11.5: Contexts - Higher Education and Work Achievement in Early Adulthood
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Learning Objectives
By the end of this section, you will be able to:
- Describe the challenges and opportunities facing young adults in preparing for work and career
- Describe formal and informal educational opportunities in early adulthood
- Explain the considerations surrounding workforce entry for the developing young adult
Moira has been attending a community college part-time for the past few years while working as a server in her uncle’s restaurant. She enjoys her work (especially as it keeps her connected to her family), but she doesn’t want to work there forever. When she thinks about transferring to a four-year school, however, she feels overwhelmed by the cost—especially when she isn’t sure what she “wants to be when she grows up.” She likes the ocean and loves marine mammals (especially dolphins and whales), but she doesn’t enjoy science courses. She also enjoys photography. She took photos of her next-door neighbor’s family, and they turned out so well that she has been asked to photograph other families. However, she doesn’t know if she’ll be able to support herself long-term by turning this hobby into a job. She wonders if she could intern with the local newspaper to get some journalistic photography experience.
One of the most profound tasks facing young adults is preparing for life after high school. Education in childhood and adolescence is focused on learning basic skills and gaining knowledge helpful to becoming an independently functioning adult. Now people must decide what they may want to do next and how to achieve those goals. High school guidance counselors sometimes use the idea of “the 4 E’s ” to describe actions students can take after graduating high school. They are enlisting in the military, enrolling in a college or vocational program, becoming employed , and/or taking a gap year to explore possibilities (Figure 11.22). In this section, we’ll examine two of the most common actions in that list: Employment and enrollment in higher education.
Work and Career Preparation
There’s no one single way to decide what you want to do for work or a career. We all take different paths to find our way to that destination, sometimes through a series of different pathways. Some of us know our interests from an early age, while others aren’t sure even after starting college or embarking on a first job. And many people decide to change their careers as they move through adulthood, as well. We should also note that additional groups of people may not pursue paid employment because they take on responsibilities such as raising children or caring for elderly relatives. Adulthood allows us to make these choices, many times changing pathways throughout the course of a person’s lifespan.
The choice of work or career as a person enters early adulthood oftentimes depends to a great extent on an individual’s identity and life circumstances, especially their skills, values, beliefs, and goals. Someone who aspires to a specific lifestyle might choose their career based on expected salary. For some people, the chance to travel, help others, or apply a specific skill like language proficiency or musical talent might be their main goal. Thus, career choices often depend on the person’s having done some self-exploration and developing a sense of identity and purpose. The process of self-exploration may also include actively investigating jobs of interest by discussing the jobs with those in the field, volunteering, interning, doing internet searches, and taking college courses or completing a vocational program or certification course.
Life circumstances may also affect work choices. As we learned at the beginning of this chapter, having a low SES may prevent a person from exploring personally fulfilling career options and instead focus on meeting immediate needs like paying for food and shelter. Larger societal factors such as the nation’s economy and the availability of jobs in one’s community may also affect a person’s options. Having children may necessitate getting a job to provide for them or prevent a person from getting a job if the cost of childcare is too prohibitive. Overall childcare costs are hard to quantify because these figures depend on area population, the age of the child, and whether the care is provided in home or in a center. However, recent U.S. Census data indicate that families in the United States can spend from 8 percent to 19 percent of their income on childcare, an annual cost of between $5,357 and $17,171 in 2022 dollars (Grundy, 2024). For some people, it may make more financial sense to not pursue a career if paying for childcare would significantly cut into their earnings.
Having the opportunity to receive education beyond high school may also affect career and job decisions. While certain careers require more than a high school diploma or GED, an individual’s career path may not need to include a 2- or 4-year college degree, or and may not need to include higher education at all. In the section that follows, we’ll cover not only a “traditional” college education (i.e., one that leads to a bachelor’s degree) but also possible alternatives, such as vocational training and apprenticeships.
College as Context in Early Adulthood
Many jobs now require some form of degree or certification beyond high school, a change that has had a profound influence in making emerging adulthood a distinct developmental stage in many countries. According to U.S. Census data (Figure 11.23), while only 24 percent of people completed high school in 1940, that number rose to 41 percent in 1960, then 55 percent by 1970 (U.S. Census Bureau, 1970). According to the latest Census data, 91 percent of people in 2021 reported that they had completed their high school degree or high school equivalent degree (GED) (U.S. Census Bureau, 2022). People completing advanced degrees such as a masters or doctorate increased from 10.9 percent to 14.4 percent in the years from 2011 to 2021 (U.S. Census Bureau, 2022).
In the Fall 2021 semester, approximately 18,580,000 people were enrolled in college in the United States (U.S. Department of Education, 2023a). Similar data from Canada indicate that approximately 1,440,000 people were enrolled in college there in the Fall 2023 semester (Universities Canada, 2023). These figures represent only people currently enrolled, not people who’ve completed a college education. It should be noted that data for both countries include international students who aren’t considered official residents of country where they attend school. This makes it somewhat challenging to determine the proportion of the population these figures represent, but ballpark estimates suggest that these figures represent roughly 5.6 percent of the United States population and 3.7 percent of the Canadian population.
Of the roughly two million bachelor’s degrees earned in in the United States during the 2021–2022 academic year, the top six most common majors were business (18.6 percent), health professions (13.1 percent), social sciences and history (7.5 percent), biology/biomedical sciences (6.5 percent), psychology (6.4 percent), and engineering (6.1 percent; U.S. Department of Education, 2023b). Perhaps not surprisingly, many of these popular majors have obvious ties to potentially rewarding career fields. Careers in fields such as health care and psychology may be intrinsically rewarding and lead to greater job satisfaction and personal fulfillment. College education is also frequently touted as a pathway to higher salaries, greater career flexibility, and more opportunities for advancement (Blagg, 2022; Chau et al., 2023; Klor de Alva, 2022; OECD, 2023; Rosenbaum et al., 2017; Warick & Melnick, 2022). In fact, one study of finance majors found that 85 percent of the sample expected a salary at the 70 th percentile or higher within 10 years of graduation, despite the statistical unlikelihood of this actually happening (Schnusenberg, 2020).).
Looking at specific demographic variables and their relationship to educational attainment, we see some interesting patterns. In terms of gender, similar numbers of men and women have a high school diploma as their highest degree earned (29.4 percent versus 26.5 percent respectively; U.S. Census Bureau, 2022). However, the gap is a bit larger when it comes to higher education. By age 25, 46.9 percent of men had completed at least a bachelor’s degree, compared to 53.1 percent of women (U.S. Census Bureau, 2022).
The picture is somewhat different when we look at race and ethnicity. Recent data indicate that Asian youth had the highest number of baccalaureate degree holders at 61 percent, followed by young adults who were White at 41.9 percent, Black (28.1 percent), and Hispanic (20.6 percent; U.S. Census Bureau, 2022). These discrepancies are likely due to factors such as economic disadvantage, language barriers, and lack of appropriate guidance regarding college education. Some of these factors may intersect with other characteristics. For example, Black and Hispanic youth are proportionately more likely than White and Asian youth to be first-generation college students, a group also more likely to have a lower median income as well as higher risk of dropping out of college (RTI International, 2019). However, the number of Black and Hispanic adults earning higher education degrees is rising. Between 2011 and 2021, the number of Black people with a college degree by the age of 25, increased from 19.9 percent to 28.1 percent, and the percentage of Hispanic youths with a degree increased from 14.1 percent to 20.6 percent (U.S. Census Bureau, 2022)(Figure 11.24).
Intersections and Contexts: Disparities in Race, Ethnicity, and Gender in STEM Fields
Research suggests there are differences in types of jobs (professional, service-oriented, STEM) and in the wages earned by people of different racial or ethnic groups. For example, people who are Hispanic, especially Hispanic women, have a higher chance of having jobs that offer lower than average wages, and White people are more likely than Black people to have better job security (Earle et al., 2014).
