15.0: Chapter 1
Your Turn
1.1.1
1. Set notation lists the elements inside curly braces, separated by commas. Your list may include different tools. Sets are usually designated with capital letters.
\(T=\{\) wrench, screwdriver, hammer, pliers }
One possible solution: \(T=\{\) wrench, screwdriver, hammer, pliers }.
1.1.2
1. A well-defined set clearly communicates whether the element is a member of the set or not. "A collection of medium-sized potatoes" is not well defined. What makes a potato medium-sized?
This is not a well-defined set.
2. A well-defined set clearly communicates whether the element is a member of the set or not. You can determine which musical artists were members of the group when it formed in 1995.
\(B=\{\) will.i.am, apl. de. ap, taboo }
This is a well-defined set.
1.1.3
1. When there are no elements that satisfy the set conditions, you represent the set with either version of the empty set notation: a zero with a slash through it or empty curly braces.
\(\emptyset\) or \(\}\)
\(\emptyset\) or \(\}\)
1.1.4
1. Be sure to include zero because the requirements include "greater than or equal to zero." You stop before 10 because 10 is not a single-digit number.
\[
\begin{array}{l}
D=\{0,1,2,3,4,5,6,7,8,9\} \\
D=\{0,1,2, \ldots, 9\}
\end{array}
\nonumber \]
1.1.5
1. This set is infinite. You represent large sets by giving enough numbers to define the pattern, and then using three dots, the ellipsis, to show that the set continues. When no number follows the ellipsis, you know the set continues infinitely.
\[
\begin{array}{l}
M=\{1,3,5, \cdots\} \\
M=\{1,3,5, \ldots\}
\end{array}
\nonumber \]
1.1.6
1. The roster method lists the elements of the set. Set builder notation uses a letter to represent a sample member of the set, a vertical bar, and then a description of a sample member. In this case, because it is not practical to list all types of cars, it is much easier to use set builder notation.
\[
\begin{array}{l}
C=\{c \mid c \text { is a type of car }\} \\
C=\{c \mid c \text { is a car }\}
\end{array}
\nonumber \]
1.1.7
1. The roster method lists the elements of the set. Set builder notation uses a letter to represent a sample member of the set, a vertical bar, and then a description of a sample member. In this case, because it is not practical to list all musical instruments, it is much easier to use set builder notation.
\(I=\{i \mid i\) is a musical instrument \(\}\)
\(I=\{i \mid i\) is a musical instrument \(\}\)
1.1.8
1. The cardinal value is the number of elements in the set. There are no prime numbers less than 2.
\[
\begin{array}{l}
n(\mathrm{P})=0 \\
n(P)=0
\end{array}
\nonumber \]
2. The cardinal value is the number of elements in the set. There are 26 lowercase letters of the English alphabet.
\[
\begin{array}{l}
n(\mathrm{P})=26 \\
n(A)=26
\end{array}
\nonumber \]
1.1.9
1. There are four elements in set \(B\), so this set is finite. finite
2. There is no largest or smallest real number, so this set is infinite. infinite
1.1.10
1. Equal sets have the same elements, without regard to order. These sets have the same elements, although in a different order.
Set \(B\) is equal to set \(A, B=A\)
2. Equal sets have the same elements, without regard to order. These sets are not equal because only one has the element \(f\). Equivalent sets have the same number of elements. One set has four elements, while the other set has five elements, so they are not equivalent. These sets are neither equal nor equivalent. neither
3. Equal sets have the same elements, without regard to order. Equivalent sets have the same number of elements. These sets are equivalent but not equal because they have the same number of elements (four), but not the same elements. One has a \(b\), while the other has a \(c\).
Set \(B\) is equivalent to set \(C, B \sim C\)
1.2.1
1. Step 1: Begin with the set itself, as every set is a subset of itself. The original set has two elements, so its cardinality is 2.
\{head, tails \}
Step 2: Next, list all the proper subsets containing one less element than the original set. That is, list the sets with one element.
\{heads\}
\{tails \}
Step 3: Continue this process, reducing the number of elements in the subset by one each step. On this step, list the subsets with zero elements, or the empty set.
\{\}
\{heads, tails \(\} ;\{\) heads \(\},\{\) tails \(\} ;\) and \(\emptyset\)
1.2.2
1. A set is a proper subset of \(L\) if every member of the subset is a member of \(L\) and the subset does not contain an element that is in \(L\). The proper subsets of \(L\) with one member include the following.
\{Articuno \}
\{Zapdos \}
\{Moltres \}
\{Mewtwo\}
A set with one member could contain any one of the following:
\{Articuno \(\},\{\) Zapdos \(\},\{\) Moltres \(\}\), or \(\{\) Mewtwo \(\}\).
2. A set is a proper subset of \(L\) if every member of the subset is a member of \(L\) and the subset does not contain an element that is in \(L\). The proper subsets of \(L\) with three members will include three elements and leave out one.
\{Articuno, Zapdos, Moltres\}
\{Articuno, Zapdos, Mewtwo\}
\{Articuno, Moltres, Mewtwo\}
\{Zapdos, Moltres, Mewtwo \}
Any of the following combinations of three members would work:
\{Articuno, Zapdos, Mewtwo\}, \{Articuno, Moltres, Mewtwo\}, or \{Zapdos, Moltres, Mewtwo\}.
3. A set is a proper subset of \(L\) if every member of the subset is a member of \(L\) and the subset does not contain an element that is in \(L\). The only proper subset of \(L\) with no members is the empty set, represented by \(\}\) or \(\emptyset\) The empty set is represented as \(\}\) or \(\emptyset\).
1.2.3
1. The set of even numbers given in this question is a subset of the set of natural numbers. The set of even numbers is a proper subset because there are members of the natural numbers ( 1 and 3 for example) that are not even numbers. The notation for "the even numbers are a proper subset of the natural numbers":
\[
E \subset \mathbb{N}
\nonumber \]
Note that there are also negative even numbers that were not a part of this question.
\[
E \subset \mathbb{N}
\nonumber \]
1.2.4
1. Use the formula for the number of subsets of set \(A\) with \(n\) elements: \(2^{n(A)}\).
Set \(T\) has nine elements.
\[
2^{n(T)}=2^9=512
\nonumber \]
1.2.5
1. Set builder notation uses a letter to represent a sample member of the set, a vertical bar, and then a description of a sample member.
\[
\begin{array}{l}
\{m \mid m=5 n \text { where } n \in \mathbb{N}\} \\
\{m \mid m=5 n \text { where } n \in \mathbb{N}\}
\end{array}
\nonumber \]
1.2.6
1. Serena also ordered a fish sandwich and chicken nuggets, because for the two sets to be equal they must contain the exact same items: \{fish sandwich, chicken nuggets\} = \{fish sandwich, chicken nuggets\}. Equal sets have the same elements. For the choices to be equal, they needed to order the same food. Serena ordered a fish sandwich and chicken nuggets.
1.2.7
1. There are multiple possible solutions. Each set must contain two players, but both players cannot be the same, otherwise the two sets would be equal, not equivalent. For example, \{Maria, Shantelle \(\}\) and \{Angie, Maria\}.
1.3.1
1. \(L\) is a subset of \(U\) if every element of \(L\) is a member of \(U\). Because all lions are cats, every member of the set of Lions is a member of the set of Cats.
\(L \subset C\)
The set of lions is a subset of the universal set of cats. In other words, the Venn diagram depicts the relationship that all lions are cats. This is expressed symbolically as \(L \subset U\).
1.3.2
1. Sets are disjoint if they have no members in common. No member of the set of Eagles is a member of the set of Canaries. No member of the set of Canaries is a member of the set of Eagles. These sets are disjoint. The set of eagles and the set of canaries are two disjoint subsets of the universal set of all birds. No eagle is a canary, and no canary is an eagle.
1.3.3
1. The universal set is the set of integers. Draw a rectangle and label it with \(U=\) Integers. Next, draw a circle in the rectangle and label with Natural numbers.
\(U=\) Integers
Natural numbers
Venn diagram with universal set, \(U=\) Integers, and subset
\(\mathbb{N}=\) Natural numbers.
2.
U
Venn Diagram with universal set, \(U\) and subset \(A\).
1.3.4
1.
\(U=\) Things That Can Fly
Venn Diagram with universal set, \(U=\) Things that can fly with disjoint subsets Airplanes and Birds.
1.3.5
1. \(A^{\prime}\) represents the complement of set \(A\). The complement of \(\operatorname{set} A\) is the set of all elements in the universal set \(U\) that are not in set \(A\). List the elements of \(U\) that are not in \(A\).
\{orange, green, indigo, violet\}
\(A^{\prime}=\{\) orange, green, indigo, violet \(\}\)
2. \(A^{\prime}\) represents the complement of set \(A\). The complement of \(\operatorname{set} A\) is the set of all elements in the universal set \(U\) that are not in set \(A\). List the elements of \(U\) that are not in \(A\). Because \(A\) includes the cats that are not lions, the complement will include the cats that are lions.
\[
\begin{array}{l}
A^{\prime}=\{c \mid c \text { is a lion }\} \\
A^{\prime}=\{c \in U \mid c \text { is a lion }\} \text { or } A^{\prime}=\{c \in U \mid c \notin A\}
\end{array}
\nonumber \]
1.4.1
1. The members that two sets have in common are included in the intersection of the sets. Only element \(a\) is in both sets.
\[
\begin{array}{l}
A \cap B=\{a\} \\
A \cap B=\{a\}
\end{array}
\nonumber \]
1.4.2
1. The members that two sets have in common are included in the intersection of the sets. The sets have no elements in common, so their intersection is the empty set. You can represent the empty set by either the zero with a slash or empty curly braces.
\[
\begin{array}{l}
A \cap B=\emptyset \text { or } A \cap B=\{ \} \\
A \cap B=\{ \}
\end{array}
\nonumber \]
1.4.3
1. The members that two sets have in common are included in the intersection of the sets. The two sets have the vowels in common.
\[
\begin{array}{l}
A \cap B=\{a, e, i, o, u\} \\
A \cap B=B=\{a, e, i, o, u\}
\end{array}
\nonumber \]
1.4.4
1. The union of two sets includes all the members of the first set and all the members of the second set. Do not list duplicates twice.
\[
\begin{array}{l}
A \cup B=\{a, d, h, p, s, y\} \\
A \cup B=\{a, d, h, p, s, y\}
\end{array}
\nonumber \]
1.4.5
1. The union of two sets includes all the members of the first set and all the members of the second set. Do not list duplicates twice.
