boddy’s relativistic explanation helps make the practice comprehensible and allows cultural outsiders to understand how it is internally culturally coherent. But the question remains. Once anthropologists understand why people practice FGC/FGM, should they accept it? Because they uncover the cultural meaning of a practice, must they maintain a neutral stance or should they fight a practice viewed as an injustice? How does an anthropologist know what is right?
Unfortunately, answers to these questions are rarely simple, and anthropologists as a group do not always agree on an appropriate professional stance and responsibility. Nevertheless, examining practices such as FGC/FGM can help us understand the debate over objectivity versus “activism” in anthropology more clearly. Some anthropologists feel that striving for objectivity in ethnography is paramount. That even if objectivity cannot be completely achieved, anthropologists’ ethnography should be free from as much subjective opinion as possible. Others take the opposite stance and produce anthropological research and writing as a means of fighting for equality and justice for disempowered or voiceless groups. The debate over how much (if any) activism is acceptable is ongoing. What is clear is that anthropologists are continuing to grapple with the contentious relationship between objectivity and activism in ethnographic research.
NOTES
5. Janice Bodd, Civilizing Women: British Crusades in Colonial Sudan (Princeton NJ: Princeton University Press, 2007). ↵
Adapted From
"Doing Fieldwork: Methods in Cultural Anthropology" by Katie Nelson, Inver Hills Community College. In Perspectives: An Open Invitation to Cultural Anthropology , 2nd Edition, Society for Anthropology in Community Colleges, 2020, under CC BY-NC 4.0 .


1.33: Comparative and Superlative Adjectives and Adverbs; Dative of Degree of Difference
Module 27
© 2021 Philip S. Peek, CC BY 4.0 https://doi.org/ 10.11647/OBP.0264. 33
Adjectives
Remember that Greek and English adjectives have the same function, though the Greek adjective has endings and the English adjective does not. Adjectives in both languages are words that describe nouns. In the phrase the happy and sad blues, the , happy, and sad describe the noun blues . The Greek adjective has endings because the endings enable it to agree in gender, case, and number with the noun it modifies.
Adjectives and Degree
English and Greek adjectives have three degrees: positive, comparative, and superlative . English creates the three degrees of the adjective through suffixes or with the adverbs more and most :
|
Positive: |
happy |
|
Comparative: |
happier, more happy |
|
Superlative: |
happiest, most happy |
Greek does the same.
Adjectives of the Comparative and Superlative Degree
Adjectives show degrees by using one of two sets of suffixes or by using the adverb μᾶλλον more for the comparative degree and the adverb μάλιστα most for the superlative degree.
The comparative degree suffixes - τερος , - τέρᾱ , - τερον decline like first and second declension adjectives.
The superlative degree suffixes - τατος , - τ άτη , - τατον decline like first and second declension adjectives.
The comparative degree suffixes - ιων , - ιον decline like third declension nouns with some alternate forms.
The superlative degree suffixes - ιστος , - ίστη , - ιστον decline like first and second declension adjectives.
An additional way to form the comparative and superlative is to use the adverbs μᾶλλον or μάλιστα , along with the positive degree of the adjective.
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μᾶλλον |
+ |
the positive degree of the adjective. |
|
μάλιστα |
+ |
the positive degree of the adjective. |
Declining Comparative and Superlative Adjectives
To form the comparative and superlative degrees of 1st and 2nd declension adjectives, obtain the stem by dropping - ος from the masculine nominative singular.
If the stem ends in a long syllable (a long vowel or diphthong or a short vowel followed by two consonants or by the double consonants - ζ , - ξ , or - ψ ), add to it - ότερος , - οτ έρᾱ , - ότερον for the comparative and - ότατος , - οτάτη , - ότατον for the superlative.
If the stem ends in a short syllable , add to it - ώτερος , - ωτέρᾱ , - ώ τερον for the comparative, and - ώτατος , - ωτάτη , - ώτατον for the superlative.
Consider the following examples.
Comparative degree
|
M |
F |
N |
|
|
δηλότερος |
δηλοτέρᾱ |
δηλότερον |
clearer, more clear |
|
σοφώτερος |
σοφωτέρᾱ |
σοφώτερον |
wiser, more wise |
Superlative degree
|
M |
F |
N |
|
|
δηλότατος |
δηλοτάτη |
δηλότατον |
clearest, most clear |
|
σοφώτατος |
σοφωτάτη |
σοφώτατον |
wisest, most wise |
Declining Comparative and Superlative Adjectives in -ης, -ες and in -ύς, -εῖα, -ύ
To form the comparative and superlative degrees of 3rd declension adjectives ending in - ης , - ες and of adjectives ending in - ύς , - εῖα , - ύ , add the suffixes directly to the stem without any intervening vowel. The stem of these adjectives is taken from the neuter singular nominative.
Comparative degree
|
M |
F |
N |
|
|
ἀκριβέσ - τερος |
ἀκριβεστέρᾱ |
ἀκριβέστερον |
preciser, more precise |
|
ἀ ληθέσ - τερος |
ἀληθέστέρᾱ |
ἀληθέστερον |
truer, more true |
|
εὐθύ - τερος |
εὐθυτέρᾱ |
εὐθύτερον |
straighter, more straight |
Superlative degree
|
M |
F |
N |
|
|
ἀκριβέσ - τατος |
ἀκριβεστάτη |
ἀκριβέστατον |
precisest, most precise |
|
ἀληθέσ - τατος |
ἀληθεστάτη |
ἀληθέστατον |
truest, most true |
|
εὐθύ - τατος |
εὐθυτάτη |
εὐθύ τατον |
straightest, most straight |
Declining Comparative and Superlative Adjectives -ίων, -ιον and -ιστος, -ίστη, -ιστον
Adjectives that take - ίων , - ιον in the comparative degree take - ιστος , - ίστη , - ιστον in the superlative degree. The comparative degree declines like a third declension noun with a few alternate forms. The superlative degree declines like 1 st and 2 nd declension adjectives.
Comparative degree
|
|
||||||||||||||||||||||||||||||||||||||||||
|
|
Superlative degree
|
M |
F |
N |
|
|
Ν |
ἥδιστος |
ἡδίστη |
ἥδιστον |
Adverbs
Unlike nouns, pronouns, and adjectives, adverbs do not have gender, number, or case. Adverbs modify adjectives, verbs, or other adverbs. Like adjectives, adverbs have three degrees: positive, comparative, and superlative .
For the positive degree English typically adds the suffix -ly . English uses the suffixes -er and -est or more and most to create the comparative and superlative degrees of the adverb:
|
Positive: |
loudly |
|
Comparative: |
loudlier, more loudly |
|
Superlative: |
loudliest, most loudly |
Greek creates the three degrees by adding a suffix.
Adverbs of the Positive Degree
Most adverbs are formed from adjectives by adding - ως to the stem, found by dropping the ending from the masculine genitive singular of the adjective.
|
|
- Adverbs in - ως accented on the ultima always have a circumflex: κακῶς .
Adverbs of the Comparative Degree
Adverbs formed from adjectives use as their comparative degree the neuter accusative singular of the comparative degree of the adjective.
|
|
Adverbs of the Superlative Degree
Adverbs formed from adjectives use as their superlative degree the neuter accusative plural of the superlative degree of the adjective.
|
|
Comparison with ἤ and the Genitive Case
When you compare two things as in these sentences,
laughter is stronger than tears,
and
the whole is greater than the part,
using the comparative degree, you may do so by using the word ἤ than or by using the genitive case without ἤ , called a genitive of comparison .