STEM fields (science, technology, engineering, and math) are one area with large discrepancies in ethnic, racial, and gender representation. According to Fry et al. (2021), many minorities are underrepresented within STEM fields, especially Hispanic (17 percent of STEM jobs) and Black (9 percent of STEM jobs). Employment within STEM fields often requires a college degree and as of 2018, Black and Hispanic individuals represented 7 percent and 12 percent respectively of bachelor’s degrees within STEM fields, with White and Asian students earning the majority of STEM degrees (Fry et al., 2021).
Though women represented just over half the STEM degrees earned in 2018, they’re still underrepresented in some areas. Most degrees women earned within STEM fields were in fields related to health care (85 percent), as opposed to degrees in engineering (22 percent) and even computer science(19 percent). Though women do represent a little over half of all STEM-related degrees, Hispanic and Black women are underrepresented, together comprising only 17 percent of the STEM workforce (Fry et al., 2021).
These discrepancies persist after these graduates get to the workplace. Not only are Black and Hispanic workers underrepresented in STEM fields, but they also often get paid less than White and Asian counterparts. Race and ethnicity also intersect with sex; Black and Hispanic women receive the lowest wages within STEM fields (Fry et al., 2021). There are groups attempting to erase these discrepancies within STEM fields by providing resources such as peer mentoring, financial support for student research activities, and summer bridge programs for rising college freshmen (Estrada et al., 2016); however, increasing inclusion in these fields and college majors is complex, as many minorities do not have access to resources, especially educational resources, helpful to increasing representation in STEM fields. STEM is not always incorporated into early education programs, which may also decrease exposure to these fields for young children.
Here are links to the National Science Foundation Initiative to increase participation in STEM fields and a list of organizations that are working to increase diversity within STEM fields from National University.
Link to Learning
The United States Department of Education “College Scorecard” website contains a search tool that allows prospective students to find information about different schools, programs of study, and degree options. It also contains a section titled “Alternative Pathways to a Career” that provides links to information about apprenticeships and training programs.
Earning a bachelor’s degree has other benefits besides potentially increased income and career satisfaction. A fundamental purpose of college is to help people gain and refine cognitive abilities including verbal skills, quantitative skills, and critical thinking skills. While these skills can also develop outside of a college education, college provides a setting in which all three of them are regularly used, with desirable consequences for successful performance. These skills are transferrable to nearly every area of life, as are the “soft” skills that emerging adults can also hone in college, including planning, organizing, taking initiative, managing time, leading, and persuading. While interacting with classmates and working cooperatively on group projects, college students also develop social skills such as interpersonal communication, perspective-taking, and conflict resolution. Having a college degree is associated with improved decision-making about issues such as online safety and with prosocial civic engagement activities such as voting and volunteer work (OECD, 2023). College can also provide an excellent opportunity for finding purpose during emerging adulthood. Self-exploration at college can help people settle on their adult values and belief systems, further shape their identity, and develop complex, long-lasting relationships both platonic and romantic.
Along those lines, it’s not unusual for students to try out different majors before following through on the education or training required to attain a job within their field of choice. According to the U.S. Department of Education, around 30 percent of students enrolled in a two- or four-year degree program change their major before graduating, and at least 10 percent change their major multiple times (Leu, 2017). This is developmentally appropriate in the context of emerging adulthood and identity formation, discussed at the beginning of the chapter. Also, as we’ll see, college is not the only route to a career.
COVID and the College Experience
In Section 11.2, we examined how the COVID-19 pandemic affected mental health in emerging adulthood. Here, we will further consider how it affected the college experience. Around the world, in-person instruction shifted to online instruction on short notice, with students, faculty, staff, and families often having only a week or two to prepare. Not surprisingly, this change – combined with the circumstances that caused it – was associated with academic challenges. For many of them, they now had to share computer time and equipment with other family members, potentially limiting their ability to complete assignments or participate in synchronous online class meetings. These students were also cut off from academic resources such as tutoring as well as the social support and companionship of their friends and student organizations, which are also important components of the college experience.
Not all students entered this transition with equal challenges. For example, Birmingham and colleagues found that while college students were generally comfortable with technology, a third of them had never taken an online class prior to the pandemic (2023). Demographic factors also appeared to affect the adjustment to online learning. Students of color tend to perform lower in online classes than in face-to-face classes (Shankar et al., 2023). While financial concerns were common across demographic groups, people of color and families with low SES were more likely to experience financial challenges (Ibarra-Mejia et al., 2023; Molock & Parchem, 2022).
However, some valuable lessons may have been learned in the process. The pandemic provided the opportunity for a “natural experiment” in that students couldn’t self-select into online versus in-person classes, therefore allowing us to study performance in online classes while minimizing the effects of self-selection bias. Online classes with synchronous meetings tend to have better student outcomes than ones that are entirely asynchronous. (Reference needed). Large introductory-level classes conducted face-to-face may not typically enable high-impact practices such as small group discussions and flipped classrooms, but these things can be accomplished online. Finally, to address concerns about potential cheating on online exams, instructors could move away from multiple-choice, memorization-based questions to application-based questions that require short answer or essay responses. These questions may encourage more student engagement and represent a better assessment of learning (Shankar et al., 2023).
Alternatives to a College Degree
While an associate or a bachelor’s degree is a common educational pathway in emerging adulthood, it’s not the only option out there. People can also choose to earn an associate’s degree by attending a community college or a vocational school such as a trade school or technical institute (Figure 11.25). Less costly than a four-year college, these institutions offer one- to two-year training or licensing programs for jobs like HVAC technician, veterinary assistant, dental assistant, electrician, plumber, and pharmacy technician. Some of these schools offer two-year courses of study—similar to the general education curriculum portion of a bachelor’s degree—leading to associate’s degrees that may be transferrable to four-year institutions. Associate’s degrees are a good alternative to a bachelor’s degree for people whose career goals don’t require one or who aren’t completely certain about going to college. They may also be a way for someone who’s earned some credits but isn’t able to complete a bachelor’s degree to avoid leaving higher education empty-handed (Institute for Higher Education Policy, 2017).
At vocational school s, courses are hands-on and intensive, with small classes focused on specific skills and preparation for any testing required for the job. In some countries, vocational training includes more than just blue-collar occupations. For example, an apprenticeship is a paid position involving a combination of on-the-job training, mentoring, and classroom learning, often leading to employment in a specific field. Apprenticeship programs often focus on skilled trades such as information technology, manufacturing, food service, or cosmetology, but not always. Germany has an apprenticeship system, offered as an alternative to college, that involves all sectors of the workforce, including civil service job s (government jobs that don’t involve political appointments or legal work, such as urban planners, budget analysts, sanitation workers, and firefighters; Deissinger & Gonon, 2021). In Japan, high school students can enroll in specialized training schools called senmon gakko , designed to prepare students to meet workforce needs in such diverse areas as agriculture, health care, apparel and homemaking, and the arts (Kogirima, 2014; Taylor, 2020). Other countries regard vocational training as a possible alternative to, but not necessarily a replacement for, college. In Québec, students enroll in either a pre-university program or a vocational training program after finishing high school, and completion of either program makes them eligible to get a university education if they want one (Taylor, 2020).
In some cases, a 2-year or 4-year degree may not be necessary. For some positions, a short certificate training course opens the path to becoming a certified personal trainer, medical transcriptionist, court reporter, or security guard. Many service-sector jobs such as bartending and construction work may also require only supervised training and a test to become certified; for these positions, a high school degree or GED may not even be needed. There’s a wide range of employment opportunities out there, with a corresponding wide range of education and training necessary (or not) for each. No one educational path or career field is objectively better than any other; each person has to find what’s best for them.
Also, not all learning occurs in the context of formal education. Informal education is education that occurs without a set curriculum. Learning is hands-on and can include talking with and listening to others, helping organize events, and sharing experiences. Participation can help people learn about others, including those whose lives are unlike their own, and can expand their perspective-taking skills along with learning more about themselves. Thus, informal learning can be a form of experiential learning , a form of learning that involves not only gaining understanding by doing, but also reflecting on the experiences. It can occur in a college setting through internships and service learning courses as well as outside college through activities like volunteering and even activism (Gross & Rutland, 2017).