\(A \cup B=\{\) red, yellow, blue, orange, green, purple \(\}\)
\(A \cup B=\{\) red, yellow, blue, orange, green, purple \(\}\)
A set with one member could contain any one of the following:
\(\{ {\text{Articuno}}\} {\text{, }}\{ {\text{Zapdos}}\} {\text{, }}\{ {\text{Moltres}}\} {\text{, or }}\{ {\text{Mewtwo}}\} \).
1.4.6
1. The union of two sets includes all the members of the first set and all the members of the second set. Do not list duplicates twice. The vowels in set \(B\) are a subset of set \(A\), so the union is equal to set \(A\).
\[
\begin{array}{l}
A \cup B=\{a, b, c, d, \ldots, z\} \\
A \cup B=\{a, b, c, \ldots, z\}=A
\end{array}
\nonumber \]
1.4.7
1. 33
Use the formula: \(n(A \cup B)=n(A)+n(B)-n(A \cap B)\)
\[
n(A \cup B)=n(A)+n(B)-n(A \cap B)=23+17-7=33
\nonumber \]
1.4.8
1. 113
Use the formula: \(n(A \cup B)=n(A)+n(B)-n(A \cap B)\)
\[
n(A \cup B)=n(A)+n(B)-n(A \cap B)=35+78-0=113
\nonumber \]
1.4.9
1. To find \(A\) or \(B\), find \(A \cup B\). The union of two sets includes all the members of the first set and all the members of the second set. Do not list duplicates twice.
\(A\) or \(B=A \cup B=\{h, a, p, y, w, e, s, o, m\}\)
\(A\) or \(B=A \cup B=\{h, a, p, y, w, e, s, o, m\}\).
2. To find \(A\) and \(C\), find \(A \cap C\). The intersection of two sets includes the elements that are in both sets.
\(A\) and \(C=A \cap C=\{a, h\}\)
\(A\) and \(C=A \cap C=\{a, h\}\).
3. To find \(B\) or \(C\), find \(B \cup C\). The union of two sets includes all the members of the first set and all the members of the second set. Do not list duplicates twice.
\(B\) or \(C=B \cup C=\{a, w, e, s, o, m, t, h\}\)
\(B\) or \(C=B \cup C=\{a, w, e, s, o, m, t, h\}\).
4. To find \((A\) and \(C)\) and \(B\), you need to follow the order of operations and evaluate the parentheses first.
First, evaluate inside the parentheses. Find \(A\) and \(C=A \cap C\). The intersection of two sets includes the elements that are in both sets.
\(A\) and \(C=A \cap C=\{h, a, p, y\} \cap\{m, a, t, h\}=\{a, h\}\)
To find \((A\) and \(C)\) and \(B=(A \cap C) \cap B\), find \(\{a, h\} \cap B\).
\(\{a, h\} \cap\{a, w, e, s, o, m\}=\{a\}\)
\((A\) and \(C)\) and \(B=(A \cap C) \cap B=\{a, h\} \cap\{a, w, e, s, o, m\}=\{a\}\).
1.4.10
1. 127
Total number of guests \(=150\)
\(n(\) neither soup nor salad \()=23\)
To find the number who had soup, salad, or both, subtract 23 from 150.
\[
150-23=127
\nonumber \]
2. 50
Total number of guests \(=150\)
\(n\) (neither soup nor salad \()=23\)
To find the number who had soup, salad, or both, subtract 23 from 150.
\[
150-23=127
\nonumber \]
This represents the number of elements in the union of the sets of Soup and Salad eaters.
| Substitute. |
\(127=92+85-n(A\cap B)\) |
|---|---|
| Simplify. | \(127=177-n(A \cap B)\) |
| Subtract 127. | \(127-127=177-n(A \cap B)-127\) |
| Simplify. | \(0=50-n(A \cap B)\) |
| Add the variable. | \(0+n(A \cap B)=50-n(A \cap B)+n(A \cap B)\) |
| Simplify. | \(n(A \cap B)=50\) |
1.4.11
1. \(A \cap B=\{3,5,7\}\).
The members that two sets have in common are included in the intersection of the sets. Look for the elements where the circles overlap.
\[
A \cap B=\{3,5,7\}
\nonumber \]
2. \(A \cup B=\{1,2,3,5,7,9\}\).
The union of two sets includes all the members of the first set and all the members of the second set. Do not list duplicates twice. List the elements that are inside the two circles.
\[
A \cup B=\{1,2,3,5,7,9\}
\nonumber \]
3. \(A \cap B^{\prime}=\{1,9\}\).
First, find \(B^{\prime}\), the complement of \(B\). The complement of set \(B\) is the set of all elements in the universal set \(U\) that are not in set \(B\). List the elements of \(U\) that are not in \(B\).
\[
B^{\prime}=\{0,1,4,6,8,9\}
\nonumber \]
Now find
\[
A \cap B^{\prime}=\{1,3,5,7,9\} \cap\{0,1,4,6,8,9\}
\nonumber \]
The intersection of two sets are the elements that are in both sets.
\[
A \cap B^{\prime}=\{1,9\}
\nonumber \]
4. \(n\left(A \cap B^{\prime}\right)=2\).
First, find \(B^{\prime}\), the complement of \(B\). The complement of set \(B\) is the set of all elements in the universal set \(U\) that are not in set \(B\). List the elements of \(U\) that are not in \(B\).
\[
B^{\prime}=\{0,1,4,6,8,9\}
\nonumber \]
Now find
\[
A \cap B^{\prime}=\{1,3,5,7,9\} \cap\{0,1,4,6,8,9\}
\nonumber \]
The intersection of two sets are the elements that are in both sets.
\(A \cap B^{\prime}=\{1,9\}\)
\(n\left(A \cap B^{\prime}\right)\) is the number of elements in the set, which is two.
1.4.12
1. \(A \cap B=\{3,5,7\}\).
The members that two sets have in common are included in the intersection of the sets. Look for the elements where the circles overlap.
\[
A \cap B=\{3,5,7\}
\nonumber \]
2. \(A \cup B=\{1,2,3,5,7,9\}\).
The union of two sets includes all the members of the first set and all the members of the second set. Do not list duplicates twice. List the elements that are inside the two circles.
\[
A \cup B=\{1,2,3,5,7,9\}
\nonumber \]
3. \(A \cap B^{\prime}=\{1,9\}\).
First, find \(B^{\prime}\), the complement of \(B\). The complement of set \(B\) is the set of all elements in the universal set \(U\) that are not in set \(B\). List the elements of \(U\) that are not in \(B\).
\[
B^{\prime}=\{0,1,4,6,8,9\}
\nonumber \]
Now find
\[
A \cap B^{\prime}=\{1,3,5,7,9\} \cap\{0,1,4,6,8,9\}
\nonumber \]
The intersection of two sets are the elements that are in both sets.
\[
A \cap B^{\prime}=\{1,9\}
\nonumber \]
4. \(n\left(A \cap B^{\prime}\right)=2\).
First, find \(B^{\prime}\), the complement of \(B\). The complement of set \(B\) is the set of all elements in the universal set \(U\) that are not in set \(B\). List the elements of \(U\) that are not in \(B\).
\[
B^{\prime}=\{0,1,4,6,8,9\}
\nonumber \]
Now find
\[
A \cap B^{\prime}=\{1,3,5,7,9\} \cap\{0,1,4,6,8,9\}
\nonumber \]
The intersection of two sets are the elements that are in both sets.
\(A \cap B^{\prime}=\{1,9\}\)
\(n\left(A \cap B^{\prime}\right)\) is the number of elements in the set, which is two.
1.5.1
1. To find \(A\) or \(B\), find \(A \cup B\). The union of two sets includes all the members of the first set and all the members of the second set. Do not list duplicates twice.
Because the two sets are disjoint, the number of elements in the union of the two sets is the sum of the number of elements in the two sets.
\[
23+17=40
\nonumber \]
2. To find \(A\) or \(B\), find \(A \cap B\). The intersection of two sets includes the elements they have in common. They have no elements in common, so the number of elements in the intersection is zero.
3. \(A^{\prime}\) is the complement of \(A\). The complement of \(\operatorname{set} A\) is the set of all elements in the universal set \(U\) that are not in set \(A\). The number of elements that are not in set \(A\) :
\[
17+10=27
\nonumber \]
1.5.2
1. \(n(B)=n\left(A B^{+}\right)+n\left(A B^{-}\right)+n\left(B^{+}\right)+n\left(B^{-}\right)=14\)
Count every blood type that has any type of \(B\) as part of the blood type.
\[
n(B)=n\left(A B^{+}\right)+n\left(A B^{-}\right)+n\left(B^{+}\right)+n\left(B^{-}\right)=3+1+8+2=14
\nonumber \]
2. \(n\left(B^{\prime}\right)=n(U)-n(B)=86\)
You could count every type that does not have a B as part of the blood type. However, if you did the previous exercise, you already know the complement of \(B\).
\[
n\left(B^{\prime}\right)=n(U)-n(B)=100-14=86
\nonumber \]
3. \(n\left(B \cup R h^{+}\right)=87\)
Count the number of people who are in either the B circle or the \(\mathrm{RH}^{+}\)circles.
\[
1+3+2+8+36+37=87
\nonumber \]
1.5.3
1. Draw a rectangle to represent the universal set. Label it with " \(U=\) Conference Attendees \(=50\)."
Inside the rectangle, draw three overlapping circles. Label one circle Soup, one Sandwich, and one Salad. In the Soup circle, write 4 in the part that does not overlap any other circle, write 2 in the part that only overlaps the Sandwich circle, write 8 in the part that overlaps all three circles, and write 3 in the part that overlaps only the Salad circle. In the Sandwich circle, write 15 in the part that does not overlap any other circle and write 10 in the part that only overlaps the Salad circle. In the Salad circle, write 8 in the part that does not overlap any other circle. Write 0 outside all the circles.