Consider these examples. The underline marks the second item in the comparison.
1. ὁ γέλως κρείττων ἢ οἰμωγή .
Laughter is stronger than tears .
2. ὁ γέλως κρείττων οἰμωγῆς .
Laughter is stronger than tears .
3. καὶ τὸ ὅλον μεῖζον τοῦ μέρους ἐστί .
The whole is greater than the part .
4. καὶ τὸ ὅλον μεῖζον ἢ τὸ μέρος ἐστί .
The whole is greater than the part .
Note that when you use ἤ , the two things being compared are always in the same case.
Dative of Degree of Difference
If you want to state the degree by which one thing is more than another, use the dative case. In this sentence,
π o λλῷ ἥδε σ o φωτέρα τῆς μητρός ἐστι ,
she is much wiser than her mother,
the π o λλῷ by much is the dative of degree of difference.
Adjectives and Adverbs in Summary
Τ he chart below contains all the information found above, and information on adjectives from other modules, but in a condensed form. Look it over, making sure that you understand how to read it.
Practice Translating Comparative and Superlative Adjectives. Translate the sentences below, which have been adapted from a variety of ancient Greek authors. Check your understanding with the translations in the Answer Key , making sure that you understand why each word translates as it does. Now go back and read each sentence two or three times, noticing with each rereading how much better your understanding of the sentence becomes. Make this a habit and you will improve quickly.
Satyros of Kallatis, Σάτυρος Κάλλατις c. 150 BCE. Satyros was a philosopher, historian, and biographer whose subjects included kings, philosophers, poets, orators, and statesmen. Fragments of his biography on Euripides were found on a papyrus scroll at Oxyrhynchus, Egypt in the early 1900s.
- τί νομίζεις τὴν δημοκρατίαν π o λλῷ ἡδίονα τυραννίδος ;
- μικροὶ γέροντι παῖδες ἡδίους πατρὶ καὶ μητρί τινου (Satyros).
- ὁ δὲ στρατηγὸς ἔφη αὐτὸς κρείττων ὕδατος .
- θάνατος ἡμᾶς ταῦτα ἀναμιμνήσκει ὅτι ὁ πλοῦτος χείρων ἢ ὑγίεια .
- τίνι τρόπῳ εἶ τρόπου ἡσυχίου ἐπε ὶ τάχιστα εἰς ὀργὴν σπεύδεις ;
- ἔ φη μέγιστον γὰρ ἕλκος πόλεως κακὸν ῥήτορα δημαγωγόν· ὃς γὰρ δῆμον πείθει κακὰ ποιέειν (Satyros).
- ταῦτα γάρ ἐστι δήπου ἃ συνέχει τὴ ν νεωτέραν κωμωιδίαν καὶ ἃ πρὸς ἄκρον ἤγαγεν Εὐριπίδης (Satyros).
- ἐν δὲ τούτῳ τῷ καιρῷ κινδυνε ύσεις ἐγεῖραί τινα ᾧ τὰ χρήματα πλεῖστα καὶ κρατίστη ἡ ἀνδραγαθία ;
- σκοπέειν δ ’ ἄξιον . τί γὰρ εὐλογώτερον ἐπεὶ τάς γε κακίας καὶ τὰς ἀρετάς , καθάπερ ἔλεγεν ὁ Σωκράτης ̣, τὰς αὐτὰς ̣ ἐν καὶ ἀνδρὶ καὶ γυναικὶ ἔστιν εὑρεῖν (Satyros);
- ἡ δὲ μεταπέμπει τὴν ἄνθρωπον καὶ ἰδοῦσα τὸ κάλλος , “ χαῖρε ,” φησίν , “ γύναι · ψευδεῖς ἄρ ’ αἱ διαβολαί · σὺ γὰρ ἐν τῷ προσώπῳ τῷ σῷ καὶ τοῖς ὀφθαλμοῖς ἔχεις φάρμακα κάλλιστα (Satyros).
Vocabulary
|
* ἄγω , ἄξω , ἤγαγον do, drive, lead; χάριν ἄγω I give thanks |
* λέγω , λέξω or ἐρέω , ἔλεξα or εἶπον say, tell, speak |
|
ἄκρον ἄκρου τό top, furthest point; mountain top |
μέγιστος , μεγίστη , μέγιστον greatest |
|
ἀναμιμνήσκω call to mind, remind |
μεταπέμπω send for, send after |
|
ἀνδραγαθία , ἀνδραγαθίας ἡ courage, bravery |
* μήτηρ , μητέρος ( μητρός ) ἡ mother |
|
* ἀνήρ , ἀνδρός ὁ man, husband |
* μικρός , μικρά , μικρόν small |
|
* ἄνθρωπος , ἀνθρώπου ἡ ὁ human, person |
νεώτερος , νεωτέρα , νεώτερον newer |
|
* ἄξιος , ἀξίᾱ , ἄξιον worthy, deserving + gen. |
* νομίζω , νομιέω , ἐνόμισα believe, think, have the custom of, hold as custom |
|
* ἀρετή , ἀρετῆς ἡ virtue; excellence |
ὀργή , ὀργῆς ἡ mood; anger, wrath |
|
γέρων , γέροντος ὁ old man |
* ὁράω , ὄψομαι , εἶδον ( ἰδεῖν ) see |
|
* γυνή , γυναικός ἡ woman, wife |
ὀφθαλμός , ὀφθαλμοῦ ὁ eye |
|
δημαγωγός , δημαγωγοῦ ὁ demagogue |
παίζω play, sport, dance |
|
δημοκρατία , δημοκρατίας ἡ democracy |
* παῖς , παιδός ἡ ὁ child |
|
* δῆμος , δήμου ὁ people |
* πατήρ , πατρός ὁ father |
|
δήπου surely |
* πείθω , πείσω , ἔπεισα persuade + inf.; (mid. or pass.) listen to, obey + dat. or gen. |
|
διαβολή διαβολῆς ἡ slander |
* πλεῖστος , πλείστη , πλεῖστον most, greatest, largest |
|
ἐγείρω , ἐγερέω , ἤγειρα awaken, rouse |
πλοῦτος , πλούτου ὁ wealth |
|
ἕλκος , ἕλκεος (- ους ) τό wound |
* ποιέω , ποιήσω , ἐποίησα do, make, cause; (mid.) consider |
|
εὐλογώτερος , εὐλογώτερον more reasonable |
* πόλις , πόλιος ( πόληος , πόλεως ) ἡ city |
|
Εὐριπίδης , Εὐριπίδου ὁ Euripides , tragic playwright from Athens, c. 450 BCE |
πρόσωπον , προσώπου τό face |
|
* εὑρίσκω , εὑρήσω , η ὗρον find |
ῥήτωρ , ῥήτορος ὁ rhetor, speaker |
|
* ἔφη he, she, it said |
* σκοπέω , σκοπήσω , ἐσκόπησα , look at; examine; consider |
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* ἔχω , ἕξω or σχ ήσω , ἔσχον have, hold; be able + inf.; καλῶς ἔχειν to be well |
* σός , σή , σόν your |
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ἡδίων , ἥ διον sweeter, pleasanter |
σπεύδω , σπεύσω , ἔσπευσα urge on, press on; hasten, be eager |
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ἡσύχιος , ἡσύχιον quiet |
συνέχω make up, contain |
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θάνατος , θαν άτου ὁ death |
Σωκράτης ̣, Σωκράτους ὁ Sokrates , Plato’s teacher and famous oral philosopher c. 450 BCE |
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ἰδοῦσα , ἰδούσης ἡ seeing, having seen |
τάχιστα most quickly |
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καθάπερ just as |
* τρόπος , τρόπου ὁ way, manner, turn; (pl.) character |
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* καιρός , καιροῦ ὁ right moment, critical time, opportunity |
τυραννίς , τυραννίδος ἡ tyranny |
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κακία , κακίας ἡ wickedness; fault |
ὑγίεια , ὑγιείᾱς ἡ health, soundness |
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* κακός , κακή , κακόν bad, evil, cowardly |
ὕδωρ , ὕδατος τό water; pond |
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κάλλιστος , καλλ ίστη , κάλλιστον most beautiful |
φάρμακον , φαρμάκου τό drugs, medicine |
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κάλλος , κάλλεος ( κάλλους ) τό beauty |
* φημί say, affirm, assert |
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κινδυνεύω , κινδυνεύσω , ἐκινδύνευσα run a risk, hazard; dare + inf.; be likely + inf. |
χαίρω be well; rejoice at, take pleasure in + dat. |
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κράτιστος , κρατίστη , κράτιστον strongest |
χείρων , χεῖρον worse, inferior |
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κρείττων , κρεῖττον better, stronger, greater |
* χρῆμα , χρήματος τό thing; (pl.) goods, money, property |
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κωμωιδία , κωμωιδίας ἡ comedy |
ψευδής , ψευδές false |
- The asterisk indicates the top 250 most frequently occurring vocabulary, which you are to memorize.