Costs and Rewards of College
College isn’t the path for everyone, but it’s still a popular choice for many young adults to explore options in life, specifically options for work. However, it comes with a literal cost that can vary greatly. According to the United States Department of Education, in the 2021–2022 academic year, the yearly cost of tuition and fees ranged from $,4000 at two-year public colleges, to $9,700 at four-year public colleges, to an average of $38,800 at private colleges (National Center for Education Statistics, 2023). We should note that these figures for public colleges represent in-state tuition rates only; out-of-state students typically pay more, oftentimes double the costs annually. Also, these figures only represent tuition and fees; they don’t include the additional costs of necessities such as textbooks and housing. These costs can vary widely depending on the program of study, geographic location, whether a college is public or private, and whether a student lives on or off campus. However, recent data indicate that college students can expect to spend between $1,000 and $1,400 per year on textbooks and materials, and between $7,000 and $13,000 per year on housing and food (Hanson, 2023).
Link to Learning
Visit this website from the U.S. government on tools for assessing the cost of college to learn more.
It also takes time to earn a degree, and many young adults have to work in order to partially or fully support themselves while pursuing their degree. Thus, finding time to attend classes and study while working can be stressful. This may cause students to reduce their working hours (and thus their earnings) to make time for classes, or to take reduced class loads (and thus take longer to graduate as well as pay more overall in fees) to accommodate their job. Either approach has financial consequences. Compared to many other countries, the United States provides more financial aid in the form of loans and scholarships, but tuition costs are higher than in many other countries (Bouchikra, 2024; Hanson, 2021; also see “Intersections and Contexts” feature box). Many students in the United States take out loans to pay for college , but the loans must be paid back, which can be a heavy financial burden on someone starting out at a job. Starting salaries for many occupations are typically much lower than the average salaries posted online, and someone expecting their first post-college job to give them enough money to immediately pay off their student loans will likely be in for a nasty surprise.
Intersections and Contexts: The Cost of College Around the World
The higher education costs we’ve seen so far only apply to colleges in the United States. There’s a lot of variability in how countries outside the United States manage the financial aspects of higher education. While it’s impossible to cover every single nation’s approach to college, we can identify some overall trends.
In many European countries, such as Denmark, Norway, and Sweden, college tuition is free. Often this benefit is just for residents of that country (or, in the case of Denmark and Sweden, the European Union; Bouchikra, 2024; Hanson, 2021); students from other countries will have to pay to go to school there, but this is reasonable considering that the primary purpose of public education is to educate the citizens of that country. Having free tuition means, obviously, that higher education is more accessible and that college graduates can enjoy higher earning potential without the burden of student loans.
In the United Kingdom, tuition costs are similar to those in the United States. However, most baccalaureate degrees are 3-year programs instead of the 4-year programs typically offered in the United States, reducing potential costs. Also, loans don’t have to be repaid until after a recipient crosses a certain income threshold depending on which payment plan they’re enrolled in (Hanson, 2021; Repaying Your Student Loan , n.d.). By contrast, in the United States, loan repayment is expected to start as soon as the recipient is no longer in school.
South and East Asian countries vary widely in their tuition costs. For example, students at public universities in India pay only $50 per year in tuition, compared to $2200 per year in China and approximately $8000 in Japan and the Republic of Korea (Bouchikra, 2024). However, the cost of living is very high in some of these countries; many Chinese students live in residence halls or even with host families because they can’t afford to rent an apartment (Hanson, 2021). In contrast, cost of living and college tuition costs in many Latin American countries are lower than in many other countries (Bouchkira, 2024; Hanson, 2021), partly because of poor economic conditions in many of those countries. Recently, Chile has instituted a taxpayer-funded gratuidad program to provide free college tuition for citizens in its lowest income bracket. In addition to increasing access to higher education, this program also attempts to compensate for poor quality college preparation in primary and secondary schools in impoverished areas (Bouchikra, 2024; Nadworny et al., 2019).
What’s the payoff of attaining a degree? There are many ways to answer that question. Financially speaking, college graduates generally earn higher salaries and have higher earning potential. In the United States, the median yearly salary of a 4-year college graduate in 2022 was around $74,464, versus $52,260 for people with an associate’s degree or a certificate from a vocational program. Those with only a high-school diploma earn around $44,356, and people without a high school diploma or GED earn around $35,464 (U.S. Bureau of Labor Statistics, 2023). However, the difference in earnings between levels of educational attainment isn’t universal. The wage gap between people with different levels of education tends to be relatively small early in employment, but widens in middle adulthood, with amount of education and earnings being positively correlated (OECD, 2023). Earnings for any job also depend on factors other than education, such as age, gender, geographic location, and demand/availability (OECD, 2023). For example, a person with a bachelor’s degree in art history may have a lot of difficulty finding work in that field if they live in a rural, low-income area. Their earnings and job prospects may not perfectly reflect the value of their degree if it doesn’t happen to fit a need in the place where they live.
However, a college degree in many arts and sciences fields, such as history, biology, and music, doesn’t limit what a person can actually do with that degree. Thanks to the soft skills and general knowledge they confer, college degrees may also increase an individual’s ability to get jobs outside the field of their degree. For instance, a college graduate who majored in English may become a technical writer for a pharmaceutical company despite not having any background in chemistry. Having a post-high school degree conveys that a person has knowledge, communication skills, and the ability to set and complete goals, all characteristics valuable to employers regardless of that person’s major (Figure 11.26).
This value is reflected in different unemployment rates for people with different amounts of education. U.S. Bureau of Labor Statistics data from 2022 indicated that people without a high school degree had an unemployment rate of 5.5 percent, while people with a high school diploma had an unemployment rate of 4 percent. For people with a four-year college degree the rate was 2.2 percent (2023). Interestingly, the unemployment rate for people with some college but no degree is 3.5 percent, compared with just 2.7 percent for people with an associate’s degree (U.S. Bureau of Labor Statistics, 2023). While these people may have completed similar numbers of college credits, and people in the “some college, no degree” group may even have completed more credits than people with associate’s degrees, employers clearly don’t regard these two statuses as equivalent. This provides more support for earlier statements about the utility of two-year degree programs as an alternative to a four-year degree.
Workplace as Context
In previous generations, it wasn't unusual to stay with a company for life, working their way up in responsibility or even building their own business over time. This pattern has changed dramatically due to multiple factors, including the increasing role of technology in every aspect of life. When Generation X (born between 1965 and 1981) joined the workforce, technology and a changing employment landscape no longer guaranteed working for one employer for life, and many people no longer expected to do so. These changes influenced attitudes about work as well. Instead of prioritizing loyalty to a company, many people now place more value on work they enjoy or find meaningful (Črešnar & Nedelko, 2020)—although, as mentioned earlier, life circumstances may not make this possible for everyone to achieve.
Other work-related views are likely influenced by life events and experiences. Research on younger Millennials and Generation Z (those born between 1995 and 2010) finds that many report feeling less “engaged” in their work (Pendell & Vender Helm, 2022). Putting this in context, Generation Z is the first cohort to live in a digital world, meaning they have been interacting with or using technology since birth (Benítez-Márquez et al., 2022). They express increased concern about environmental issues such as climate change and societal issues such as equal rights, and place importance on working for a company that plays a positive role in addressing these issues (Barhate & Dirani, 2022; Črešnar & Nedelko, 2020; Whitney Gibson et al., 2011). They’ve also witnessed global phenomena such as financial crises and a world-wide pandemic; not surprisingly, this cohort of workers also reports feeling increasingly stressed, and working for employers that offer a good work-life balance or who care about the well-being of their employees is important in choosing a job (Črešnar & Nedelko, 2020; Pendell & Vender Helm, 2022). As a result, members of Generation Z actively desire careers, but those that allow flexibility and enhance personal well-being (Črešnar & Nedelko, 2020; Whitney Gibson et al., 2011).