Venn diagram – Attendees at a conference with sets: Soup, Sandwich, Salad – Complete Solution
1.5.4
1. \(A \cap(B \cap C)\)
First, evaluate the operation inside the parentheses.
\[
B \cap C=\{0,2,4,6,8,10,12\} \cap\{0,3,6,9,12\}
\nonumber \]
The intersection of two sets includes the elements that are common to both sets.
\[
\begin{array}{l}
B \cap C=\{0,6,12\} \\
A \cap(B \cap C)=\{0,1,2,3,4,5,6\} \cap\{0,6,12\} \\
A \cap(B \cap C)=\{0,6\} \\
A \cap(B \cap C)=\{0,1,2,3,4,5,6\} \cap\{0,6,12\}=\{0,6\}
\end{array}
\nonumber \]
2. \((A \cap B) \cup(A \cap C)\)
Evaluate the expression in the left parentheses.
\(A \cap B=\{0,1,2,3,4,5,6\} \cap\{0,2,4,6,8,10,12\}\)
The intersection of two sets includes the elements that are common to both sets.
\(A \cap B=\{0,2,4,6\}\)
Evaluate the expression in the right parentheses.
\(A \cap C=\{0,1,2,3,4,5,6\} \cap\{0,3,6,9,12\}\)
\(A \cap C=\{0,3,6\}\)
Now, you can find \((A \cap B) \cup(A \cap C)\).
\[
\begin{array}{l}
(A \cap B) \cup(A \cap C)=\{0,2,4,6\} \cup\{0,3,6\} \\
(A \cap B) \cup(A \cap C)=\{0,2,3,4,6\} \\
(A \cap B) \cup(B \cap C)=\{0,2,4,6\} \cup\{0,3,6\}=\{0,2,3,4,6\}
\end{array} \nonumber \]
3. \(\left(A \cup C^{\prime}\right) \cap\left(B \cup C^{\prime}\right)\)
\(C^{\prime}\), the complement of \(C\), includes the elements of \(U\) that are not in \(C\).
\[
\begin{array}{l}
C=\{0,3,6,9,12\} \text { and } U=\{0,1,2,3,4,5,6,7,8,9,10,11,12\} \\
C^{\prime}=\{1,2,4,5,7,8,10,11\}
\end{array}
\nonumber \]
Evaluate the expression in the left parentheses.
\[
A \cup C^{\prime}=\{0,1,2,3,4,5,6\} \cap\{1,2,4,5,7,8,10,11\}
\nonumber \]
The intersection of two sets includes the elements that are common to both sets.
\[
A \cup C^{\prime}=\{0,1,2,3,4,5,6,7,8,10,11\}
\nonumber \]
Evaluate the expression in the right parentheses.
\[
\begin{array}{l}
B \cup C^{\prime}=\{0,2,4,6,8,10,12\} \cap\{1,2,4,5,7,8,10,11\} \\
B \cup C^{\prime}=\{0,1,2,4,5,6,7,8,10,11,12\}
\end{array}
\nonumber \]
Now, you can find \(\left(A \cup C^{\prime}\right) \cap\left(A \cup C^{\prime}\right)\).
\[
\begin{array}{l}
\left(A \cup C^{\prime}\right) \cap\left(B \cup C^{\prime}\right)=\{0,1,2,3,4,5,6,7,8,10,11\} \cap\{0,1,2,4,5,6,7,8,10,11,12\} \\
\left(A \cup C^{\prime}\right) \cap\left(B \cup C^{\prime}\right)=\{0,1,2,4,5,6,7,8,10,11\} \\
\left(A \cup C^{\prime}\right) \cap\left(B \cup C^{\prime}\right)=\{0,1,2,3,4,5,6,7,8,10,11\} \cap\{0,1,2,4,5,6,7,8,10,11,12\}=\{0,1,2,4,5,6,7,8
\end{array}
\nonumber \]
1.5.5
1. First, draw the left side of the equation.
For \(A \cap B\), draw a rectangle to represent the universal set \(U\). Label the rectangle " \(U\)." Inside the rectangle, draw overlapping circles to represent two sets. Label one circle " \(A\) " and the other circle " \(B\)." Shade the overlap.
Now, draw the complement of the intersection. Draw the same Venn diagram, but shade everything except the overlap to represent \((A \cap B)^{\prime}\).
Now, draw the right side of the equation: \(A^{\prime} \cup B^{\prime}\).
First, draw \(A^{\prime}\). Draw a rectangle to represent the universal set \(U\). Label the rectangle " \(U\)." Inside the rectangle, draw a circle and label the circle " \(A\)." Shade everything outside the circle.
To the right of the rectangle, draw the \(U\) symbol.
Now, draw \(B^{\prime}\). Draw a rectangle to represent the universal set \(U\). Label the rectangle " \(U\)." Inside the rectangle, draw a circle and label the circle " \(B\)." Shade everything outside the circle.
To the right of the second rectangle, draw the equal symbol: =.
For \(A^{\prime} \cup B^{\prime}\), draw a rectangle to represent the universal set \(U\). Label the rectangle " \(U\)." Inside the rectangle, draw overlapping circles to represent two sets. Label one circle " \(A\) " and the other circle " \(B\)." Shade everything except for the overlap.
The left side of the equation is:
Venn diagram of intersection of two sets and its complement.
The right side of the equation is given by:
Venn diagram of union of the complement of two sets.
Check Your Understanding
5. equivalent, but not equal 6. finite
7. The roster method lists the individual members of the set inside curly braces, separated by commas. Once you have established the pattern, you can use the ellipsis to avoid needing to list them all. Be sure to list the final member of the set (Z) to avoid giving the appearance of an infinite set.
\[
A=\{A, B, C, D, \cdots, Z\}
\nonumber \]
Set builder notation uses a letter to represent a sample member of the set, a vertical bar, and then a description of a sample member.
\(A=\{a \mid a\) is a capital letter of the English alphabet \(\}\)
Roster method: \(\{\mathrm{A}, \mathrm{B}, \mathrm{C}, \ldots, \mathrm{Z}\}\), and set builder notation:
\(\{x \mid x\) is a capital letter of the English alphabet \(\}\)
8. subset
9. To be a subset of a set, every member of the subset must also be a member of the set. To be a proper subset, there must be at least one member of the set that is not also in the subset.
10. empty
11. true
12. Use the formula for the number of subsets of set \(A\) with \(n\) elements: \(2^{n(A)}\). Set \(A\) has ten elements.
\[
\begin{array}{l}
2^{n(A)}=2^{10}=1024 \\
2^{10}=1024
\end{array}
\nonumber \]
13. Equivalent sets have the same number of elements. Because \(n(A)=n(B)\), the sets are equivalent. equivalent
14. equal. Equal sets have the same elements. For each set, there can be no element that is not an element of the other set. Sets \(A\) and \(B\) are equal because every member of \(A\) is in \(B\) and every member of \(B\) is in \(A\).
15. relationship
16. universal
17. disjoint or non-overlapping
18. complement
19. disjoint
20. To help you remember, think of the intersection of two streets. The intersection is the pavement that the two streets have in common.
intersection
21. To help you remember, think of the union of the United States. A union joins things together. union
22. \(A \cup B\)
It is very helpful that the union symbol \((\cup)\) looks somewhat like a \(U\).
23. \(A \cap B\)
24. The intersection of two sets includes the elements they have in common. Because \(A\) is a subset of \(B\), all the elements of \(A\) are also in \(B\). There can be no other elements in \(B\) that are in \(A\). Thus, \(A \cap B=A\).
25. The union of two sets includes all the members of the first set and all the members of the second set. Do not list duplicates twice.
Because \(A\) is a subset of \(B\), all the elements of \(A\) are also in \(B\). There can be no elements in \(A\) that are not in \(B\). Thus, \(A \cup B=B\).
26. Disjoint sets have no elements in common. The intersection of two sets includes the elements they have in common. Because set \(A\) and set \(B\) are disjoint, their intersection is the empty set. empty
27. \(n(A \cup B)=n(A)+n(B)-n(A \cap B)\)
28. overlap
By beginning with the region involving the most overlap, you start with the regions involving the most sets and the most complexity. Work toward the simpler parts of the diagram.
29. central
30. intersection of all three sets, \(A \cap B \cap C\)
31. parentheses, complement
32. equation, true


14.1: Prelude to Sequences, Series and Binomial Theorem
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A strange charge suddenly appears on your credit card. But your card is in your wallet—it’s not even lost or stolen. Sadly, you may have been a victim of cyber crime. In this day and age, most transactions take advantage of the benefit of computers in some way. Cyber crime is any type of crime that uses a computer or computer network. Thankfully, many people are working to prevent cyber crime. Sometimes known as cryptographers, these people develop complex patterns in computer codes that block access to would-be thieves as well as write codes to intercept and decode information from them so that they may be identified. In this chapter, you will explore basic sequences and series related to those used by computer programmers to prevent cyber crime.


6.2: Phase
Learning Objectives
- Explain why phase is a scalar
Phase is a scalar
In section 1.3, we defined a (Lorentz) invariant as a quantity that was unchanged under rotations and Lorentz boosts. A measurement such as \(c \to e = 24\) is an invariant because it is simply a count. We’ve counted the number of periods. In fact, a count is not just invariant under rotations and boosts but under any well-behaved change of coordinates — the technical condition being that each coordinate in each set is a differentiable function of each coordinate in the other set. Such a change of coordinates is called a diffeomorphism . For example, a uniform scaling of the coordinates \((t,x,y,z) \to (kt,kx,ky,kz)\), which is analogous to a change of units, 1 is all right as long as \(k\) is nonzero. A quantity that stays the same under any diffeomorphism is called a scalar . Since a Lorentz transformation is a diffeomorphism , every scalar is a Lorentz invariant. Not every Lorentz invariant is a scalar .
Example \(\PageIndex{1}\): The determinant of the metric
Minkowski coordinates can be defined as coordinates in which the metric has the standard form \(g = diag(1,-1,-1,-1)\). If we rescale these coordinates according to \((t,x,y,z) \to (kt,kx,ky,kz)\), then the metric changes according to \(g \to k^{-2} g\). To keep track of how “un-Minkowski” this scaling is, we could use the determinant of the metric \(det(g) = -k^{-8}\). This determinant tells us how many coordinate-grid boxes fit in a unit volume, and it is of interest in a more general context than this example of uniform rescaling, e.g., it serves a similar function when converting from Cartesian coordinates to polar coordinates.
Under a Lorentz transformation or a rotation, the metric retains its standard form, and therefore \(det(g)\) is Lorentz invariant. Another way of seeing this is that spacetime volume is Lorentz invariant, so that a Lorentz transformation doesn’t change how many coordinate-grid boxes fit in a unit volume.