Practice Translating. Translate the sentences below, which have been adapted from Homer’s Odyssey ( Ὀδύσσεια ). Remember the meanings and functions of the cases presented in Module 7 . Use your memory to identify endings and their functions. If you forget an ending or a function, consult the Adjective, Adverb, Noun, and Pronoun Chart in Appendix VIII and the Case and Function Chart in Appendix I. Check your understanding with the translations in the Answer Key , making sure that you understand why each word translates as it does. Now go back and read each sentence two or three times, noticing with each rereading how much better your understanding of the sentence becomes. Make this a habit and you will improve quickly.
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Πηνελόπεια : δοιαὶ γάρ τε πύλαι ἀμενηνῶν εἰσὶν ὀνείρων · αἱ μὲν γὰρ κεράεσι τεύχονται , αἱ δ ’ ἐλέφαντι . αὐτῶν οἱ μὲν ἔρχονται διὰ πριστοῦ ἐλέ φαντος . οὗτοι ῥ ’ ἐλεφαίρονται καὶ ἔπε ’ ἀκράαντα φέ ρουσιν . οἱ δὲ διὰ ξεστῶν κεράων ἔρχονται θύραζε . οὗτοι ῥ ’ ἔτυμα κραίνουσι , βροτῶν εἴ τις ὁρά ει . ἀλλʼ ἐμοὶ οὐκ ἐντεῦθεν οἴομαι α ἰνὸν ὄνειρον ἐλθεῖν . εἰ οὕτως , ἦ ἀσπαστὸν ἐμοὶ καὶ παιδὶ γενήσεται . ἄλλο δέ σοι ἐρέω· ἥδε δὲ ἠὼς ἐλεύσεται δυσώνυμος , ἣ μ ’ Ὀδυσσέως οἴκου ἀποσχήσει . νῦν γὰρ καταθή σω ἄεθλον , τοὺς πελέκεας , οὓς κεῖνος ἐν μεγάροις ἵστησιν ἑξείης , δρυόχους ὥς , δώδεκα πάντας . ἵστησιν δ ’ ὅ γε πολὺ ἄνευθε καὶ διαρριπτάσκει οἰστόν . νῦν δὲ μνηστήρεσιν ἄεθλον τοῦτον ἐφήσω· ὃς δὲ ῥηΐτατʼ ἐντανύει βιὸν ἐν παλάμαις καὶ διοϊστεύει πελέκεων δυοκαίδεκα π άντων , τούτῳ αὐτὴ ἅμ ’ ἕψομαι . κα ὶ νοφίσομαι τόδε δῶμα κουρίδιον , μά λα καλόν , ἐνίπλειον βιότου , οὗ ποτὲ μνήσεσθαι οἴομαι ἔν περ ὀνείρῳ . |
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Ὀδυσσεύς ξεῖνος : ὦ γύναι αἰδοίη Λαερτι άδου Ὀδυσσέως , δεῖ μηκέτι νῦν ἀ ναβάλλειν δόμοις ἔνι τοῦτον ἄεθλον . πρὶ ν γὰρ πολύμητις ἐλεύσεται ἐνθάδ ’ Ὀδυσσεύς , πρὶν τούτους τόδε τό ξον ἐΰξοον ἀμφαφάειν τε νευρήν τ ’ ἐνταν ύσαι διοϊστεῦσαί τε σιδήρου . |
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Πηνελόπεια : εἰ ἐ θέλεις παρά μοι , ξεῖνε , ἧσθαι ἐν μεγ άροις καὶ τέρπειν , οὐκ οἴομαί μοι ὕπνον ἐπὶ βλεφάροις χυθήσεσθαι . |
Adverbs and Verbs
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ἀμφαφ άω touch, feel, handle |
ἦ in truth, verily |
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ἀναβάλλω , - βαλέω , - έβαλον strike up; throw up; delay |
ἧμαι ( ἧσαι , ἧσται , ἥμεθα , ἧστε , ἧνται ; ἧσθαι ) sit |
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ἄ νευθε far away, distant |
θύραζε to the doors, out of doors |
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ἀπέχω keep off or away from |
* ἵστημι , στήσω , ἔστησα or ἔστην stand, make stand , place |
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* γίγνομαι , γενήσομαι , ἐγενόμην be, be born |
κατατίθημι place, put, lay down |
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* δεῖ it is necessary + ‘x’ in gen. or dat. or acc. + inf., δεῖ ἐλθεῖν it is necessary to come |
κραίνω , κρανέω , ἔκρανα accomplish, fulfil, bring to pass |
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διαρριπτάσκω shoot through, cast |
* λ έγω , λέξω or ἐρέω , ἔλεξα or ε ἶπον say, tell, speak |
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διοϊστεύω shoot an arrow through + gen. |
μάλα very |
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* εἰμί , ἔσομαι be, be possible |
μηκέτι no longer |
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* ἐθέλω , ἐθελήσω , ἠθέ λησα wish, be willing |
μιμνήσκω , μνήσω , ἔμνησα remember + gen. |
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ἐλεφαίρομαι cheat with empty hopes |
νοφίζω , νοφιέω νοφίσομαι , ἐνόφισα turn away, shrink back; forsake, abandon |
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ἐνθάδε here, there; hither, thither |
* οἴομαι ( οἶμαι ), οἰήσομαι , ᾠσάμην think, suppose, believe |
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ἐνί = ἐν |
* ὁράω , ὄψομαι , εἶδον ( ἰδεῖν ) see |
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ἐντανύω stretch, bend, string |
ῥηΐτατα most easily |
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ἐντεῦθεν ( ἐνθεῦτεν ) thence, hence |
τέρπω , τέρψω , ἐτερψα delight, cheer ; enjoy + dat. |
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ἑξεί ης in order, in a row |
τεύχω , τεύξω , ἔτευξα make ready, make, produce; cause, bring to pass |
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* ἕπομαι (imp. εἱπόμην ), ἕψομαι , ἑσπόμην follow + dat. |
* φέρω , οἴσω , ἤνεγκα or ἤνεγκον bring, bear, carry; endure |
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* ἔρχομαι , ἐλεύσομαι , ἦ λθον ( ἐλθεῖν ) come, go |
χέω , χέω , ἔχεα , κέχυκα , κέχυμαι , ἐχύθην pour; (pass.) be heaped up |
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ἐφίημι , - ήσω , - ῆκα send on, against; let go, yield; (mid.) command, give orders; (mid.) aim at, long for + gen.; (mid.) allow, permit + ‘x’ in dat. + inf. |
Adjectives, Nouns, Pronouns
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ἄεθλον , ἀέθλου τό prize, contest |
ἠώς , ἠούς ἡ dawn |
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αἰδοῖος , αἰδοία , αἰδοῖον : deserving of awe, respect; valuable, excellent; bashful, modest; τὰ αἰδοῖα private parts |
* καλός , καλή , καλόν beautiful, good, noble |
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αἰνός , αἰνή , αἰνόν dread, dire, grim |
κεῖνος , κείνου ὁ he, that one |
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ἀκράαντος , ἀκράαντον unfulfilled, fruitless |
κέρας , κέραος τό horn of an animal |
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* ἄλλος , ἄλλη , ἄλλο another, other |
κουρίδιος , κουριδία , κουρίδιον wedded, nuptial, bridal |
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ἀμενηνός , ἀμενηνόν powerless, fleeting, feeble |
Λαερτιάδης , Λαερτιάδου ὁ son of Laertes |