Though our views of work might be changing, Pew Research data on workers of all ages show that most appear happy with their job, if not so satisfied with their pay or ability to be promoted. Workers over 65 tended to feel most positively toward their employer and reported the least amount of stress at work, finding their work fulfilling, while younger people reported feeling more stress and less fulfillment. People with higher incomes also reported more satisfaction with their benefits, pay, and promotion opportunities than people with lower incomes (Horowitz & Parker, 2023).
What can be done for those who are unhappy at work, especially about finding a better work-life balance ? It’s hard to define what work-life balance means to all people. For some, it means having a job that allows them to spend enough time with family, while for others it involves being able to manage health problems while still performing their job effectively (Gragnano et al., 2020; Sirgy & Lee, 2018).
Other valued aspects of the work environment include flexibility (being able to choose whether to work from home or in the office, along with having flexible hours at work), access to childcare, and resources for enhancing physical and mental health. Research has shown that employers that emphasize or have programs that address? these things see increased productivity, morale, and physical and mental health in their workforce, along with lower employee turnover (Kalev & Dobbin, 2022).
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Institute for Higher Education Policy (IHEP), Wheatle, K., Taylor, J., Bragg, D., & Ajinkya, J. (2017). The potential of degree reclamation: A path to reclaiming the nation’s unrecognized students and degrees . Institute for Higher Education Policy. http://files.eric.ed.gov/fulltext/ED591575.pdf
Kalev, A., & Dobbin, F. (2022, August 15). The surprising benefits of work/life support. Harvard Business Review . https://hbr.org/2022/09/the-surprisi...k-life-support
Klor de Alva, J. (2022). Accounting for demographics, selectivity, and risk in postcollege earnings. In J. D. Delisle (Ed.), Student outcomes and earnings in higher education policy . American Enterprise Institute. http://files.eric.ed.gov/fulltext/ED621363.pdf
Kogirima, A. (2014). Senmon gakko (professional training colleges) in Japan . National Institute for Educational Policy Research. www.nier.go.jp/English/educa.../201403SMG.pdf
Leu, K. (2017). Beginning college students who change their majors within 3 years of enrollment . U.S. Department of Education. https://nces.ed.gov/pubs2018/2018434.pdf
Molock, S. D., & Parchem, B. (2022). The impact of COVID-19 on college students from communities of color. Journal of American College Health, 70 (8), 2399–2405. doi.org/10.1080/07448481.2020.1865380
Nadworny, E., Marcus, J., & Gutierrez, M. (2019, November 25). What the U.S. can learn from free college in Chile . National Public Radio. https://www.npr.org/2019/11/25/77601...lege-in-chile/
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Pendell, R., & Vander Helm, S. (2022, November 11). Generation disconnected: Data on Gen Z in the workplace . Gallup.com. https://www.gallup.com/workplace/404...workplace.aspx
Repaying your student loan . (n.d.) UK.GOV. https://www.gov.uk/repaying-your-student-loan/print
Rosenbaum, J. E., Ahearn, C. E, & Rosenbaum, J. E. (2017). Bridging the gaps: College pathways to career success . Russell Sage Foundation. https://eric.ed.gov/?redir=http%3a%2...ridging-gaps-0
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11: Molecular Biology of Gene Expression
-
- 11.1: Introduction
- The three letters “DNA” have now become synonymous with crime solving, paternity testing, human identification, and genetic testing. DNA can be retrieved from hair, blood, or saliva. Each person’s DNA is unique, and it is possible to detect differences between individuals within a species on the basis of these unique features.
-
- 11.2: The Structure of DNA
- The DNA molecule is a polymer of nucleotides. Each nucleotide is composed of a nitrogenous base, a five-carbon sugar (deoxyribose), and a phosphate group. There are four nitrogenous bases in DNA, two purines (adenine and guanine) and two pyrimidines (cytosine and thymine). A DNA molecule is composed of two strands. Each strand is composed of nucleotides bonded together covalently between the phosphate group of one and the deoxyribose sugar of the next. From this backbone extend the bases.
-
- 11.3: DNA Replication
- When a cell divides, it is important that each daughter cell receives an identical copy of the DNA. This is accomplished by the process of DNA replication. The replication of DNA occurs during the synthesis phase, or S phase, of the cell cycle, before the cell enters mitosis or meiosis.
-
- 11.4: Transcription
- The second function of DNA (the first was replication) is to provide the information needed to construct the proteins necessary so that the cell can perform all of its functions. To do this, the DNA is “read” or transcribed into an mRNA molecule. The mRNA then provides the code to form a protein by a process called translation. Through the processes of transcription and translation, a protein is built with a specific sequence of amino acids that was originally encoded in the DNA.
-
- 11.5: Translation
- The synthesis of proteins is one of a cell’s most energy-consuming metabolic processes. In turn, proteins account for more mass than any other component of living organisms (with the exception of water), and proteins perform a wide variety of the functions of a cell. The process of translation, or protein synthesis, involves decoding an mRNA message into a polypeptide product. Amino acids are covalently strung together in lengths ranging from approximately 50 amino acids to more than 1,000.
-
- 11.6: How Genes Are Regulated
- All organisms and cells control or regulate the transcription and translation of their DNA into protein. The process of turning on a gene to produce RNA and protein is called gene expression. Whether in a simple unicellular organism or in a complex multicellular organism, each cell controls when and how its genes are expressed. For this to occur, there must be a mechanism to control when a gene is expressed to make RNA and protein, how much of the protein is made, and when it is time to stop.
Thumbnail: A digital rendering of a DNA double helix. (CC0 License; Sangharsh Lohakare , via Unsplash )


17.10.2: Thought Questions
Thought Questions
If the star Sirius emits 23 times more energy than the Sun, why does the Sun appear brighter in the sky?
14.
How would two stars of equal luminosity—one blue and the other red—appear in an image taken through a filter that passes mainly blue light? How would their appearance change in an image taken through a filter that transmits mainly red light?
15.
Table 17.2 lists the temperature ranges that correspond to the different spectral types. What part of the star do these temperatures refer to? Why?
16.
Suppose you are given the task of measuring the colors of the brightest stars, listed in Appendix J, through three filters: the first transmits blue light, the second transmits yellow light, and the third transmits red light. If you observe the star Vega, it will appear equally bright through each of the three filters. Which stars will appear brighter through the blue filter than through the red filter? Which stars will appear brighter through the red filter? Which star is likely to have colors most nearly like those of Vega?
17.
Star X has lines of ionized helium in its spectrum, and star Y has bands of titanium oxide. Which is hotter? Why? The spectrum of star Z shows lines of ionized helium and also molecular bands of titanium oxide. What is strange about this spectrum? Can you suggest an explanation?
18.
The spectrum of the Sun has hundreds of strong lines of nonionized iron but only a few, very weak lines of helium. A star of spectral type B has very strong lines of helium but very weak iron lines. Do these differences mean that the Sun contains more iron and less helium than the B star? Explain.
19.
What are the approximate spectral classes of stars with the following characteristics?
- Balmer lines of hydrogen are very strong; some lines of ionized metals are present.
- The strongest lines are those of ionized helium.
- Lines of ionized calcium are the strongest in the spectrum; hydrogen lines show only moderate strength; lines of neutral and metals are present.
- The strongest lines are those of neutral metals and bands of titanium oxide.
20.
Look at the chemical elements in Appendix K. Can you identify any relationship between the abundance of an element and its atomic weight? Are there any obvious exceptions to this relationship?
21.
Appendix I lists some of the nearest stars. Are most of these stars hotter or cooler than the Sun? Do any of them emit more energy than the Sun? If so, which ones?
22.
Appendix J lists the stars that appear brightest in our sky. Are most of these hotter or cooler than the Sun? Can you suggest a reason for the difference between this answer and the answer to the previous question? (Hint: Look at the luminosities.) Is there any tendency for a correlation between temperature and luminosity? Are there exceptions to the correlation?
23.
What star appears the brightest in the sky (other than the Sun)? The second brightest? What color is Betelgeuse? Use Appendix J to find the answers.