But although \(det(g)\) is a Lorentz invariant, it is not a scalar, because it changes under the transformation described above.
In birdtracks notation, any expression that has no external arrows at all represents a scalar. Since the expression \(c \to e = 24\) has no external arrows, only internal ones, it represents a scalar. Another way of describing this measurement is as a phase. If we prefer to measure the phase \(φ\) in units of cycles, then we have
\[φ = c\to e\]
If we like radians, we can use
\[φ = 2πc\to e\]
Scaling
A convenient way of summarizing all of our categories of variables is by their behavior when we rescale our coordinates. If we switch our time unit from hours to minutes, the number of apples in a bowl is unchanged, the earth’s period of rotation gets \(60\) times bigger, and the frequency of the cuckoo clock changes by a factor of \(1/60\). In other words, a quantity u under rescaling of coordinates by a factor α becomes \(α^p u\), where the exponents \(-1\), \(0\), and \(+1\) correspond to covectors, scalars, and vectors, respectively. We can therefore see that these distinctions are of interest even in one dimension, contrary to what one would have expected from the freshman-physics concept of a vector as something transforming in a certain way under rotations.
In
section 1.3, we defined an invariant as a quantity that did not change under rotations or Lorentz boosts, i.e., one that was independent of the frame of reference. For a scalar we have the even more restrictive condition that it must not change under any change of coordinates. For example, area in \(1 + 1\)-dimensional spacetime is an invariant, but it’s not a scalar; it changes when we rescale our coordinates.


2.3: Formation and evolution of the Earth
The solar system is believed to have formed about 5 billion years ago as a result of aggregation of cosmic dust and interstellar atoms in a region of space in which the density of such material happened to be greater than average. Over 99.8% of this mass, which consisted mostly of hydrogen, collapsed into a proto-sun; the gravitational energy released in this process raised the temperature sufficiently to initiate the hydrogen fusion reactions discussed above.
The planets
The remaining material probably formed a disk that rotated around the sun. As the temperature dropped to around 2000K, some of the most stable combinations of the elements began to condense out. These substances might have been calcium aluminum silicates, followed by the more volatile iron-nickel system, and then magnesium silicates. The further aggregation of these materials, together with the other constituents of the cooling disk, is now believed to be the origin of the planets. Density estimates indicate that the planets closest to the sun are predominantly rocky in nature, and probably condensed first. The outer planets (Uranus, Neptune and Pluto) appear to consist largely of water ice, methane, and ammonia, with a smaller rocky core.
Formation of the Earth
The Earth formed by accretion of solid and particulate material that remained after the much more massive amounts of hydrogen and helium present in the original protoplanets had been dispersed out of the solar system. Gradually, the heat produced by decay of radioactive elements brought about partial melting of the silicate rocks; these lower density molten materials migrated upward, leaving the more dense, iron-containing minerals below. This process, which took about 2 million years, was the first of the three stages into which the chemical evolution of the earth is usually divided:
- Primary differentiation of the elements between the core and mantle.
- Secondary differentiation of the elements, reflecting relative ionic sizes, bonding properties, and solubilities (influencing phase behavior such as fractional crystallization, etc.)
- Tertiary differentiation , still operative, involving the interaction of the crust with the hydrosphere and atmosphere.
The above listing should not be taken too literally; all three kinds of processes have probably proceeded simultaneously, and over a number of cycles. Since the earth is losing approximately four times as much heat as is generated by radioactive decay, the principal driving force of primary and secondary differentiation has gradually slowed down. Partial melting of the upper mantle brought about further fractionation as silicon-containing materials of low density migrated outward to form a crust. In its early stages the stronger granitic rocks had not yet appeared, and the crust was mechanically weak. Upwelling flows of lava would break the surface, and the weight of the solidified lava would cause the crust to subside. In some places, magma would solidify underground, forming low-density rock (batholiths) that would eventually rise by buoyancy and push up overlying crust. These mountain-building periods probably occurred in 6-8 major episodes, each lasting about 800 million years.
Rain
At the same time, outgassing of solids released large amounts of HCl, CO, CO 2 , H 2 S, CH 4 , SO 2 , and SO 3 into the primitive atmosphere. Large amounts of water were present in the primeval rocks in the form of hydrates, which were broken down as the result of the heating. Eventually when the outer crust cooled enough to permit condensation of the water vapor as rain, a new stage of chemical evolution began. The rain was initially highly acidic, equivalent to about 1M HCl; this reacted readily with the basic rocks having high contents of K, Na, Mg, and Ca, leaching them away and forming what would eventually evolve into the oceans. The partial dissolution of the rocks also resulted in large amounts of sediments, which played their own role in the transformation of the earth’s surface.
The continents
Within the crust, the lighter materials, being in isostatic equilibrium with the upper mantle, floated higher, and gradually became the nuclei of continents, which grew by accumulating similar material around their boundaries. This picture of continental development is supported by isotopic ratio studies which indicate that the nucleus of the North American continent, the Canadian Shield, is over 2.5 billion years old, while the peripheral parts are less than 0.6 billion years of age.
Primary differentiation of the elements
The more traditional geochemical view of primary differentiation begins with the assumption that the core of the earth is in a chemically reduced state, while the metallic elements constituting the mantle are almost entirely oxidized to their lower free energy cationic forms. Oxygen and sulfur acted as the major electron acceptors in this process, but the abundance of these elements was insufficient to oxidize much of the nickel or iron.
Iron as a reductant
Iron itself is believed to have played a crucial role in the primary differentiation of other metals and of oxidized metallic elements that iron is able to reduce. As the dense molten iron migrated in toward the core, it dissolved (formed a liquid alloy with) any other metals with which it came in contact, and it reduced (donated electrons to) those metallic cations that are less “active” metals than iron under these conditions. The resulting metal would then mix with more of the migrating liquid iron, and be carried along with it into the core.
Redox power of the elements
Accordingly, elements whose reduction potentials are more positive than iron (i.e., are lower-free energy electron sinks) are called siderophiles ; these elements have a low abundance in the crust and upper mantle. The other two important classes of solid-forming elements are lithophile and chalcophile (see below.) These generally have more negative reduction potentials than iron, and are distinguished mainly by their relative affinity for oxygen or sulfur. The chalcophiles, of which Cu, Cd, and Sb are examples, tend to form larger, more polarizable ions which can associate with the sulfide ion. The lithophiles comprise those elements such as K, Al, Mn, and Si, which have smaller ions and which combine preferentially with oxygen. This broad classification is reflected in the dominant forms in which many of these elements occur in nature.
Secondary differentiation of the elements
The differential distribution of the elements within one of the main regions of the earth has been studied in detail only in that portion that is accessible, namely the upper crust. It is clear that fractional crystallization from the cooling magma has played an important role. The relative temperatures at which minerals crystallize is determined in large part by their lattice energies, which are in turn related to ionic sizes and charges. Minerals with small, highly charged ions will have higher melting points and should crystallize first. Thus the sodium-containing feldspar albite (NaAlSi 2 O 8 ) is found nearer the surface than is its calcium analog anorthite (CaAlSi 2 O 8 ). The less abundant elements often do not form minerals of their own, but may replace the ion of a more abundant mineral in its crystal lattice. This is known as isomorphous replacement , and it naturally depends on the relative ionic radii. Some ion pairs that undergo isomorphous replacement in minerals are K + and Ba 2 + , Si 4 + and Ge 4 + .
Phase behavior
The Phase Rule can be invoked to explain in a very rough way the differentiation of the elements into distinct solid phases.
P = C + 2 – F
Taking the degrees of freedom as 2 (fixed temperature and pressure), the six major elemental components (O, Si, Al, Fe, Mg and Na) can form up to six phases. Actually, more than 99% of igneous rocks comprise seven principal mineral phases. These are: the silica minerals, feldspars, feldspathoids, olivine, pyroxenes, amphiboles and micas.
The differential deposition of minerals is also influenced by the temperature-composition phase relations as exemplified by the ordinary two-component phase diagram. If the mineral that is rich in one component and which first crystallizes out is also more dense, then the richer ore will occur near the bottom of the deposit, while a more mixed ore (approaching the eutectic) will remain near the top.
Geochemical classification of the elements
Whether an element is concentrated in the crust or elsewhere depends on its chemical behavior and on the physical properties of its stable compounds. Geochemists have found it convenient to establish the following general classifications:
- lithophiles ("rock-loving") elements are those such as Fe, Al, and Si which tend to occur as oxides (and to a lesser extent as chlorides and carbonates.) Elements in this, the largest of all the groups, are concentrated in the crust.
- chalcophiles also occur in the crust, but mainly in combination with sulfur and the other chalcogen elements of Groups 15-16.
- siderophiles refer to the elements such as Ni which have concentrated in the core along with Fe.
-
atmosphiles
consist of N, H and their volatile compounds
and the noble gases which concentrate in the atmosphere.
References
"TheTalk" site contains numerous links to discussions of geochronology and other evidence for the age of the earth, and on the pseudoscientific concept of a "young Earth".
Unravelling geological time (Columbia University)
Formation of the Earth's core - description of theories based on high pressure studies
Page last modified: 21.01.2008
© 1998 by Stephen Lower
For information about this Web site or to contact the author,
please see the
Chem1 Virtual Textbook home page
.
This work is licensed under a
Creative Commons Attribution-NonCommercial 2.5 License
.


3.4: Beowulf Sections 11-15
XI
THEN from the moorland, by misty crags,
with God’s wrath laden, Grendel came.
The monster was minded of mankind now
sundry to seize in the stately house.
Under welkin he walked, till the wine-palace there,
gold-hall of men, he gladly discerned,
flashing with fretwork. Not first time, this,
that he the home of Hrothgar sought, —
yet ne’er in his life-day, late or early,
such hardy heroes, such hall-thanes, found!
To the house the warrior walked apace,
parted from peace;
[1]
the portal opended,
though with forged bolts fast, when his fists had
struck it,
and baleful he burst in his blatant rage,
the house’s mouth. All hastily, then,
o’er fair-paved floor the fiend trod on,
ireful he strode; there streamed from his eyes
fearful flashes, like flame to see.