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ἀσπαστ ός , ἀσπαστή , ἀσπαστόν welcome |
μέγαρον , μεγάρου τό megaron, great hall |
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βιός , βιοῦ ὁ bow |
μνηστήρ , μνηστῆρος ὁ (epic dat. pl. μνηστήρεσσιν ) suitor |
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βίοτος , βιότου ὁ life |
νευρή , νευρῆς ἡ sinew, bow-string; cord |
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βλέφαρα , βλεφ άρων τά eyelids, eyes |
ξεῖνος ( ξένος ), ξείνου ( ξένου ) ὁ stranger; guest-friend |
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βροτός , βροτοῦ ὁ mortal |
ξεστός , ξεστή , ξεστόν smooth, polished, wrought |
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* γυνή , γυναικ ός ἡ woman, wife |
Ὀδυσσεύς , Ὀδυσσέως ὁ Odysseus |
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δοιός , δοιά , δοιόν two |
οἶκος , οἴκου ὁ house, palace |
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δόμο ς , δόμου ὁ house |
οἰστός , οἰστοῦ ὁ arrow |
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δρύοχοι , δρυόχων οἱ props, stays, shores (on which the frame of a ship is propped) |
ὄνειρος , ὀνείρου ὁ dream |
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δυοκαίδεκα twelve |
* παῖς , παιδός ἡ ὁ child |
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δυσώνυμος , δυσώνυμον bearing an ill name, ill-omened |
παλάμη , παλάμης ἡ palm, hand |
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δώδεκα twelve |
πέλεκυς , πελέκεως ὁ axe |
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δῶμα , δώματος τό house |
πολύμητις , πολυμήτιος of many counsels |
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ἐλέφας , ἐλέφαντος ὁ elephant; elephant’s tusk; ivory |
πριστός , πριστή , πριστόν sawn |
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ἐνίπλειος , ἐνίπλειον full of + gen. |
πύλη , πύλης ἡ gate |
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ἔπος , ἔπεος ( ἔπους ) τό word, speech; song; saying; verse |
σίδηρος , σιδήρου ὁ iron |
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ἔτυμος , ἔτυμον true, real, actual |
τόξον , τόξου τό bow |
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ἐΰξοος , ἐΰξοον polished |
ὕπνος , ὕπνου ὁ sleep |
- The asterisk indicates the top 250 most frequently occurring vocabulary, which you are to memorize.
Practice Parsing Greek Sentences. Parse each word of the sentences found below. For nouns and pronouns, give their case and function. For verbs, their give person, number, tense, mood, and voice. For adverbs and conjunctions, identify them. For prepositional phrases, give the preposition and the preposition’s object. For adjectives, tell what noun they agree with in gender, number, and case.
ο ὖν τι βούλῃ περὶ τῶν ἐκεῖ φράζειν ἐμοί ἢ ἄπειμι ;
εἴ ἐθέλεις παρά μοι , ξεῖνε , ἧσθαι ἐν μεγάροις καὶ τ έρπειν , οὐκ οἴομαί μοι ὕπνον ἐπὶ βλεφάροις χυθήσεσθαι .
Check your answers with those in the Answer Key .
Boustrophedon
Boustrophedon is a way of writing that starts in the way that we are used to but upon reaching the end of the line the writing turns back on itself with the letters facing in the new direction, much as a farmer, plowing a field, turns his cow at row’s end.
In cultures where books are expensive and difficult to acquire, one book serves many readers and people learn how to read words from many different angles, including upside down. For them, reading upside down is as rightside up as reading rightside up is to us. As you consider the many ways that people differ from each other, also consider how culture determines much of what we assume, believe, desire, think about, and value. These differences represent the vast range of human potential and possibility.
Module 27 Top 250 Vocabulary to be Memorized. Like learning the alphabet and endings, memorizing vocabulary is essential to acquiring language. The better you memorize the top 250 most frequently occurring vocabulary words the greater mastery of the language you will have.
Adjectives
λοιπός , λοιπή , λοιπόν left, remaining
μέσος , μέση , μέσον middle, middle of + gen.
μικρός , μικρά , μικρόν small, little, short
Verbs
εἶμι come, go
ἔρομαι ( εἴρομαι ), ἐρήσομαι ( εἰρήσομαι ), ἠρόμην , -----, -----, ----- ask, ask ‘x’ in acc. about ‘y’ in acc.
εὑρίσκω , εὑρήσω , ηὗρον , ηὕρηκα , ηὕρημαι , ηὑρέθην find out, discover
ἥκω , ἥξω , ------, ------, ------, ------ have come, be present
πάρειμι go in, enter; pass by


13.5: Summary
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Key Takeaways
The equation of motion for the position, \(x(t)\) , of the mass in a one-dimensional spring-mass system with no friction can be written:
\[\begin{aligned} \frac{d^2x}{dt^2}=-\sqrt{\frac{k}{m}}x = -\omega^2 x\end{aligned}\]
and has a solution:
\[\begin{aligned} x(t) = A\cos(\omega t + \phi)\end{aligned}\]
where \(A\) is the amplitude of the motion, \(\phi\) is the phase, which depends on our choice of initial conditions (when we choose time \(t=0\) ), and \(\omega\) :
\[\begin{aligned} \omega = \sqrt{\frac{k}{m}}\end{aligned}\]
is the angular frequency of the motion. The mass will oscillate about an equilibrium position with a period, \(T\) , and frequency, \(f\) , given by:
\[\begin{aligned} T&=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{k}}\\[4pt] f&=\frac{1}{T}=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}\end{aligned}\]
The velocity and acceleration of the mass are found by taking the time derivatives of the position \(x(t)\) :
\[\begin{aligned} x(t)&= A \cos(\omega t + \phi)\\[4pt] v(t)&=\frac{d}{dt}x(t) = -A\omega\sin(\omega t + \phi)\\[4pt] a(t)&= \frac{d^2}{dt^2}x(t) =\frac{d}{dt}\left( -A\omega\sin(\omega t + \phi)\right)= -A\omega^2\cos(\omega t + \phi)\end{aligned}\]
The total mechanical energy of the mass, at some position \(x\) , is given by:
\[\begin{aligned} E =U+K=\frac{1}{2}kx^2+\frac{1}{2}mv^2= \frac{1}{2}kA^2\end{aligned}\]
and is conserved.