24.
Suppose hominids one million years ago had left behind maps of the night sky. Would these maps represent accurately the sky that we see today? Why or why not?
25.
Why can only a lower limit to the rate of stellar rotation be determined from line broadening rather than the actual rotation rate? (Refer to Figure 17.14.)
26.
Why do you think astronomers have suggested three different spectral types (L, T, and Y) for the brown dwarfs instead of M? Why was one not enough?
27.
Sam, a college student, just bought a new car. Sam’s friend Adam, a graduate student in astronomy, asks Sam for a ride. In the car, Adam remarks that the colors on the temperature control are wrong. Why did he say that?
28.
Would a red star have a smaller or larger magnitude in a red filter than in a blue filter?
29.
Two stars have proper motions of one arcsecond per year. Star A is 20 light-years from Earth, and Star B is 10 light-years away from Earth. Which one has the faster velocity in space?
30.
Suppose there are three stars in space, each moving at 100 km/s. Star A is moving across (i.e., perpendicular to) our line of sight, Star B is moving directly away from Earth, and Star C is moving away from Earth, but at a 30° angle to the line of sight. From which star will you observe the greatest Doppler shift? From which star will you observe the smallest Doppler shift?
31.
What would you say to a friend who made this statement, “The visible-light spectrum of the Sun shows weak hydrogen lines and strong calcium lines. The Sun must therefore contain more calcium than hydrogen.”?


12.2: Ring and Moon Systems Introduced
By the end of this section, you will be able to:
- Briefly describe the system of moons around each of the jovian planets
- Describe the basic composition of each jovian planet’s ring system
The rings and moons (see the moons in Figure 12.2) of the outer solar system are not composed of the same materials as the mostly rocky objects in the inner solar system. We should expect this, since they formed in regions of lower temperature, cool enough so that large quantities of water ice were available as building materials. Most of these objects also contain dark, organic compounds mixed with their ice and rock. Don’t be surprised, therefore, to find that many objects in the ring and moon systems are both icy and dark.
Roughly a third of the moons in the outer solar system are in direct or regular orbits; that is, they revolve about their parent planet in a west-to-east direction and in the plane of the planet’s equator. The majority are irregular moons that orbit in a retrograde (east-to-west) direction or else have orbits of high eccentricity (more elliptical than circular) or high inclination (moving in and out of the planet’s equatorial plane). These irregular moons are mostly located relatively far from their planet; they were probably formed elsewhere and subsequently captured by the planet they now orbit.
The Jupiter System
Jupiter has 79 known moons (that’s the number as we write) and a faint ring. These include four large moons— Callisto , Ganymede , Europa , and Io (see Figure 12.1)—discovered in 1610 by Galileo and therefore often called the Galilean moons . The smaller of these, Europa and Io, are about the size of our Moon, while the larger, Ganymede and Callisto, are about the same size as the planet Mercury. Most of Jupiter’s moons are much smaller. The majority are in retrograde orbits more than 20 million kilometers from Jupiter; these are very likely small captured asteroids.
The Saturn System
Saturn has at least 82 known moons in addition to a magnificent set of rings. The largest of the moons, Titan , is almost as big as Ganymede in Jupiter’s system, and it is the only moon with a substantial atmosphere and lakes or seas of liquid hydrocarbons (such as methane and ethane) on the surface. Saturn has six other large regular moons with diameters between 400 and 1600 kilometers, a collection of small moons orbiting in or near the rings, and many captured strays similar to those of Jupiter. Mysteriously, one of Saturn’s smaller moons, Enceladus , has active geysers of water being expelled into space.
The rings of Saturn, one of the most impressive sights in the solar system, are broad and flat, with a few major and many minor gaps. They are not solid, but rather a huge collection of icy fragments, all orbiting the equator of Saturn in a traffic pattern that makes rush hour in a big city look simple by comparison. Individual ring particles are composed primarily of water ice and are typically the size of ping-pong balls, tennis balls, and basketballs.
The Uranus System
The ring and moon system of Uranus is tilted at 98°, just like the planet itself. It consists of 11 rings and 27 currently known moons. The five largest moons are similar in size to the six regular moons of Saturn, with diameters of 500 to 1600 kilometers. Discovered in 1977, the rings of Uranus are narrow ribbons of dark material with broad gaps in between. Astronomers suppose that the ring particles are confined to these narrow paths by the gravitational effects of numerous small moons, many of which we have not yet glimpsed.
The Neptune System
Neptune has 14 known moons. The most interesting of these is Triton , a relatively large moon in a retrograde orbit—which is unusual. Triton has a very thin atmosphere, and active eruptions were discovered there by Voyager in its 1989 flyby. To explain its unusual characteristics, astronomers have suggested that Triton may have originated beyond the Neptune system, as a dwarf planet like Pluto. The rings of Neptune are narrow and faint. Like those of Uranus, they are composed of dark materials and are thus not easy to see.


1.3: LBA Catalysts
The combination of Lewis acids and chiral Brønsted acids affords LBA catalysts. In this system, the coordination of the Lewis acids to the heteroatom of the chiral Brønsted acid results in increase the acidity of the latter. For examples, the LBA, derived from optically active monoalkylated-1,2-diaryl ethane- 1,2-diol and SnCl 4 , has been found to be an effective catalyst for the enantioselective protonation of silyl enol ethers and ketene disilyl acetals (Scheme \(\PageIndex{1}\)).
Examples:
Based on the related X-ray crystal structure, the following transition states , controlled by a linear O-H— π bonding interaction, are proposed for the stereochemical course of the reactions (Scheme \(\PageIndex{2}\)).
The chiral catechol-derived LBA 1 has been employed as an artificial cyclaze for the cyclization of various 2-(polyprenyl)phenol derivatives with good yield and enantioselectivity. For example, a short total synthesis of (-)-chromazonarol can be accomplished with 88% enantioselectivity (Scheme \(\PageIndex{3}\)).
In addition, LBAs have been used as powerful catalysts for allylation reactions. For examples, LBA 2 has been used as an effective catalyst for allylation of aldehydes with high diasterofacial selectivity (Scheme \(\PageIndex{4}\)).


17.8: Islam to the Mamluks
-
- 17.8.2: Introduction
- Known for this Cyclical Theory of History, Ibn Khaldun posited that, seduced by the wiles of urban culture, the dominant group would over time become soft and enter into a period of decay, until a new group of desert peoples conquered them, when the process would begin anew. This theory applies to the development of Islamic history discussed throughout this chapter.
-
- 17.8.5: Geography of the Middle East
- Academics have not reached a consensus on the geographical boundaries of the Middle East. However, for the purposes of this chapter, this region will encompass the broadly defined areas known as Persia, Mesopotamia, the Levant, Asia Minor, and the Arabian Peninsula. The Middle East straddles three continents, including Asia, Africa, and Europe. The geography of the area promoted cultural diffusion by facilitating the spread of peoples, ideas, and goods along overland and maritime trade routes.
-
- 17.8.6: Rise of Islam
- As a religion of the Abrahamic faith, Islam holds much in common with Judaism and Christianity. Islam grew out of the Judeo-Christian tradition, a link which helped to legitimize the new religion. In fact, Muslims believe in the same God, or Allah in Arabic, as the Jewish and Christian God. Although Muslims trust that the People of the Book had received the word of God, they believe that it had become distorted over time, so God sent the Angel Gabriel to deliver His word to Muhammad.
-
- 17.8.7: The Expansion of Islam
- The Prophet Muhammad started publically preaching his strict brand of monotheism in the year 613, by reciting the Quran, quickly convincing some of the commoners of Mecca to believe in him. Most of his early converts belonged to groups of people who had failed to achieve any significant social mobility, which, of course, included many of the poor. His followers memorized his recitations and message that called for the powerful to take care of the weak, a message that resonated with many.