He spied in hall the hero-band,
kin and clansmen clustered asleep,
hardy liegemen. Then laughed his heart;
for the monster was minded, ere morn should dawn,
savage, to sever the soul of each,
life from body, since lusty banquet
waited his will! But Wyrd forbade him
to seize any more of men on earth
after that evening. Eagerly watched
Hygelac’s kinsman his cursed foe,
how he would fare in fell attack.
Not that the monster was minded to pause!
Straightway he seized a sleeping warrior
for the first, and tore him fiercely asunder,
the bone-frame bit, drank blood in streams,
swallowed him piecemeal: swiftly thus
the lifeless corse was clear devoured,
e’en feet and hands. Then farther he hied;
for the hardy hero with hand he grasped,
felt for the foe with fiendish claw,
for the hero reclining, — who clutched it boldly,
prompt to answer, propped on his arm.
Soon then saw that shepherd-of-evils
that never he met in this middle-world,
in the ways of earth, another wight
with heavier hand-gripe; at heart he feared,
sorrowed in soul, — none the sooner escaped!
Fain would he flee, his fastness seek,
the den of devils: no doings now
such as oft he had done in days of old!
Then bethought him the hardy Hygelac-thane
of his boast at evening: up he bounded,
grasped firm his foe, whose fingers cracked.
The fiend made off, but the earl close followed.
The monster meant — if he might at all —
to fling himself free, and far away
fly to the fens, — knew his fingers’ power
in the gripe of the grim one. Gruesome march
to Heorot this monster of harm had made!
Din filled the room; the Danes were bereft,
castle-dwellers and clansmen all,
earls, of their ale. Angry were both
those savage hall-guards: the house resounded.
Wonder it was the wine-hall firm
in the strain of their struggle stood, to earth
the fair house fell not; too fast it was
within and without by its iron bands
craftily clamped; though there crashed from sill
many a mead-bench — men have told me —
gay with gold, where the grim foes wrestled.
So well had weened the wisest Scyldings
that not ever at all might any man
that bone-decked, brave house break asunder,
crush by craft, — unless clasp of fire
in smoke engulfed it. — Again uprose
din redoubled. Danes of the North
with fear and frenzy were filled, each one,
who from the wall that wailing heard,
God’s foe sounding his grisly song,
cry of the conquered, clamorous pain
from captive of hell. Too closely held him
he who of men in might was strongest
in that same day of this our life.
XII
NOT in any wise would the earls’-defence
[2]
suffer that slaughterous stranger to live,
useless deeming his days and years
to men on earth. Now many an earl
of Beowulf brandished blade ancestral,
fain the life of their lord to shield,
their praised prince, if power were theirs;
never they knew, — as they neared the foe,
hardy-hearted heroes of war,
aiming their swords on every side
the accursed to kill, — no keenest blade,
no farest of falchions fashioned on earth,
could harm or hurt that hideous fiend!
He was safe, by his spells, from sword of battle,
from edge of iron. Yet his end and parting
on that same day of this our life
woful should be, and his wandering soul
far off flit to the fiends’ domain.
Soon he found, who in former days,
harmful in heart and hated of God,
on many a man such murder wrought,
that the frame of his body failed him now.
For him the keen-souled kinsman of Hygelac
held in hand; hateful alive
was each to other. The outlaw dire
took mortal hurt; a mighty wound
showed on his shoulder, and sinews cracked,
and the bone-frame burst. To Beowulf now
the glory was given, and Grendel thence
death-sick his den in the dark moor sought,
noisome abode: he knew too well
that here was the last of life, an end
of his days on earth. — To all the Danes
by that bloody battle the boon had come.
From ravage had rescued the roving stranger
Hrothgar’s hall; the hardy and wise one
had purged it anew. His night-work pleased him,
his deed and its honor. To Eastern Danes
had the valiant Geat his vaunt made good,
all their sorrow and ills assuaged,
their bale of battle borne so long,
and all the dole they erst endured
pain a-plenty. — ’Twas proof of this,
when the hardy-in-fight a hand laid down,
arm and shoulder, — all, indeed,
of Grendel’s gripe, — ’neath the gabled roof.
XIII
MANY at morning, as men have told me,
warriors gathered the gift-hall round,
folk-leaders faring from far and near,
o’er wide-stretched ways, the wonder to view,
trace of the traitor. Not troublous seemed
the enemy’s end to any man
who saw by the gait of the graceless foe
how the weary-hearted, away from thence,
baffled in battle and banned, his steps
death-marked dragged to the devils’ mere.
Bloody the billows were boiling there,
turbid the tide of tumbling waves
horribly seething, with sword-blood hot,
by that doomed one dyed, who in den of the moor
laid forlorn his life adown,
his heathen soul, and hell received it.
Home then rode the hoary clansmen
from that merry journey, and many a youth,
on horses white, the hardy warriors,
back from the mere. Then Beowulf’s glory
eager they echoed, and all averred
that from sea to sea, or south or north,
there was no other in earth’s domain,
under vault of heaven, more valiant found,
of warriors none more worthy to rule!
(On their lord beloved they laid no slight,
gracious Hrothgar: a good king he!)
From time to time, the tried-in-battle
their gray steeds set to gallop amain,
and ran a race when the road seemed fair.
From time to time, a thane of the king,
who had made many vaunts, and was mindful of verses,
stored with sagas and songs of old,
bound word to word in well-knit rime,
welded his lay; this warrior soon
of Beowulf’s quest right cleverly sang,
and artfully added an excellent tale,
in well-ranged words, of the warlike deeds
he had heard in saga of Sigemund.
Strange the story: he said it all, —
the Waelsing’s wanderings wide, his struggles,
which never were told to tribes of men,
the feuds and the frauds, save to Fitela only,
when of these doings he deigned to speak,
uncle to nephew; as ever the twain
stood side by side in stress of war,
and multitude of the monster kind
they had felled with their swords. Of Sigemund grew,
when he passed from life, no little praise;
for the doughty-in-combat a dragon killed
that herded the hoard:
[3]
under hoary rock
the atheling dared the deed alone
fearful quest, nor was Fitela there.
Yet so it befell, his falchion pierced
that wondrous worm, — on the wall it struck,
best blade; the dragon died in its blood.
Thus had the dread-one by daring achieved
over the ring-hoard to rule at will,
himself to pleasure; a sea-boat he loaded,
and bore on its bosom the beaming gold,
son of Waels; the worm was consumed.
He had of all heroes the highest renown
among races of men, this refuge-of-warriors,
for deeds of daring that decked his name
since the hand and heart of Heremod
grew slack in battle. He, swiftly banished
to mingle with monsters at mercy of foes,
to death was betrayed; for torrents of sorrow
had lamed him too long; a load of care
to earls and athelings all he proved.
Oft indeed, in earlier days,
for the warrior’s wayfaring wise men mourned,
who had hoped of him help from harm and bale,
and had thought their sovran’s son would thrive,
follow his father, his folk protect,
the hoard and the stronghold, heroes’ land,
home of Scyldings. — But here, thanes said,
the kinsman of Hygelac kinder seemed
to all: the other
[4]
was urged to crime!
And afresh to the race,
[5]
the fallow roads
by swift steeds measured! The morning sun
was climbing higher. Clansmen hastened
to the high-built hall, those hardy-minded,
the wonder to witness. Warden of treasure,
crowned with glory, the king himself,
with stately band from the bride-bower strode;
and with him the queen and her crowd of maidens
measured the path to the mead-house fair.
XIV
HROTHGAR spake, — to the hall he went,
stood by the steps, the steep roof saw,
garnished with gold, and Grendel’s hand: —
“For the sight I see to the Sovran Ruler
be speedy thanks! A throng of sorrows
I have borne from Grendel; but God still works
wonder on wonder, the Warden-of-Glory.
It was but now that I never more
for woes that weighed on me waited help
long as I lived, when, laved in blood,
stood sword-gore-stained this stateliest house, —
widespread woe for wise men all,
who had no hope to hinder ever
foes infernal and fiendish sprites
from havoc in hall. This hero now,
by the Wielder’s might, a work has done
that not all of us erst could ever do
by wile and wisdom. Lo, well can she say
whoso of women this warrior bore
among sons of men, if still she liveth,
that the God of the ages was good to her
in the birth of her bairn. Now, Beowulf, thee,
of heroes best, I shall heartily love
as mine own, my son; preserve thou ever
this kinship new: thou shalt never lack
wealth of the world that I wield as mine!
Full oft for less have I largess showered,
my precious hoard, on a punier man,
less stout in struggle. Thyself hast now
fulfilled such deeds, that thy fame shall endure
through all the ages. As ever he did,
well may the Wielder reward thee still!”
Beowulf spake, bairn of Ecgtheow: —
“This work of war most willingly
we have fought, this fight, and fearlessly dared
force of the foe. Fain, too, were I
hadst thou but seen himself, what time
the fiend in his trappings tottered to fall!
Swiftly, I thought, in strongest gripe
on his bed of death to bind him down,
that he in the hent of this hand of mine
should breathe his last: but he broke away.
Him I might not — the Maker willed not —
hinder from flight, and firm enough hold
the life-destroyer: too sturdy was he,
the ruthless, in running! For rescue, however,
he left behind him his hand in pledge,
arm and shoulder; nor aught of help
could the cursed one thus procure at all.
None the longer liveth he, loathsome fiend,
sunk in his sins, but sorrow holds him
tightly grasped in gripe of anguish,
in baleful bonds, where bide he must,
evil outlaw, such awful doom
as the Mighty Maker shall mete him out.”
More silent seemed the son of Ecglaf
[6]
in boastful speech of his battle-deeds,
since athelings all, through the earl’s great prowess,
beheld that hand, on the high roof gazing,
foeman’s fingers, — the forepart of each
of the sturdy nails to steel was likest, —
heathen’s “hand-spear,” hostile warrior’s
claw uncanny. ’Twas clear, they said,
that him no blade of the brave could touch,
how keen soever, or cut away
that battle-hand bloody from baneful foe.
XV
THERE was hurry and hest in Heorot now
for hands to bedeck it, and dense was the throng
of men and women the wine-hall to cleanse,
the guest-room to garnish. Gold-gay shone the hangings
that were wove on the wall, and wonders many
to delight each mortal that looks upon them.
Though braced within by iron bands,
that building bright was broken sorely;
[7]
rent were its hinges; the roof alone
held safe and sound, when, seared with crime,
the fiendish foe his flight essayed,
of life despairing. — No light thing that,
the flight for safety, — essay it who will!