Any system that can be described by the equation of motion:
\[\begin{aligned} \frac{d^2x}{dt^2}= -\omega^2 x\end{aligned}\]
is said to be a simple harmonic oscillator, and its position will be described by:
\[\begin{aligned} x(t) = A\cos(\omega t + \phi)\end{aligned}\]
A simple harmonic oscillator will always oscillate about an equilibrium position, where the net force on the oscillator is zero. The net force on a simple harmonic oscillator is always directed towards the equilibrium position, and has a magnitude proportional to the distance of the oscillator from its equilibrium position. The force is called a restoring force. A vertical spring-mass system, and a mass attached to two springs will both undergo simple harmonic motion about their respective equilibrium position.
A simple pendulum will undergo simple harmonic oscillations, if the amplitude of the oscillations is small. The angular frequency for the oscillations of a simple pendulum only depends on the length of the pendulum:
\[\begin{aligned} \omega = \sqrt{\frac{g}{L}}\end{aligned}\]
This is valid in the small angle approximation, where:
\[\begin{aligned} \sin\theta \approx \theta\end{aligned}\]
A physical pendulum of mass \(m\) which oscillates about an axis through the object will also undergo simple harmonic oscillation in the small angle approximation. The angular frequency of the oscillations for a physical pendulum is given by:
\[\begin{aligned} \omega = \sqrt{\frac{mgh}{I}}\end{aligned}\]
where \(h\) is the distance between the center of mass and the axis of rotation, and \(I\) is the moment of inertia of the object about the rotation axis.
Important Equations
Position, velocity, and acceleration for SHM:
\[\begin{aligned} x(t)&= A \cos(\omega t + \phi)\\[4pt] v(t)&=\frac{d}{dt}x(t) = -A\omega\sin(\omega t + \phi)\\[4pt] a(t)&= \frac{d^2}{dt^2}x(t) = -A\omega^2\cos(\omega t + \phi)\end{aligned}\]
Period and frequency:
\[\begin{aligned} \omega &= \sqrt{\frac{k}{m}}\\[4pt] T&=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{k}}\\[4pt] f&=\frac{1}{T}=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}\end{aligned}\]
Mechanical energy:
\[\begin{aligned} E =U+K=\frac{1}{2}kx^2+\frac{1}{2}mv^2= \frac{1}{2}kA^2\end{aligned}\]
Simple pendulum (small angles):
\[\begin{aligned} \omega = \sqrt{\frac{g}{L}}\end{aligned}\]
Physical pendulum (small angles):
\[\begin{aligned} \omega = \sqrt{\frac{mgh}{I}}\end{aligned}\]
Important Definitions
Definition
Angular frequency: is related to a usual frequency by a factor of \(2\pi\) . For an object rotating around a circle at constant speed, the angular frequency of the rotation is the same as the angular speed (the rate of change of a position angle). SI units: \([\text{rad/s}]\). Common variable(s): \(\omega\) .


2.9: Aromatic Heterocycles - Pyridine and Pyrrole
After completing this section, you should be able to
- draw the structure of the common aromatic heterocycles pyridine and pyrrole.
- use the Hückel 4 n + 2 rule to explain the aromaticity of each of pyridine and pyrrole.
- draw a diagram to show the orbitals involved in forming the conjugated six‑pi‑electron systems present in aromatic heterocycles such as pyridine, pyrrole, etc.
Aromatic Heterocycles
Many cyclic compounds have an element other than carbon present in the ring. Cyclic compounds that include one or more elements other than carbon are called a heterocycle. The most common heterocyclic compounds contain carbon along with nitrogen, oxygen, or sulfur. Because some heterocyclic compounds are aromatic, it is important to discuss how the inclusion of non-carbon atoms affects the determination of aromaticity.
Pyridine & Pyrimidine
Pyridine is an example of a six-membered aromatic heterocycle and has an electronic structure similar to benzene. In the bonding picture of pyridine the five carbons and single nitrogen are all sp 2 hybridized. All six of these atoms have a p orbital perpendicular to the plane of the ring and each contains one pi electron which allows the ring to be fully conjugated. This makes the pi electron count for pyridine 6 pi electrons. In much the same fashion as benzene, the 6 pi electrons follow the 4n + 2 rule and completely fill the bonding molecular orbitals which fulfills the electronic requirement for aromaticity. The lone pair electrons on pyridine's nitrogen are contained in a sp 2 orbital that lies in the same plane as the ring and does not overlap with the p orbitals of the ring. The lone pair electrons are not part of the aromatic system and do not affect the p electron count. Pyridine's electrostatic potential map shows that the lone pair electrons, shown in a yellow/red color, are locked onto the nitrogen and not distributed around the ring.
Pyrimidine is an another six-membered aromatic heterocycle that is analogous to pyridine. Pyrimidine has four carbons atoms and two nitrogen atoms that are sp 2 hybridized. Each of these atoms contributes a p orbital and a pi election allowing pyrimidine to be fully conjugated and aromatic. Both of the nitrogens in pyrimidine have lone pair electrons contained in a sp 2 orbitals and are not involved in the aromatic system. An electrostatic potential map of pyrimidine shows that neither set of lone pair electrons is distributed around the ring.
Pyrrole
Pyrrole is a five-membered heterocyclic ring which has 5 p orbitals and six pi electrons contributing to its aromaticity. Each carbon in pyrrole contributes one p orbital and pi electron. The nitrogen in pyrrole contributes two pi electrons by becoming sp 2 hybridized and placing its lone pair electrons into a p orbital. The electrostatic potential map of pyrrole show the nitrogen's lone pair electrons are distributed in the ring. Because the nitrogen lone pair is part of the aromatic sextet the electrons are very stable and are much less available for bonding to a proton (and if they do pick up a proton, the molecule loses aromaticity). Pyrrole is a very weak base: the conjugate acid, the pyrrolium ion, is a strong acid with a pK a of 0.4.
Imidazole
Imidazole is another five-membered heterocyclic ring which has 6 pi electrons and is aromatic. Both of the nitrogen atoms are sp 2 hybridized. One nitrogen is pyridine-like because it is part of a double bond and adds one pi electron to the aromatic pi ring. The other nitrogen is not part of a double bond making it pyrrole-like allowing it to contribute 2 pi electrons from it lone pair electrons. When looking at the electrostatic potential map of imidazole, the lone pair electrons on the pyrrole-like nitrogen are distributed around the ring while the lone pair electrons on the pyridine-like nitrogen are non-conjugated and locked into place.
Pyrimidine and imidazole rings are particularly important in biological chemistry. Pyrimidine, for instance, is the parent ring system in cytosine, thymine, and uracil, three of the five heterocyclic amine bases found in nucleic acids. An aromatic imidazole ring is present in histidine, one of the
Determine if furan is aromatic. Draw an orbital diagram to support your answer.
- Answer
- For a molecule to be aromatic it must: Be cyclic, be planar, be fully conjugated , and have 4n+2 π electrons.
- Furan is obviously cyclic and can be assumed to be planar due to the constrains of having a five membered ring with two double bonds. To be fully conjugated each atom in the ring must have a p orbital. Although the oxygen has 4 electron groups and appears to be sp 3 hybridized, it will become sp 2 hybridized to gain the stability of conjugation. Becoming sp 2 hybridized allows the oxygen atom to put a set of lone pair electrons into a p orbital allowing it to be part of the cyclic conjugation. These two pi electrons along with the 4 pi electrons provided by the two double bonds gives furan 6 pi electrons total. This follows the 4n + 2 rule making furan aromatic.