-
- 17.8.8: The Rashidun Caliphs
- The Prophet Muhammad started publically preaching his strict brand of monotheism in the year 613, by reciting the Quran, quickly convincing some of the commoners of Mecca to believe in him. Most of his early converts belonged to groups of people who had failed to achieve any significant social mobility, which, of course, included many of the poor. His followers memorized his recitations and message that called for the powerful to take care of the weak, a message that resonated with many.
-
- 17.8.9: The Umayyad Caliphate
- The Prophet Muhammad started publically preaching his strict brand of monotheism in the year 613, by reciting the Quran, quickly convincing some of the commoners of Mecca to believe in him. Most of his early converts belonged to groups of people who had failed to achieve any significant social mobility, which, of course, included many of the poor. His followers memorized his recitations and message that called for the powerful to take care of the weak, a message that resonated with many.
-
- 17.8.10: The 'Abbasid Caliphate
- The Prophet Muhammad started publically preaching his strict brand of monotheism in the year 613, by reciting the Quran, quickly convincing some of the commoners of Mecca to believe in him. Most of his early converts belonged to groups of people who had failed to achieve any significant social mobility, which, of course, included many of the poor. His followers memorized his recitations and message that called for the powerful to take care of the weak, a message that resonated with many.
-
- 17.8.11: The Fatimid Caliphate
- The Prophet Muhammad started publically preaching his strict brand of monotheism in the year 613, by reciting the Quran, quickly convincing some of the commoners of Mecca to believe in him. Most of his early converts belonged to groups of people who had failed to achieve any significant social mobility, which, of course, included many of the poor. His followers memorized his recitations and message that called for the powerful to take care of the weak, a message that resonated with many.
-
- 17.8.12: The Crusades
- The Prophet Muhammad started publically preaching his strict brand of monotheism in the year 613, by reciting the Quran, quickly convincing some of the commoners of Mecca to believe in him. Most of his early converts belonged to groups of people who had failed to achieve any significant social mobility, which, of course, included many of the poor. His followers memorized his recitations and message that called for the powerful to take care of the weak, a message that resonated with many.
-
- 17.8.13: The Mamluk Sultanate
- The Prophet Muhammad started publically preaching his strict brand of monotheism in the year 613, by reciting the Quran, quickly convincing some of the commoners of Mecca to believe in him. Most of his early converts belonged to groups of people who had failed to achieve any significant social mobility, which, of course, included many of the poor. His followers memorized his recitations and message that called for the powerful to take care of the weak, a message that resonated with many.
-
- 17.8.14: Conclusion
- The Prophet Muhammad started publically preaching his strict brand of monotheism in the year 613, by reciting the Quran, quickly convincing some of the commoners of Mecca to believe in him. Most of his early converts belonged to groups of people who had failed to achieve any significant social mobility, which, of course, included many of the poor. His followers memorized his recitations and message that called for the powerful to take care of the weak, a message that resonated with many.
-
- 17.8.15: Works Consulted and Further Reading
- The Prophet Muhammad started publically preaching his strict brand of monotheism in the year 613, by reciting the Quran, quickly convincing some of the commoners of Mecca to believe in him. Most of his early converts belonged to groups of people who had failed to achieve any significant social mobility, which, of course, included many of the poor. His followers memorized his recitations and message that called for the powerful to take care of the weak, a message that resonated with many.
-
- 17.8.16: Links to Primary Sources
- The Prophet Muhammad started publically preaching his strict brand of monotheism in the year 613, by reciting the Quran, quickly convincing some of the commoners of Mecca to believe in him. Most of his early converts belonged to groups of people who had failed to achieve any significant social mobility, which, of course, included many of the poor. His followers memorized his recitations and message that called for the powerful to take care of the weak, a message that resonated with many.
Thumbnail: The Kaaba in Makkah. Image used wtih permission (GFDL Free Art License; Muhammad Mahdi Karim).


12.8: Resources for Further Learning
Bierley, Paul E. The Incredible Band of John Philip Sousa . University of Illinois Press, 2010.
Dils, Ann and Ansley Cooper Albright. Moving History/Dancing Cultures: A Dance History Reader. Wesleyan University Press, 2001.
Jamison, Phil. Hoedowns, Reels, and Frolics: Roots and Branches of Southern Appalachian Dance . University of Illinois Press, 2015.
Lott, Eric. Love & Theft: Blackface Minstrelsy and the American Working Class . 20th anniversary edition. Oxford University Press, 2013.
Malone, Jacqui. Steppin’ on the Blues: The Visible Rhythms of African American Dance. University of Illinois Press, 1996.
Siblin, Eric. The Cello Suites: J. S. Bach, Pablo Casals, and the Search for a Baroque Masterpiece. Grove Press, 2011.
Taruskin, Richard. Music in the Seventeenth and Eighteenth Centuries: The Oxford History of Western Music . Oxford University Press, 2009.
Turino, Thomas. Music as Social Life: The Politics of Participation . University of Chicago Press, 2008.


5.6: The Challenges Ahead
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Furthering the advance of m-learning in developing countries is governed by a combination of five key constraints, four of which are technical. (Other nontechnical constraints such as literacy, culture, and language, are not covered here). Depending on the target area, none or all of these may apply. I consider these issues to be as follows.
Although growing at a phenomenal rate, mobile ownership in many developing countries is still relatively low, and nowhere close to the near 100 percent penetration rates that we see in many mature markets. If educational establishments begin to embrace mobile technology to any significant extent, then issues of ownership and access to handsets by students need to be addressed to ensure that, in the words of a recent American president, “no child is left behind”. Putting learning tools in the hands of children in developing countries is a key objective of MIT’s One Laptop Per Child project. Many people believe that the mobile phone would be a better tool to work with. The debate continues.
Where pupils do own, or have access to, mobile phones, more often than not—and this is particularly the case in many rural areas—these phones will be either older models, or lower-end handsets with limited functionality. In order to develop appropriate teaching tools, the reality of the target market needs to be considered. The wider community should most likely consider ownership and use of PDAs and Pocket PCs as non-existent.
Higher-end handsets with data capability are only useful in areas where the mobile network can service them, and where costs of data access are not prohibitive. In many cases neither of these are a safe bet. By way of an example, during a recent one-month visit to Uganda working with Grameen, I was able to connect to the Internet using my phone approximately 10 percent of the time.
Mobile phones may be ubiquitous, highly portable, shareable, immediate, and always-on, but there also limitations that challenge even the most talented mobile applications developers—small (and generally low-resolution) displays, awkward text input methods, slow data access (if at all), and issues of battery life, among others. On top of all that, the mobile industry has historically suffered from a lack of standards, with different manufacturers supporting different video and audio formats, no standard screen size and resolution, lack of regular support for Java and/or Flash, incompatible browsers (if at all), and a wide array of memory sizes. All of this fragments the platform landscape, making developing an m-learning application a real challenge.
“A project is complete when it starts working for you, rather than you working for it”. – Scott Allen


7.2: Introduction to Cosets and Normal Subgroups
Definition
Given a group \(G\) with subgroup \(H\text{,}\) we define \(\sim_L\) on \(G\) by
\begin{equation*} a\sim_L b \text{ if and only if } a^{-1}b\in H \end{equation*}
and \(\sim_R\) on \(G\) by
\begin{equation*} a\sim_R b \text{ if and only if } ab^{-1}\in H. \end{equation*}
Theorem \(\PageIndex{1}\)
\(\sim_L\) and \(\sim_R\) are equivalence relations on \(G\text{.}\)
- Proof
-
First, let \(a∈G\). Then \(a^{−1}a=e∈H\), so \(a \sim_La\). Thus, \(\sim_L\) is reflexive.
Next, let \(a,b∈G\) with \(a\sim_Lb\). Then \(a^{−1}b∈H\), so, since \(H\) is a subgroup of \(G\), \((a^{−1}b)^{−1}∈H\). But \((a^{−1}b)^{−1}=b^{−1}(a^{−1})^{−1}=b^{−1}a\); thus, \(b \sim_La\), and so \(\sim_L\) is symmetric.