Forced of fate, he shall find his way
to the refuge ready for race of man,
for soul-possessors, and sons of earth;
and there his body on bed of death
shall rest after revel.
Arrived was the hour
when to hall proceeded Healfdene’s son:
the king himself would sit to banquet.
Ne’er heard I of host in haughtier throng
more graciously gathered round giver-of-rings!
Bowed then to bench those bearers-of-glory,
fain of the feasting. Featly received
many a mead-cup the mighty-in-spirit,
kinsmen who sat in the sumptuous hall,
Hrothgar and Hrothulf. Heorot now
was filled with friends; the folk of Scyldings
ne’er yet had tried the traitor’s deed.
To Beowulf gave the bairn of Healfdene
a gold-wove banner, guerdon of triumph,
broidered battle-flag, breastplate and helmet;
and a splendid sword was seen of many
borne to the brave one. Beowulf took
cup in hall:
[8]
for such costly gifts
he suffered no shame in that soldier throng.
For I heard of few heroes, in heartier mood,
with four such gifts, so fashioned with gold,
on the ale-bench honoring others thus!
O’er the roof of the helmet high, a ridge,
wound with wires, kept ward o’er the head,
lest the relict-of-files
[9]
should fierce invade,
sharp in the strife, when that shielded hero
should go to grapple against his foes.
Then the earls’-defence
[10]
on the floor
[11]
bade lead
coursers eight, with carven head-gear,
adown the hall: one horse was decked
with a saddle all shining and set in jewels;
’twas the battle-seat of the best of kings,
when to play of swords the son of Healfdene
was fain to fare. Ne’er failed his valor
in the crush of combat when corpses fell.
To Beowulf over them both then gave
the refuge-of-Ingwines right and power,
o’er war-steeds and weapons: wished him joy of them.
Manfully thus the mighty prince,
hoard-guard for heroes, that hard fight repaid
with steeds and treasures contemned by none
who is willing to say the sooth aright.
-
That is, he was a “lost soul,” doomed to hell.
↵
-
Kenning for Beowulf.
↵
-
“Guarded the treasure.”
↵
-
Sc. Heremod.
↵
-
The singer has sung his lays, and the epic resumes its story. The time-relations are not altogether good in this long passage which describes the rejoicings of “the day after”; but the present shift from the riders on the road to the folk at the hall is not very violent, and is of a piece with the general style.
↵
-
Unferth, Beowulf’s sometime opponent in the flyting.
↵
-
There is no horrible inconsistency here such as the critics strive and cry about. In spite of the ruin that Grendel and Beowulf had made within the hall, the framework and roof held firm, and swift repairs made the interior habitable. Tapestries were hung on the walls, and willing hands prepared the banquet.
↵
-
From its formal use in other places, this phrase, to take cup in hall, or “on the floor,” would seem to mean that Beowulf stood up to receive his gifts, drink to the donor, and say thanks.
↵
-
Kenning for sword.
↵
-
Hrothgar. He is also the “refuge of the friends of Ing,” below. Ing belongs to myth.
↵
-
Horses are frequently led or ridden into the hall where folk sit at banquet: so in Chaucer’s Squire’s tale, in the ballad of King Estmere, and in the romances.
↵
- Beowulf. Authored by : Trans. Gummere. Provided by : Project Gutenberg. Located at : http://www.gutenberg.org/files/981/981-h/981-h.htm#linkfootnote2b . License : Public Domain: No Known Copyright


22.11: Burmese Art
Burmese Art
Burmese art has been influenced primarily by Theravada Buddhism, with additional influences from India, Thailand, and China.
Discuss the art and architecture of the Burmese Pyu culture, the Pagan Empire, the Innwa style of the Ava Kingdom, the Mandalay style of the Konbaung dynasty, and the Shan style
Key Points
- Buddhist art and architecture first began to flourish in Burma under the Pyu city-states (2nd century BCE—mid-11th century CE). The Pyu imported Buddhism from India and built solid stupas that served as inspiration for later Burmese temples.
- Under the Pagan Empire (849–1297 CE), two types of Buddhist temples developed: the stupa- style solid temple and the gu-style hollow temple.
- Innovations in temple architecture of the Pagan period include pointed arches , vaulted chambers, and the pentagonal temple plan.
- There are three popular styles of Buddha images in Burma: the Ava style, the Mandalay style, and the Shan style.
Key Terms
- mudra : Any of several formal symbolic hand postures used in classical dance of India and in Hindu and Buddhist iconography.
- jataka tales : A voluminous body of literature native to India concerning the previous births (jāti) of the Buddha, in both human and animal form.
- pagoda : An Asian religious building, especially a multistory Buddhist tower, erected as a shrine or temple.
- bodhisattva : In Buddhism, an enlightened being—existing beyond the Western scientific thought of physical reality—resembling the Western notion of angels but with a wealth of its own nuances and expectations.
Overview
The Republic of the Union of Myanmar, also known as Burma, is a sovereign state in Southeast Asia bordered by Bangladesh, India, China, Laos, and Thailand. Both the names “Burma” and “Myanmar” are derived from the name of the Bamar ethnic group. The name Burma has been in use in English since the 18th century; however in 1989, the country’s military government officially changed the country’s name from Burma to Myanmar. The renaming remains a contested issue, and many political and ethnic opposition groups and countries continue to use “Burma” because they do not recognize the legitimacy of the ruling military government. Burma continues to be used in English by the governments of many countries, such as Australia, Canada, and the United Kingdom. Official United States policy retains Burma as the country’s name, although both are sometimes used interchangeably.
Burmese art has been influenced primarily by Theravada Buddhism and the culture of the Mon people, with additional influences from India, Thailand, and China. Themes are commonly drawn from Buddhist and Hindu cosmology and myths. Burma is particularly renowned for its richness of Buddhist architecture, and is justifiably called “The Land of Pagodas ,” as Buddhist monasteries and gilded pagodas dot the landscape.
Background: Pyu Art
Buddhist art and architecture began to flourish in Burma under the Pyu city-states (2nd century BCE—mid-11th century CE). Pyu culture was heavily influenced by trade with India, and Buddhism was imported along with various Indian cultural and architectural concepts. While Theravada Buddhism was the primary religion of the Pyu kingdoms, archaeological finds indicate that Tantric Buddhism, Mahayana Buddhism, and Hinduism were also present. Figures from the Mahayana and Tantric pantheons were represented in Pyu art, including Avalokiteshwara (a bodhisattva who embodies compassion) and Tara (a female Tantric meditation deity ); Hindu deities, including the Hindu holy trinity of Brahma, Vishnu, and Shiva; and the goddess Lakshmi.
Starting from the 4th century CE, the Pyu began to build stupas and other religious buildings. The styles, ground plans, and construction techniques used are very similar to those of Amaravati and Nagarjunakonda in the Andhra region in present-day south-eastern India. By the 7th century, the tall cylindrical solid stupas, characteristic of the Pyu reign, had emerged in the city of Sri Ksetra. These solid stupas served as models for later pagodas.
The Pagan Dynasty (849–1297 CE)
The Pyu city-states declined in the early 9th century CE when a new group of ethnic Burmans from the north entered and raided their lands, establishing the Pagan dynasty and eventually overtaking the Pyu realm in the late 10th century. The Pagan state lasted from 849 until 1297 CE, laying the foundations for Burmese language and culture and the spread of Theravada Buddhism through all of Burma. The Pagan Empire is famous for its religious architecture: over 2,000 temples from the period survive to the present day. In addition to setting the model for subsequent Buddhist stupas and temples, the Pagan period also witnessed the production of beautiful jeweled statues of the Buddha.
Architecture during this time was built in two broad styles: the stupa-style solid temple inherited from the Pyu city-states, and the gu-style hollow temple.
Stupa-style Temples
During the Pagan period, the stupa developed into a longer, more cylindrical form than its Pyu predecessors at Sri Ksetra. By the 11th century, it had assumed a bell-like form, surmounted by a series of increasingly smaller rings placed one on top of the other and rising to a point. A lotus or banana-bud topped these rings. Three or four rectangular terraces served as the base for a pagoda, often decorated by a gallery of terracotta tiles depicting the Jataka tales . One of the earliest examples of Pagan temple architecture is the Shwezigon Pagoda in Nyaung-U. It is considered a prototype of Burmese stupas and consisted of a bell-shaped goldleaf-gilded stupa surrounded by smaller temples and shrines.
Gu-style Temples
The hollow gu-style temple comes in two basic styles: the one-face design with one main entrance, and the four-face design with four main entrances. The primary features of these temples are pointed arches and vaulted chambers, both of which became larger and grander during the Pagan period. Vaulting was an innovation that developed independently in Pagan itself around the 11th century, and the masonry of these vaulted buildings is astonishingly perfect and durable. Another architectural innovation arising in Pagan is the Buddhist temple with the pentagonal floor in order to honor the Maitreya Buddha, a future Buddha foretold of in Buddhist mythology.
Later Kingdoms
The Pagan Empire was succeeded by the Ava Kingdom, which ruled Upper Burma from 1364 to 1555 CE. Ava rule was marked by the popularization of the Innwa style of Buddha, in which the Buddha is depicted with large protruding ears, exaggerated eyebrows that curve upward, half-closed eyes, thin lips, and a hair bun that is pointed at the top. Ava statues of the Buddha are also usually portrayed in the bhumisparsha mudra , in which the figure is seated with the right hand reaching for the ground, palm facing inward.
The Innwa style continued to be popular in Burma until the end of the 18th century, when a markedly new style of Buddha image, the Mandalay style, emerged under the Konbaung dynasty (1752–1885). The face of the Buddha in the Mandalay style was natural and fleshy with realistically slanted eyebrows, slightly slanted eyes, thicker lips, and a round hair bun. These statues were portrayed wearing flowing, draped robes. Three postures were commonly depicted: reclining, standing, or sitting. Another common style of the Buddha image is the Shan style, inherited from the Shan people who inhabit the highlands of Burma. Buddhas in the Shan style have angular features, large and prominently-pointed noses, and small, thin mouths.