Exercises
1) Draw the orbitals of thiophene to show that it is aromatic.
2) The thiazolium ring is a five-membered sulfur containing aromatic ring system which is found in biological systems, such as thiamine diphosphate (ThDP). Describe how thiazolium ring is aromatic.
- Answer
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1) This drawing shows thiophene has 6 electrons in the pi-orbital.
2) Similar to the last question, the drawing shows that there is only 6 electrons in the pi-system.
Draw the orbitals of thiophene to show that is aromatic.
- Answer
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This drawing shows it has 6 electrons in the pi-orbital.
The following ring is called a thiazolium ring. Describe how it is aromatic.
- Answer
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Similar to the last question, the drawing shows that there is only 6 electrons in the pi-system.


1.1: “And” “Or”
To communicate mathematics you will need to understand and abide by the conventions of mathematicians. In this chapter we review some of these conventions.
Statements are declarative sentences; that is, a statement is a sentence which is true or false. Mathematicians make mathematical statements - sentences about mathematics which are true or false. For instance, the statement:
"All prime numbers, except the number 2, are odd."
is a true statement. The statement: \[" 3<2 . "\] is false.
We use natural language connectives to combine mathematical statements. The connectives "and" and "or" have a particular usage in mathematical prose. Let \(P\) and \(Q\) be mathematical statements. The statement \[P \text { and } Q\] is the statement that both \(P\) and \(Q\) are true.
Mathematicians use what is called the "inclusive or". In everyday usage the statement " \(P\) or \(Q\) " can sometimes mean that exactly one (but not both) of the statements \(P\) and \(Q\) is true. In mathematics, the statement \[P \text { or } Q\] is true when either or both statements are true, i.e. when any of the following hold:
\(P\) is true and \(Q\) is false.
\(P\) is false and \(Q\) is true.
\(P\) is true and \(Q\) is true.


7.7: Topologies. Borel Sets. Borel Measures
I. Our theory of set families leads quite naturally to a generalization of metric spaces. As we know, in any such space \((S, \rho),\) there is a family \(\mathcal{G}\) of open sets, and a family \(\mathcal{F}\) of all closed sets. In Chapter 3, §12, we derived the following two properties.
(i) \(\mathcal{G}\) is closed under any (even uncountable) unions and under finite intersections (Chapter 3, §12, Theorem 2). Moreover,
\[\emptyset \in \mathcal{G} \text { and } S \in \mathcal{G},\]
(ii) \(\mathcal{F}\) has these properties, with "unions" and "intersections" interchanged (Chapter 3, §12, Theorem 3). Moreover, by definition,
\[A \in \mathcal{F} \text { iff }-A \in \mathcal{G}.\]
Now, quite often, it is not so important to have distances (i.e., a metric) defined in \(S,\) but rather to single out two set families, \(\mathcal{G}\) and \(\mathcal{F},\) with properties (i) and (ii), in a suitable manner. For examples, see Problems 1 to 4 below. Once \(\mathcal{G}\) and \(\mathcal{F}\) are given, one does not need a metric to define such notions as continuity, limits, etc. (See Problems 2 and 3.) This leads us to the following definition.
A topology for a set \(S\) is any set family \(\mathcal{G} \subseteq 2^{S},\) with properties (i).
The pair \((S, \mathcal{G})\) then is called a topological space. If confusion is unlikely, we simply write \(S\) for \((S, \mathcal{G}).\)
\(\mathcal{G}\)-sets are called open sets; their complements form the family \(\mathcal{F}\) (called cotopology) of all closed sets in \(S; \mathcal{F}\) satisfies (ii) (the proof is as in Theorem 3 of Chapter 3, §12).
Any metric space may be treated as a topological one (with \(\mathcal{G}\) defined as in Chapter 3, §12), but the converse is not true. Thus \((S, \mathcal{G})\) is more general.
Note 1. By Problem 15 in Chapter 4, §2, a map
\[f :(S, \rho) \rightarrow\left(T, \rho^{\prime}\right)\]
is continuous iff \(f^{-1}[B]\) is open in \(S\) whenever \(B\) is open in \(T\).
We adopt this as a definition, for topological spaces \(S, T\).
Many other notions (neighborhoods, limits, etc.) carry over from metric spaces by simply treating \(G_{p}\) as "an open set containing \(p.\)" (See Problem 3.)
Note 2. By (i), \(\mathcal{G}\) is surely closed under countable unions. Thus by Note 2 in §3,
\[\mathcal{G}=\mathcal{G}_{\sigma}.\]
Also, \(\mathcal{G}=\mathcal{G}_{d}\) and
\[\mathcal{F}_{\delta}=\mathcal{F}=\mathcal{F}_{s},\]
but not
\[\mathcal{G}=\mathcal{G}_{\delta} \text { or } \mathcal{F}=\mathcal{F}_{\sigma}\]
in general.
\(\mathcal{G}\) and \(\mathcal{F}\) need not be rings or \(\sigma\)-rings (closure fails for differences). But by Theorem 2 in §3, \(\mathcal{G}\) and \(\mathcal{F}\) can be "embedded" in a smallest \(\sigma\)-ring. We name it in the following definition.
The \(\sigma\)-ring \(\mathcal{B}\) generated by a topology \(\mathcal{G}\) in \(S\) is called the Borel field in \(S.\) (It is a \(\sigma\)-field, as \(S \in \mathcal{G} \subseteq \mathcal{B}.)\)
Equivalently, \(\mathcal{B}\) is the least \(\sigma\)-ring \(\supseteq \mathcal{F}.\) (Why?)
\(\mathcal{B}\)-sets are called Borel sets in \((S, \mathcal{G})\).
As \(\mathcal{B}\) is closed under countable unions and intersections, we have not only
\[\mathcal{B} \supseteq \mathcal{G} \text { and } \mathcal{B} \supseteq \mathcal{F},\]
but also
\[\mathcal{B} \supseteq \mathcal{G}_{\delta}, \mathcal{B} \supseteq \mathcal{F}_{\sigma}, \mathcal{B} \supseteq \mathcal{G}_{\delta \sigma}\left[\text { i.e. },\left(\mathcal{G}_{\delta}\right)_{\sigma}\right], \mathcal{B} \supseteq \mathcal{F}_{\sigma \delta}, \text { etc.}\]
Note that
\[\mathcal{G}_{\delta \delta}=\mathcal{G}_{\delta}, \mathcal{F}_{\sigma \sigma}=\mathcal{F}_{\sigma}, \text { etc. (Why?)}\]
II . Special notions apply to measures in metric and topological spaces.
A measure \(m : \mathcal{M} \rightarrow E^{*}\) in \((S, \mathcal{G})\) is called topological iff \(\mathcal{G} \subseteq \mathcal{M},\) i.e., all open sets are measurable; \(m\) is a Borel measure iff \(\mathcal{M}=\mathcal{B}\).
Note 3. If \(\mathcal{G} \subseteq \mathcal{M}\) (a \(\sigma\)-ring), then also \(\mathcal{B} \subseteq \mathcal{M}\) since \(\mathcal{B}\) is, by definition, the least \(\sigma\)-ring \(\supseteq \mathcal{G}.\)
Thus \(m\) is topological iff \(\mathcal{B} \subseteq \mathcal{M}\) (hence surely \(\mathcal{F} \subseteq \mathcal{M}, \mathcal{G}_{\delta} \subseteq \mathcal{M}, \mathcal{F}_{\sigma} \subseteq \mathcal{M}\), etc.).