Finally, let \(a,b,c∈G\) with \(a\sim_Lb\) and \(b\sim_Lc\). Then \(a^{−1}b\) and \(b^{−1}c\) are in \(H\). Since \(H\) is a subgroup of \(G\), we must then have \((a^{−1}b)(b^{−1}c)∈H\); but \((a^{−1}b)(b^{−1}c)\) equals \(a^{−1}c\). Thus, \(a\sim_Lc\), and so \(\sim_L\) is transitive.
Thus, \(\sim_L\) is an equivalence relation on \(G\). The proof that \(\sim_R\) is an equivalence relation is left as an exercise for the reader.
Remark
Of course, different subgroups \(H\) and \(K\) in a group \(G\) will give rise to different relations \(\sim_L\) and \(\sim_R\) on \(G\text{;}\) that is, these relations are really defined with respect to a particular subgroup of \(G\text{.}\)
From now on, whenever we discuss \(\sim_L\) or \(\sim_R\) on a group, assume that it is with respect to a particular subgroup \(H\) of \(G\text{.}\)
Now, as equivalence relations on a group \(G\text{,}\) each of \(\sim_L\) and \(\sim_R\) gives rise to a partition of \(G\text{.}\) What are the cells of those partitions?
Definition: Left and Right Cosets
Given \(a\in G\text{,}\) we define
\begin{equation*} aH = \{ah\,:\, h\in H\} \end{equation*}
and
\begin{equation*} Ha=\{ha\,:\,h\in H\}. \end{equation*}
W e call \(aH\) and \(Ha\text{,}\) respectively, the left and right cosets of \(H\) containing \(a\) .
If we know that \(G\) is abelian, with operation denoted by \(+\text{,}\) we may denote these left and right cosets by \(a+H\) and \(H+a\text{,}\) respectively.
Note
In the following, we use the notation \(\Leftrightarrow\) to denote the phrase “if and only if.”
Theorem \(\PageIndex{2}\)
Let \(a\in G\text{.}\) Then under \(\sim_L\text{,}\) \([a]=aH\) while under \(\sim_R\text{,}\) \([a]=Ha\text{.}\)
- Proof
-
Let \(b∈G\). Then \(b \sim_La⇔a \sim_Lb⇔a^{−1}b∈H⇔a^{−1}b=h\) for some \(h∈H⇔b=ah\) for some \(h∈H⇔b∈aH\). So under \(\sim_L\) we have \([a]=aH\). Similarly, under \(\sim_R\) we have \([a]=Ha\).
We next summarize some facts about the left and right cosets of a subgroup \(H\) of a group \(G\text{:}\)
Theorem \(\PageIndex{3}\)
Let \(G\) be a group with \(H\leq G\) and \(a,b\in G\text{.}\)
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The left [right] cosets of \(H\) in \(G\) partition \(G\text{.}\)
-
\begin{equation*} b\in aH \Leftrightarrow aH=bH \Leftrightarrow a\in bH \end{equation*}
and
\begin{equation*} b\in Ha \Leftrightarrow Ha=Hb \Leftrightarrow a\in Hb. \end{equation*}
In particular,
\begin{equation*} a\in H \Leftrightarrow aH=H \Leftrightarrow Ha=H. \end{equation*}
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\(H\) is the only left or right coset of \(H\) that is a subgroup of \(G\text{.}\)
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\(|aH|=|H|=|Ha|\text{.}\)
- Proof
-
Statements 1 and 2 hold because the left and right cosets of \(H\) in \(G\) are equivalence classes. Statement 3 holds because no left or right coset of \(H\) other than \(H\) itself can contain \(e\), since the left [right] cosets of \(H\) are mutually disjoint. For Statement 4: Define \(f:H→aH\) by \(f(h)=ah\). It is straightforward to show that \(f\) is a bijection, so \(|H|=|aH|\). Similarly, \(|Ha|=|H|\).
Remark
We can use Statements 2 and 3, above, to save some time when computing left and right cosets of a subgroup of a group.
Example \(\PageIndex{1}\)
Find the left and right cosets of \(H=\langle (12)\rangle\) in \(S_3\text{.}\)
The left cosets are
\begin{equation*} eH=H=(12)H, \end{equation*} \begin{equation*} (13)H=\{(13),(123)\}=(123)H, \end{equation*} \begin{equation*} \text{ and } (23)H=\{(23),(132)\}=(132)H, \end{equation*}
and the right cosets are
\begin{equation*} He=H=H(12), \end{equation*} \begin{equation*} H(13)=\{(13),(132)\}=H(132), \end{equation*} \begin{equation*} \text{ and } H(23)=\{(23),(123)\}=H(123). \end{equation*}
Thus, \(\sim_L\) partitions \(S_3\) into \(\{H,\{(13),(123)\},\{(23), (132)\}\}\) and \(\sim_R\) partitions \(S_3\) into \(\{H,\{(13),(132)\},\{(23), (123)\}\}\text{.}\)
Example \(\PageIndex{2}\)
Find the left and right cosets of \(H=\langle f\rangle\) in \(D_4\text{.}\)
This example is left as an exercise for the reader.
We now draw attention to some very important facts:
Note
For \(a,b\in G\text{:}\)
-
In general, \(aH \neq Ha\text{!}\)
-
\(aH=bH\) does not necessarily imply \(a=b\) or that there exists an \(h\in H\) with \(ah=bh\text{;}\) similarly, \(Ha=Hb\) does not necessarily imply \(a=b\) or that there exists an \(h\in H\) with \(ha=hb\text{.}\)
Example \(\PageIndex{3}\)
We saw above that in \(S_3\) with \(H=\langle (12)\rangle\text{,}\)
\begin{equation*} (13)H=\{(13),(123)\} \neq \{(13),(132)\}=H(13). \end{equation*}
Also, \((13)H=(123)H\) but \((13)e\neq (123)e\) and \((13)(12)\neq (123)(12)\text{.}\)
It turns out that subgroups \(H\) for which \(aH=Ha\) for all \(a\in G\) will be very important to us.
Definition: Normal Subgroup
We say that subgroup \(H\) of \(G\) is normal in \(G\) (or is normal subgroup of \(G\)) if \(aH=Ha\) for all \(a\in G\text{.}\) We denote that fact that \(H\) is normal in \(G\) by writing \(H\unlhd G\text{.}\)
Remark
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If \(H\) is normal in \(G\text{,}\) we may refer to the left and right cosets of \(G\) as simply cosets .
-
Of course, if \(G\) is abelian, every subgroup of \(G\) is normal in \(G\text{.}\) But there can also be normal subgroups of nonabelian groups: for instance, the trivial and improper subgroups of every group are normal in that group.
Example \(\PageIndex{4}\)
Find the cosets of \(5\mathbb{Z}\) in \(\mathbb{Z}\text{.}\)
Notice that in additive notation, the statement “\(a^{-1}b\in H\)” becomes \(-a+b\in H\text{.}\) So for \(a,b\in \mathbb{Z}\text{,}\) \(a\sim_L b\) if and only if \(-a+b \in 5\mathbb{Z}\text{;}\) that is, if and only if \(5\) divides \(b-a\text{.}\) In other words, \(a\sim_L b\) if and only if \(a\equiv_5 b\text{.}\) So in this case, \(\sim_L\) is just congruence modulo \(5\text{.}\) Thus, the cosets of \(5\mathbb{Z}\) in \(\mathbb{Z}\) are
\begin{align*} 5\mathbb{Z}& =\{\ldots,-5,0,5,\ldots\}\\ 1+5\mathbb{Z}& =\{\ldots,-4, 1, 6,\ldots\},\\ 2+5\mathbb{Z}& =\{\ldots,-3,2, 7, \ldots\},\\ 3+5\mathbb{Z}& =\{\ldots,-2,3, 8, \ldots\},\\ 4+5\mathbb{Z}&=\{\ldots,-1, 4, 9, \ldots\}. \end{align*}
Do you see how this example would generalize for \(n\mathbb{Z}\) (\(n \in \mathbb{Z}^+\)) in \(\mathbb{Z}\text{?}\)
Example \(\PageIndex{5}\)
Find the cosets of \(H=\langle 12\rangle\) in \(4\mathbb{Z}\text{.}\)
They are
\begin{align*} H& =\{\ldots, -12,0,12\ldots\},\\ 4+H & = \{\ldots,-8,4,16,\ldots\},\\ 8+H& =\{\ldots, -4,8,20,\ldots\}. \end{align*}
Example \(\PageIndex{6}\)
Find the cosets of \(H=\langle 4\rangle\) in \(\mathbb{Z}_{12}\text{.}\)
They are
\begin{align*} H&=\{0,4,8\},\\ 1+H & = \{1,5,9\},\\ 2+H& =\{2,6,10\}\\ 3+H& =\{3,7,11\}. \end{align*}
We now consider the set of all left cosets of a subgroup of a group.