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
- Buddha 00004. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Buddha_00004.JPG. License : Public Domain: No Known Copyright
- Shwezigon Pagoda. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Shwezigon_Pagoda. License : Public Domain: No Known Copyright
- Bagan. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Bagan. License : Public Domain: No Known Copyright
- Buddhist art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Buddhist_art. License : CC BY-SA: Attribution-ShareAlike
- Shwezigon Pagoda. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Shwezigon_Pagoda. License : CC BY-SA: Attribution-ShareAlike
- Pagodas in Burma. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Pagodas_in_Burma. License : CC BY-SA: Attribution-ShareAlike
- Culture of Burma. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Culture_of_Burma. License : CC BY-SA: Attribution-ShareAlike
- Pyu city-states. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Pyu_city-states. License : CC BY-SA: Attribution-ShareAlike
- Pagan Kingdom. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Pagan_Kingdom. License : CC BY-SA: Attribution-ShareAlike
- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//art-history/definition/jataka-tales. License : CC BY-SA: Attribution-ShareAlike
- pagoda. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/pagoda. License : CC BY-SA: Attribution-ShareAlike
- mudra. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/mudra. License : CC BY-SA: Attribution-ShareAlike
- bodhisattva. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/bodhisattva. License : CC BY-SA: Attribution-ShareAlike
- Myanmar. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Myanmar. License : CC BY-SA: Attribution-ShareAlike


7.2: What Is Cognition?
Learning Objectives
- Describe cognition
- Distinguish concepts and prototypes
- Explain the difference between natural and artificial concepts
Imagine all of your thoughts as if they were physical entities, swirling rapidly inside your mind. How is it possible that the brain is able to move from one thought to the next in an organized, orderly fashion? The brain is endlessly perceiving, processing, planning, organizing, and remembering—it is always active. Yet, you don’t notice most of your brain’s activity as you move throughout your daily routine. This is only one facet of the complex processes involved in cognition. Simply put, cognition is thinking, and it encompasses the processes associated with perception, knowledge, problem solving, judgment, language, and memory. Scientists who study cognition are searching for ways to understand how we integrate, organize, and utilize our conscious cognitive experiences without being aware of all of the unconscious work that our brains are doing (for example, Kahneman, 2011).
Cognition
Upon waking each morning, you begin thinking—contemplating the tasks that you must complete that day. In what order should you run your errands? Should you go to the bank, the cleaners, or the grocery store first? Can you get these things done before you head to class or will they need to wait until school is done? These thoughts are one example of cognition at work. Exceptionally complex, cognition is an essential feature of human consciousness, yet not all aspects of cognition are consciously experienced.
Cognitive psychology is the field of psychology dedicated to examining how people think. It attempts to explain how and why we think the way we do by studying the interactions among human thinking, emotion, creativity, language, and problem solving, in addition to other cognitive processes. Cognitive psychologists strive to determine and measure different types of intelligence, why some people are better at problem solving than others, and how emotional intelligence affects success in the workplace, among countless other topics. They also sometimes focus on how we organize thoughts and information gathered from our environments into meaningful categories of thought, which will be discussed later.
Concepts and Prototypes
The human nervous system is capable of handling endless streams of information. The senses serve as the interface between the mind and the external environment, receiving stimuli and translating it into nervous impulses that are transmitted to the brain. The brain then processes this information and uses the relevant pieces to create thoughts, which can then be expressed through language or stored in memory for future use. To make this process more complex, the brain does not gather information from external environments only. When thoughts are formed, the brain also pulls information from emotions and memories (See figure below). Emotion and memory are powerful influences on both our thoughts and behaviors.
In order to organize this staggering amount of information, the brain has developed a file cabinet of sorts in the mind. The different files stored in the file cabinet are called concepts. Concepts are categories or groupings of linguistic information, images, ideas, or memories, such as life experiences. Concepts are, in many ways, big ideas that are generated by observing details, and categorizing and combining these details into cognitive structures. You use concepts to see the relationships among the different elements of your experiences and to keep the information in your mind organized and accessible.
Concepts are informed by our semantic memory (you learned about this concept when you studied memory) and are present in every aspect of our lives; however, one of the easiest places to notice concepts is inside a classroom, where they are discussed explicitly. When you study United States history, for example, you learn about more than just individual events that have happened in America’s past. You absorb a large quantity of information by listening to and participating in discussions, examining maps, and reading first-hand accounts of people’s lives.  In the process, your brain gathers details that inform and refine your understanding of related concepts like democracy, power, and freedom.
Concepts can be complex and abstract, like justice, or more concrete, like types of birds. In psychology, for example, Piaget’s stages of development are abstract concepts. Some concepts, like tolerance, are agreed upon by many people, because they have been used in various ways over many years. Other concepts, like the characteristics of your ideal friend or your family’s birthday traditions, are personal and individualized. In this way, concepts touch every aspect of our lives, from our many daily routines to the guiding principles behind the way governments function.
Another technique used by your brain to organize information is the identification of prototypes for the concepts you have developed. A prototype is the best example or representation of a concept. For example, for the category of civil disobedience, your prototype could be Rosa Parks. Her peaceful resistance to segregation on a city bus in Montgomery, Alabama, is a recognizable example of civil disobedience. Or your prototype could be Mohandas Gandhi, sometimes called Mahatma Gandhi (“Mahatma” is an honorific title).
Mohandas Gandhi served as a nonviolent force for independence for India while simultaneously demanding that Buddhist, Hindu, Muslim, and Christian leaders—both Indian and British—collaborate peacefully. Although he was not always successful in preventing violence around him, his life provides a steadfast example of the civil disobedience prototype (Constitutional Rights Foundation, 2013). Just as concepts can be abstract or concrete, we can make a distinction between concepts that are functions of our direct experience with the world and those that are more artificial in nature.
Natural and Artificial Concepts
In psychology, concepts can be divided into two categories, natural and artificial. Natural concepts are created “naturally” through your experiences and can be developed from either direct or indirect experiences. For example, if you live in Essex Junction, Vermont, you have probably had a lot of direct experience with snow. You’ve watched it fall from the sky, you’ve seen lightly falling snow that barely covers the windshield of your car, and you’ve shoveled out \(18\) inches of fluffy white snow as you’ve thought, “This is perfect for skiing.” You’ve thrown snowballs at your best friend and gone sledding down the steepest hill in town. In short, you know snow. You know what it looks like, smells like, tastes like, and feels like. If, however, you’ve lived your whole life on the island of Saint Vincent in the Caribbean, you may never have actually seen snow, much less tasted, smelled, or touched it. You know snow from the indirect experience of seeing pictures of falling snow—or from watching films that feature snow as part of the setting. Either way, snow is a natural concept because you can construct an understanding of it through direct observations or experiences of snow.
An artificial concept , on the other hand, is a concept that is defined by a specific set of characteristics. Various properties of geometric shapes, like squares and triangles, serve as useful examples of artificial concepts. A triangle always has three angles and three sides. A square always has four equal sides and four right angles. Mathematical formulas, like the equation for area (length × width) are artificial concepts defined by specific sets of characteristics that are always the same. Artificial concepts can enhance the understanding of a topic by building on one another. For example, before learning the concept of “area of a square” (and the formula to find it), you must understand what a square is. Once the concept of “area of a square” is understood, an understanding of area for other geometric shapes can be built upon the original understanding of area. The use of artificial concepts to define an idea is crucial to communicating with others and engaging in complex thought. According to Goldstone and Kersten (2003), concepts act as building blocks and can be connected in countless combinations to create complex thoughts.
 There are many different types of schemata, and they all have one thing in common: schemata are a method of organizing information that allows the brain to work more efficiently. When a schema is activated, the brain makes immediate assumptions about the person or object being observed.
There are several types of schemata. A role schema makes assumptions about how individuals in certain roles will behave (Callero, 1994). For example, imagine you meet someone who introduces himself as a firefighter. When this happens, your brain automatically activates the “firefighter schema” and begins making assumptions that this person is brave, selfless, and community-oriented. Despite not knowing this person, already you have unknowingly made judgments about him. Schemata also help you fill in gaps in the information you receive from the world around you. While schemata allow for more efficient information processing, there can be problems with schemata, regardless of whether they are accurate: Perhaps this particular firefighter is not brave, he just works as a firefighter to pay the bills while studying to become a children’s librarian.
An event schema , also known as a cognitive script , is a set of behaviors that can feel like a routine. Think about what you do when you walk into an elevator (See figure \(\PageIndex{4}\) below). First, the doors open and you wait to let exiting passengers leave the elevator car. Then, you step into the elevator and turn around to face the doors, looking for the correct button to push. You never face the back of the elevator, do you? And when you’re riding in a crowded elevator and you can’t face the front, it feels uncomfortable, doesn’t it? Interestingly, event schemata can vary widely among different cultures and countries. For example, while it is quite common for people to greet one another with a handshake in the United States, in Tibet, you greet someone by sticking your tongue out at them, and in Belize, you bump fists (Cairns Regional Council, n.d.)
Because event schemata are automatic, they can be difficult to change. Imagine that you are driving home from work or school. This event schema involves getting in the car, shutting the door, and buckling your seatbelt before putting the key in the ignition. You might perform this script two or three times each day.  Typically, the event schema that occurs when you hear your phone ringing involves locating the phone and answering it or responding to your latest text message. So without thinking, you reach for your phone, which could be in your pocket, in your bag, or on the passenger seat of the car. This powerful event schema is informed by your pattern of behavior and the pleasurable stimulation that a phone call or text message gives your brain. Because it is a schema, it is extremely challenging for us to stop reaching for the phone, even though we know that we endanger our own lives and the lives of others while we do it (Neyfakh, 2013).
Remember the elevator? It feels almost impossible to walk in and not face the door. Our powerful event schema dictates our behavior in the elevator, and it is no different with our phones. Current research suggests that it is the habit, or event schema, of checking our phones in many different situations that makes refraining from checking them while driving especially difficult (Bayer & Campbell, 2012). Because texting and driving has become a dangerous epidemic in recent years, psychologists are looking at ways to help people interrupt the “phone schema” while driving. Event schemata like these are the reason why many habits are difficult to break once they have been acquired. As we continue to examine thinking, keep in mind how powerful the forces of concepts and schemata are to our understanding of the world.
Summary
In this section, you were introduced to cognitive psychology, which is the study of cognition, or the brain’s ability to think, perceive, plan, analyze, and remember. Concepts and their corresponding prototypes help us quickly organize our thinking by creating categories into which we can sort new information. We also develop schemata, which are clusters of related concepts. Some schemata involve routines of thought and behavior, and these help us function properly in various situations without having to “think twice” about them. Schemata show up in social situations and routines of daily behavior.