It also follows that any topological measure can be restricted to \(\mathcal{B}\) to obtain a Borel measure, called its Borel restriction.
A measure \(m : \mathcal{M} \rightarrow E^{*}\) in \((S, \mathcal{G})\) is called regular iff it is regular with respect to \(\mathcal{M} \cap \mathcal{G},\) the measurable open sets; i.e.,
\[(\forall A \in \mathcal{M}) \quad m A=\inf \{m X | A \subseteq X \in \mathcal{M} \cap \mathcal{G}\}.\]
If \(m\) is topological \((\mathcal{G} \subseteq \mathcal{M}),\) this simplifies to
\[m A=\inf \{m X | A \subseteq X \in \mathcal{G}\},\]
i.e., \(m\) is \(\mathcal{G}\)-regular (Definition 5 in §5).
A measure \(m\) is strongly regular iff for any \(A \in \mathcal{M}\) and \(\varepsilon>0,\) there is an open set \(G \in \mathcal{M}\) and a closed set \(F \in \mathcal{M}\) such that
\[F \subseteq A \subseteq G, \text { with } m(A-F)<\varepsilon \text { and } m(G-A)<\varepsilon;\]
thus \(A\) can be "approximated" by open supersets and closed subsets, both measurable. As is easily seen, this implies regularity.
A kind of converse is given by the following theorem.
If a measure \(m : \mathcal{M} \rightarrow E^{*}\) in \((S, \mathcal{G})\) is regular and \(\sigma\)-finite (see Definition 4 in §5), with \(S \in \mathcal{M},\) then \(m\) is also strongly regular.
- Proof
-
Fix \(\varepsilon>0\) and let \(m A<\infty\).
By regularity,
\[m A=\inf \{m X | A \subseteq X \in \mathcal{M} \cap \mathcal{G}\};\]
so there is a set \(X \in \mathcal{M} \cap \mathcal{G}\) (measurable and open), with
\[A \subseteq X \text { and } m X<m A+\varepsilon.\]
Then
\[m(X-A)=m X-m A<\varepsilon,\]
and \(X\) is the open set \(G\) required in (2).
If, however, \(m A=\infty,\) use \(\sigma\)-finiteness to obtain
\[A \subseteq \bigcup_{k=1}^{\infty} X_{k}\]
for some sets \(X_{k} \in \mathcal{M}, m X_{k}<\infty;\) so
\[A=\bigcup_{k}\left(A \cap X_{k}\right).\]
Put
\[A_{k}=A \cap X_{k} \in \mathcal{M}.\]
(Why?) Then
\[A=\bigcup_{k} A_{k},\]
and
\[m A_{k} \leq m X_{k}<\infty.\]
Now, by what was proved above, for each \(A_{k}\) there is an open measurable \(G_{k} \supseteq A_{k},\) with
\[m\left(G_{k}-A_{k}\right)<\frac{\varepsilon}{2^{k}},\]
Set
\[G=\bigcup_{k=1}^{\infty} G_{k}.\]
Then \(G \in \mathcal{M} \cap \mathcal{G}\) and \(G \supseteq A.\) Moreover,
\[G-A=\bigcup_{k} G_{k}-\bigcup_{k} A_{k} \subseteq \bigcup_{k}\left(G_{k}-A_{k}\right).\]
(Verify!) Thus by \(\sigma\)-subadditivity,
\[m(G-A) \leq \sum_{k} m\left(G_{k}-A_{k}\right)<\sum_{k=1}^{\infty} \frac{\varepsilon}{2^{k}}=\varepsilon,\]
as required.
To find also the closed set \(F,\) consider
\[-A=S-A \in \mathcal{M}.\]
As shown above, there is an open measurable set \(G^{\prime} \supseteq-A,\) with
\[\varepsilon>m\left(G^{\prime}-(-A)\right)=m\left(G^{\prime} \cap A\right)=m\left(A-\left(-G^{\prime}\right)\right).\]
Then
\[F=-G^{\prime} \subseteq A\]
is the desired closed set, with \(m(A-F)<\varepsilon. \quad \square\)
If \(m : \mathcal{M} \rightarrow E^{*}\) is a strongly regular measure in \((S, \mathcal{G}),\) then for any \(A \in \mathcal{M},\) there are measurable sets \(H \in \mathcal{F}_{\sigma}\) and \(K \in \mathcal{G}_{\delta}\) such that
\[H \subseteq A \subseteq K \text { and } m(A-H)=0=m(K-A);\]
hence
\[m A=m H=m K.\]
- Proof
-
Let \(A \in \mathcal{M}.\) By strong regularity, given \(\varepsilon_{n}=1 / n,\) one finds measurable sets
\[G_{n} \in \mathcal{G} \text { and } F_{n} \in \mathcal{F}, \quad n=1,2, \ldots,\]
such that
\[F_{n} \subseteq A \subseteq G_{n}\]
and
\[m\left(A-F_{n}\right)<\frac{1}{n} \text { and } m\left(G_{n}-A\right)<\frac{1}{n}, \quad n=1,2, \ldots.\]
Let
\[H=\bigcup_{n=1}^{\infty} F_{n} \text { and } K=\bigcap_{n=1}^{\infty} G_{n}.\]
Then \(H, K \in \mathcal{M}, H \in \mathcal{F}_{\sigma}, K \in \mathcal{G}_{\delta},\) and
\[H \subseteq A \subseteq K.\]
Also, \(F_{n} \subseteq H\) and \(G_{n} \supseteq K\).
Hence
\[A-H \subseteq A-F_{n} \text { and } K-A \subseteq G_{n}-A;\]
so by (4),
\[m(A-H)<\frac{1}{n} \rightarrow 0 \text { and } m(K-A)<\frac{1}{n} \rightarrow 0.\]
Finally,
\[m A=m(A-H)+m H=m H,\]
and similarly \(m A=m K\).
Thus all is proved.\(\quad \square\)


1.3: Calculations of TOA radiance and TOA reflectance
Sensor calibration typically results in a simple linear (or almost linear) conversion between the radiance that the sensor is exposed to and the voltage created by the detecting elements. This means that if we record the voltage created by each detecting element, along with the location on the Earth’s surface from which the radiation that created the current came, and we arrange all the voltages measured by each of the detecting elements in a spatial pattern according to their locations on the Earth’s surface, we can create a map-like image in which the values in each pixel are the voltages measured from that area. In essence that is how satellite images are created.
Quantization
However, the Digital Numbers that we see when we load a satellite image into a software do not equal the measured voltages. There are two related reasons for this – reducing the amount of computer memory needed to store the measurements, while still providing data with adequate precision. Measured voltages take the form of decimal numbers, like 3.560194 mV, which by default require 32 bits of computer memory to store (or 64 bits if a large number of decimals is required). But in reality the measurements of voltage in satellite sensors are not accurate to six decimals (as in the example above) or some other large number of decimals, so there is no need to use 32 bits of memory to store the voltage for each detecting element. What is typically done instead is that during, or even before, sensor development, the necessary precision of the measurements is determined, and the instrument is built to make measurements with that precision (which is then tested later when calibrating the instrument). With the precision of the measurements known, and adding information on the range of radiances/voltages that the instrument should be sensitive to, the number of bits necessary to contain the measurements can then be determined, and the voltage can be converted to a Digital Number that can be stored as an integer, which requires much less computer memory to store.