Definition: G mod H
We let \(G/H\) be the set of all left cosets of subgroup \(H\) in \(G\text{.}\) We read \(G/H\) as “\(G\) mod \(H\text{.}\)”
(We may denote the set of all right cosets of subgroup \(H\) in \(G\) by \(H\backslash G\text{,}\) but we will not use that notation in this class.)


1.8: Themes and Concepts of Biology - Levels of Organization of Living Things
- Describe the biological levels of organization from the smallest to highest level
Living things are highly organized and structured, following a hierarchy that can be examined on a scale from small to large. The atom is the smallest and most fundamental unit of matter. It consists of a nucleus surrounded by electrons. Atoms form molecules which are chemical structures consisting of at least two atoms held together by one or more chemical bonds.  An example of a macromolecule is deoxyribonucleic acid (DNA), which contains the instructions for the structure and functioning of all living organisms.
From Organelles to Biospheres
Macromolecules can form aggregates within a cell that are surrounded by membranes; these are called organelles. Organelles are small structures that exist within cells. Examples of these include: mitochondria and chloroplasts, which carry out indispensable functions. Mitochondria produce energy to power the cell while chloroplasts enable green plants to utilize the energy in sunlight to make sugars. All living things are made of cells, and the cell itself is the smallest fundamental unit of structure and function in living organisms. (This requirement is why viruses are not considered living: they are not made of cells. To make new viruses, they have to invade and hijack the reproductive mechanism of a living cell; only then can they obtain the materials they need to reproduce. ) Some organisms consist of a single cell and others are multicellular. Cells are classified as prokaryotic or eukaryotic. Prokaryotes are single-celled or colonial organisms that do not have membrane-bound nuclei; in contrast, the cells of eukaryotes do have membrane-bound organelles and a membrane-bound nucleus.
In larger organisms, cells combine to make tissues, which are groups of similar cells carrying out similar or related functions. Organs are collections of tissues grouped together performing a common function. Organs are present not only in animals but also in plants. An organ system is a higher level of organization that consists of functionally related organs. Mammals have many organ systems. For instance, the circulatory system transports blood through the body and to and from the lungs; it includes organs such as the heart and blood vessels. Furthermore, organisms are individual living entities. For example, each tree in a forest is an organism. Single-celled prokaryotes and single-celled eukaryotes are also considered organisms and are typically referred to as microorganisms.
All the individuals of a species living within a specific area are collectively called a population. For example, a forest may include many pine trees. All of these pine trees represent the population of pine trees in this forest. Different populations may live in the same specific area. For example, the forest with the pine trees includes populations of flowering plants and also insects and microbial populations. A community is the sum of populations inhabiting a particular area. For instance, all of the trees, flowers, insects, and other populations in a forest form the forest’s community. The forest itself is an ecosystem. An ecosystem consists of all the living things in a particular area together with the abiotic, non-living parts of that environment such as nitrogen in the soil or rain water. At the highest level of organization, the biosphere is the collection of all ecosystems, and it represents the zones of life on earth. It includes land, water, and even the atmosphere to a certain extent. Taken together, all of these levels comprise the biological levels of organization, which range from organelles to the biosphere.
Key Points
- The atom is the smallest and most fundamental unit of matter. The bonding of at least two atoms or more form molecules.
- The simplest level of organization for living things is a single organelle, which is composed of aggregates of macromolecules.
- The highest level of organization for living things is the biosphere; it encompasses all other levels.
- The biological levels of organization of living things arranged from the simplest to most complex are: organelle, cells, tissues, organs, organ systems, organisms, populations, communities, ecosystem, and biosphere.
Key Terms
- molecule : The smallest particle of a specific compound that retains the chemical properties of that compound; two or more atoms held together by chemical bonds.
- macromolecule : a very large molecule, especially used in reference to large biological polymers (e.g. nucleic acids and proteins)
- polymerization : The chemical process, normally with the aid of a catalyst, to form a polymer by bonding together multiple identical units (monomers).


5.8: Bass
The bass is the lowest male voice and has the lowest tessitura of all the voices. Cultural influence and individual variation create a wide variation in range and quality of bass singers. Parts for basses have included notes as low as the B-flat two octaves and a tone below middle C (B ♭ 1 ), for example in Gustav Mahler’s Symphony No. 2 and the Rachmaninov Vespers . Many basso profondos have trouble reaching those notes, and the use of them in works by Slavic composers has led to the colloquial term “Russian bass” for an exceptionally deep-ranged basso profondo who can easily sing those notes. Many British composers such as Benjamin Britten have written parts for bass (such as the first movement of his choral work Rejoice in the Lamb ) that center far higher than the bass tessitura implied by the clef.
This video shows Maxim Kuzmin-Karavaev singing a piece from Gioachino Rossini’s opera The Barber of Seville .
Basses are often divided into different subcategories based on range, vocal color or timbre, the weight of the voice, and dexterity of the voice. Basses are often broken down into six subcategories: basso profondo, basso buffo, bel canto bass, basso cantante, dramatic bass, and bass-baritone.
- Basso profondo— Basso profondo (lyric low bass) is the lowest bass voice type. According to J. B. Steane in Voices, Singers & Critics , the basso profondo voice “derives from a method of tone-production that eliminates the more Italian quick vibrato. In its place is a kind of tonal solidity, a wall-like front, which may nevertheless prove susceptible to the other kind of vibrato, the slow beat or dreaded wobble.” Dramatic basso profondo – English equivalent: dramatic low bass. Dramatic basso profondo is a powerful basso profondo voice.
Listen to basso profondo Mikhail Zlatopolsky sing with the Russian Choir.
- Basso buffo/bel canto/lyric buffo— Buffo , literally “funny”, basses are lyrical roles that demand from their practitioners a solid coloratura technique, a capacity for patter singing and ripe tonal qualities if they are to be brought off to maximum effect. They are usually the blustering antagonist of the hero/heroine or the comic-relief fool in bel canto operas. Equivalent terms are Schwerer Spielbass (German) and dramatic buffo (English).
- Basso cantante/lyric high bass/lyric bass-baritone— Basso cantante means “singing bass.” Basso cantante is a higher, more lyrical voice. It is produced using a more Italianate vocal production, and possesses a faster vibrato, than its closest Germanic/Anglo-Saxon equivalent, the bass-baritone.
In a SATB four-part mixed chorus, the bass is the lowest vocal range, below the soprano, alto, and tenor. Voices are subdivided into first bass and second bass, no distinction being made between bass and baritone voices, in contrast to the three-fold (tenor–baritone–bass) categorization of solo voices. The exception is in arrangements for male choir (TTBB) and barbershop quartets (TLBB), which sometimes label the lowest two parts baritone and bass.
Contributors and Attributions
- Bass (Voice type). Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Voice_type . License : CC BY-SA: Attribution-ShareAlike
- Maxim Kuzmin-Karavaev(bass)-Aria of Don Basilio(G.Rossini-Il barbiere di Siviglia). Authored by : u041au0430u043du0430u043b u043fu043eu043bu044cu0437u043eu0432u0430u0442u0435u043bu044f helgaopera. Located at : https://youtu.be/XGG33jIZ7YM?t=1s . License : All Rights Reserved . License Terms : Standard YouTube License