Glossary
- artificial concept
- concept that is defined by a very specific set of characteristics
- cognition
- thinking, including perception, learning, problem solving, judgment, and memory
- cognitive psychology
- field of psychology dedicated to studying every aspect of how people think
- concept
- category or grouping of linguistic information, objects, ideas, or life experiences
- cognitive script
- set of behaviors that are performed the same way each time; also referred to as an event schema
- event schema
- set of behaviors that are performed the same way each time; also referred to as a cognitive script
- natural concept
- mental groupings that are created “naturally” through your experiences
- prototype
- best representation of a concept
- role schema
- set of expectations that define the behaviors of a person occupying a particular role
- schema
- (plural = schemata) mental construct consisting of a cluster or collection of related concepts
Contributors and Attributions
-
Rose M. Spielman with many significant contributors. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>.Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


26: Hydropower Dams Can Harm Coastal Areas Far Downstream (Ezcurra & Aburto)
By Paula Ezcurra & Octavio Aburto
Thousands of hydroelectric dams are under construction around the world, mainly in developing countries . These enormous structures are one of the world’s largest sources of renewable energy, but they also cause environmental problems.
Hydropower dams degrade water quality along rivers . Water that flows downstream from the dams is depleted of oxygen, which harms many aquatic animals. The reservoirs above dams are susceptible to harmful algal blooms, and can leach toxic metals such as mercury from submerged soil.
We wanted to know whether dams also impact river systems farther away, at the coastlines where rivers flow into the sea. So we performed a natural experiment comparing four rivers along Mexico’s Pacific coast – two that are dammed and two that remain free-flowing. We found that damming rivers has measurable negative ecologic and economic effects on coastal regions more than 60 miles (100 kilometers) downstream.
Feeding or starving coastlines
We studied four river outflows along the Pacific Coast of Mexico in the states of Sinaloa and Nayarit. Two of these were from the San Pedro and Acaponeta rivers, which are relatively unrestricted, with over 75% of their flow unobstructed.
The other two outflows came from the nearby Santiago and Fuerte rivers, which have over 95% of their flow retained in reservoirs. In addition to restricting water flow, these reservoirs trap sediments – over 1 million tons per year along the two rivers combined.
In unobstructed rivers, sediment flows downstream and is eventually deposited along the coast, helping to stabilize the shoreline and sometimes even to build it up. We found that this was happening along the free-flowing Acaponeta and San Pedro rivers.
However, because the sediment from the dammed Santiago and Fuerte rivers is no longer carried downstream, wave action takes over at the coast. At the mouths of these two rivers, we found that waves were eroding up to 33 hectares of combined land – equivalent to about 62 football fields – each year, with widespread ecologic and economic effects on the surrounding regions.
The ecology of healthy coasts
Our field research clearly showed that coastal instability resulting from sediment loss at the mouths of the dammed rivers was harming ecosystems along the shore. For example, we found that coastal regions downstream of free-flowing rivers had significantly more plant diversity. Many of these plants were found only in coastal areas, and therefore had high conservation value.
Coastal erosion due to lack of sediment input from the rivers also reduces critical nursery habitat, such as mangrove forest, where many commercially important fish species spend their juvenile stage . We found that fishing activity at the mouth of the free-flowing San Pedro River was much higher than around the mouth of the dammed Fuerte River. This loss of fishing potential comes at a cost of around US$1.3 million every year.
Reduced sediment flow also deprives coastal estuaries of nutrients. Lucrative shrimp and oyster fisheries in the region we studied rely heavily on nutrient inputs from rivers. In the San Pedro River region, these fisheries generate around $5.8 million yearly; near the dammed rivers, they have been all but abandoned.
Coastal mangrove wetlands also protect shorelines from hurricanes and tropical storms, and serve as recreational areas and conservation habitat for wildlife. Knowing this, we calculated that the loss of these ecosystem services around the dammed rivers totals $3.9 million annually.
Still another valuable function that mangrove wetlands perform is storing “blue carbon” in plant tissue and soils, reducing the effects of climate change. But when coastlines recede and mangroves are destroyed, this carbon is released. We calculated that mangrove loss in our study region represented a loss of around $130,000 in annual carbon trading potential for this region.
Adding up all of the ecological services that coastal ecosystems provide, we estimate that the economic consequences of shoreline loss around the Santiago and Fuerte rivers related to hydroelectric damming totaled well over $10 million yearly.
Letting more sediment flow
Because sediments are so essential to areas around river mouths, reducing sediment trapping behind dams could mitigate some harmful impacts on coastal areas. There are several ways to do this – notably, sediment bypassing, or diverting a portion of the sediments flowing from upriver around dams and allowing it to rejoin the river downstream.
This strategy can be included in new construction or incorporated into existing dams. In addition to reducing dams’ environmental impacts, it also increases dams’ service lives by reducing the rate at which their reservoirs fill up with silt.
To date, environmental impact assessments of large inland dams have often failed to properly analyze the impacts that these dams will have downriver on coastlines, estuaries, deltas and lagoons. Our study shows how important it is to fully account for dams’ environmental and economic impacts along coasts and basins.
Critics are challenging a proposed dam on the San Pedro River at Las Cruces.
Mexico may be at a juncture in its approach to hydropower. The Mexican government recently contracted with Hydro-Quebec, the world’s largest hydroelectric power producer, to revamp existing dams across the country . And a recent study by a Mexican nongovernment organization, SuMar-Voces por la Naturaleza, reported that a long-disputed proposal to build a new hydroelectric dam at Las Cruces is neither financially feasible nor needed to meet energy demand for the region, prompting national groups to call for the final cancellation of the project.
We believe that Mexico and all nations working to develop efficient, low-impact energy sources should take a holistic approach to future dam-related projects, so they can weigh their potentially harmful consequences. The coastal effects that we documented should be part of those reviews.
____________________
Paula Ezcurra is a Digital Communications Specialist, Gulf of California Marine Program, University of California San Diego. Octavio Aburto is an Assistant Professor of Marine Biology, Scripps Institute of Oceanography, University of California San Diego.
Hydropower Dams Can Harm Coastal Areas Far Downstream by Paula Ezcurra & Octavio Aburto is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License .


27.5: Microbial Ecology
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- Last updated
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Learning Objectives
- Give examples of the beneficial roles played by prokaryotes in different ecosystems
Role of Prokaryotes in Ecosystems
Prokaryotes are ubiquitous: There is no niche or ecosystem in which they are not present. Prokaryotes play many roles in the environments they occupy, but the roles they play in the carbon and nitrogen cycles are vital to life on earth.
Prokaryotes and the Carbon Cycle
Carbon is one of the most important macronutrients. Prokaryotes play an important role in the carbon cycle. Carbon is cycled through earth’s major reservoirs: land, the atmosphere, aquatic environments, sediments and rocks, and biomass. The movement of carbon is via carbon dioxide, which is removed from the atmosphere by land plants and marine prokaryotes and is returned to the atmosphere via the respiration of chemoorganotrophic organisms, including prokaryotes, fungi, and animals. Although the largest carbon reservoir in terrestrial ecosystems is in rocks and sediments, that carbon is not readily available.
A large amount of available carbon is found in land plants, which are producers that use carbon dioxide from the air to synthesize carbon compounds. Related to this, one very significant source of carbon compounds is humus, which is a mixture of organic materials from dead plants and prokaryotes that have resisted decomposition. Consumers such as animals use organic compounds generated by producers, releasing carbon dioxide to the atmosphere. Then, bacteria and fungi, collectively called decomposers, carry out the breakdown (decomposition) of plants and animals and their organic compounds. The most important contributor of carbon dioxide to the atmosphere is microbial decomposition of dead material (dead animals, plants, and humus).
In aqueous environments and their anoxic sediments, there is another carbon cycle taking place. In this case, the cycle is based on one-carbon compounds. In anoxic sediments, prokaryotes, mostly archaea, produce methane (CH 4 ). This methane moves into the zone above the sediment, which is richer in oxygen and supports bacteria called methane oxidizers that oxidize methane to carbon dioxide, which then returns to the atmosphere.
Prokaryotes and the Nitrogen Cycle
Nitrogen is a very important element for life because it is part of proteins and nucleic acids. As a macronutrient in nature, it is recycled from organic compounds to ammonia, ammonium ions, nitrate, nitrite, and nitrogen gas by myriad processes, many of which are carried out solely by prokaryotes; they are key to the nitrogen cycle. The largest pool of nitrogen available in the terrestrial ecosystem is gaseous nitrogen from the air, but this nitrogen is not usable by plants, which are primary producers. Gaseous nitrogen is transformed, or “fixed,” into more-readily available forms such as ammonia through the process of nitrogen fixation by natural means, especially by microorganisms (prokayotes) in the soil. Ammonia can then be used by plants or converted to other forms.
Another source of ammonia is ammonification, the process by which ammonia is released during the decomposition of nitrogen-containing organic compounds. Ammonia released to the atmosphere, however, represents only 15 percent of the total nitrogen released; the rest is as N 2 and N 2 O. Ammonia is catabolized anaerobically by some prokaryotes, yielding N 2 as the final product. Nitrification is the conversion of ammonium to nitrite and nitrate. Nitrification in soils is carried out by bacteria belonging to the genera Nitrosomas , Nitrobacter , and Nitrospira . The bacteria perform the reverse process, the reduction of nitrate from the soils to gaseous compounds such as N 2 O, NO, and N 2 , a process called denitrification.
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Key Points
- Carbon and nitrogen are both macronutrients that are necessary for life on earth; prokaryotes play vital roles in their cycles.
- The carbon cycle is maintained by prokaryotes that remove carbon dioxide and return it to the atmosphere.
- Prokaryotes play a major role in the nitrogen cycle by fixing atomspheric nitrogen into ammonia that plants can use and by converting ammonia into other forms of nitrogen sources.
Key Terms
- carbon cycle : the physical cycle of carbon through the earth’s biosphere, geosphere, hydrosphere, and atmosphere that includes such processes as photosynthesis, decomposition, respiration and carbonification
- nitrogen cycle : the natural circulation of nitrogen, in which atmospheric nitrogen is converted to nitrogen oxides and deposited in the soil, where it is used by organisms or decomposed back to elemental nitrogen
- nitrogen fixation : the conversion of atmospheric nitrogen into ammonia and organic derivatives, by natural means, especially by microorganisms in the soil, into a form that can be assimilated by plants