For example, imagine that you want to measure how tall people are, and you say you want to measure it to the nearest cm. Using units of cm, and knowing that any adult measured in the history of such measurements has been somewhere between 54cm and 272cm tall, you realize that you really only need to be able to measure 219 different values (both extremes included): 54cm, 55cm, 56cm, …. 272cm. To store 219 different values, you would only need 8 bits, because 2 8 = 256, which even leaves you some extra values you don’t need. You then decide to encode a height of 54cm with a value of ‘0’, 55cm with a value of ‘1’, and so on (code values are simply 54 less than the corresponding heights in cm). Now, when you measure someone (with your super-accurate measuring tape) to be 173.234cm tall, you tell yourself that you only need cm precision so you round it off to 173cm, and you encode it as 173 – 54 = ‘119’. This is the equivalent of the Digital Number that you see in a pixel in a given band in an image. When you know how the code (‘119’) relates back to the actual measurement (173cm), the value becomes more meaningful. This process of encoding is called quantization .
Digital Numbers to TOA Radiance
When you open a satellite image in a software package, the initial value you see in each pixel is called a Digital Number (DN), which is equivalent to the code (‘119’) in the example above. While the DNs are sufficient to produce a nice visualization of the image, if you want to treat the image as a series of radiometric measurements you need to convert it into a physical quantity (like the 173cm that was the height of the person measured in that example). In remote sensing, the translation of a Digital Number value into a radiance measurement is really a two-step process. In passive optical remote sensing, the calibration of the sensor has resulted in a known relationship between the radiance that the sensor is exposed to, and the voltage generated by the radiation, and quantization has then been used to encode the measure voltage as a Digital Number. To convert this Digital Number (DN) to radiance thus in theory requires first a conversion to voltage and then a conversion to radiance. Luckily it is easy to do these two conversions at the same time. We will look at how this is done for Landsat data as an example, but the same principle is relevant to data from all other passive optical satellite sensors.
Landsat metadata contain radiometric calibration coefficients (Figure 20) that allow you to directly convert DN values to Top-of-Atmosphere (TOA) radiance. The coefficients contain a ‘multiplication’ term and an ‘addition’ term, each of which is unique to each band, and they are used like this: L λ =mult λ *DN+add λ , where L is the radiance and λ, the common notation for wavelength, indicates a given band.
20: Calibration coefficients used to convert DN values to TOA radiance values for Landsat 8. By Anders Knudby, CC BY 4.0 .
In order to produce a raster data layer in which each cell value is a measure of TOA radiance, you can therefore use the ‘Raster Calculator’ available in all GIS / remote sensing software, and some software packages even have dedicated ‘Landsat radiometric calibration’ tools that find the relevant information in the metadata and apply it for you.
Digital Numbers to TOA Reflectance
Converting DN values to a measure of radiance is an important step toward using remote sensing data in a quantitative way – in which the values in each cell have an actual physical meaning. However, there is one principal problem with radiance measurements: The radiance coming from an area depends both on the physical characteristics of that area, such as what is on the surface, but it also depends on the amount of radiation incident on that area. It is easy to confirm this with a single experiment: Move to a light switch near you, make sure it is on, and focus your gaze on a wall. Note how bright it is (a rough human approximation of the amount of radiance coming from it). Now turn off the light, and note whether the wall got darker or lighter. It got darker, of course, which means that there is now less radiance coming from it. And this happens although the wall itself did not change at all! In the remote sensing context, radiance measurements are therefore rarely used directly, but are typically used as a step toward a radiometric quantity that is more closely linked to the surface it is coming from: reflectance. For isotropic radiation, the reflectance (R) of a given area is the ratio of outgoing vs. incoming radiance, also often expressed on the basis of exitance (E) and irradiance (I):
In the wall example, when you turn off the light both the incoming and the outgoing radiance is reduced, and the reflectance of the wall is unchanged. That’s because the reflectance is a physical property of the wall that is independent of illumination. This makes it uniquely suitable for use in remote sensing, because multiple observations with varying illumination can be compared, for uses such as tracking change of an area through time, or for comparing different areas on Earth.
If we convert TOA radiance values to reflectance values, we get what’s called TOA reflectance, which is the reflectance of the entire Earth-Atmosphere system. To think about what that means, imagine that you are an astronaut and you put a hula-hoop ring horizontally right at the level of the top of the atmosphere (if the top of the atmosphere were possible to define and find!), and you then measure the amount of radiation passing through the ring in upward and downward directions. Dividing one by the other gives you TOA reflectance.
Calculating TOA reflectance is done a bit differently for different data types. For Landsat 8, there is another set of calibration coefficients that you can apply exactly like you can apply those for TOA radiance (Figure 21).
21: Calibration coefficients used to convert DN values to TOA reflectance values for Landsat 8. Note that values for bands 10 and 11 are not provided, because these bands do not measure reflected solar radiation (except to a negligible degree). Also note that coefficients are the same for all bands – for Landsat 8 the DN values have been specifically generated for each scene to ensure constant TOA reflectance calibration coefficients. By Anders Knudby, CC BY 4.0 .
For older Landsat data, as well as for several other types of sensors, it is necessary to explicitly calculate the amount of incoming solar radiation at the top of the atmosphere, and divide the radiance by this value while correcting for the difference between radiance and irradiance and the angle of incoming solar radiation. This is done generically with the following equation (rearranged for pedagogical reasons from equation 4, page 79, in the Landsat 7 handbook ), explained in more detail below:
where R TOA is the top-of-atmosphere reflectance, L is (upward) radiance, d is the Earth-Sun distance in astronomical units, I Sun is the mean extraterrestrial solar irradiance, and θ Sun is the solar zenith angle – the angle between the direction toward the Sun and the normal of the Earth’s surface. The equation is a division between the outgoing radiation in all upward directions (i.e. the exitance, in the numerator) and the incoming radiation in all downward directions (i.e. the irradiance, in the denominator).
The numerator is relatively simple: the upward radiance, L, which is quantified as the radiation per exposed unit surface area moving in directions within a specified solid angle, is multiplied by π to convert it to a value that quantifies radiation moving in all upward directions instead, to make it comparable to the incoming solar irradiance in the denominator.
In the denominator, we have I Sun , the average incoming solar irradiance at the top of the atmosphere. Because the distance between the Sun and the Earth varies throughout the year and the amount of solar irradiance varies with it, we need to correct this by the factor 1/d 2 , where d is the actual distance between the Sun and the Earth when the image was acquired. If the Sun were at zenith (directly overhead), this would quantify the incoming irradiance per unit surface area. In cases when the Sun is not at zenith, we must multiply this value by the cosine of the solar zenith angle to arrive at the irradiance per exposed unit surface area.
Note: In the above two equations, the λ subscript indicating that these calculations must be performed on a band-by-band basis has been omitted for clarity, but keep this in mind. L varies by band, as does I Sun , so both must be calculated specifically for each band to be correct.
With these calculations, you are able to convert the DN values in each pixel to a measure of TOA reflectance, which is a physical property of the Earth-Atmosphere system. This fact, that TOA reflectance is not an arbitrary number dependent on sensor calibration and illumination, but rather a physical property of the parts of the Earth seen in the image, allows us to use images captured by different sensors, at different times, and for different areas, and directly compare the values in each pixel. The one remaining problem in this approach is that most researchers are more interested in the reflectance of the Earth’s surface than in the reflectance at the top of the atmosphere, but the state of the atmosphere and the state of the surface both influence TOA reflectance. The next step to further process the data from each pixel is thus to remove the influence that the atmosphere has on these values, to arrive at a measure of surface reflectance!