14.2: Labor Market Equilibrium and Wage Determinants
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Conditions of Equilibrium
Equilibrium in the labor market requires that the marginal revenue product of labor is equal to the wage rate, and that \(\mathrm{\frac{MPL}{PL}=\frac{MPK}{PK}}\).
learning objectives
- Employ the marginal decision rule to determine the equilibrium cost of labor
The labor market differs somewhat from the market for goods and services because labor demand is a derived demand; labor is not desired for its own sake but rather because it aids in producing output. Firms determine their demand for labor through a lens of profit maximization, ultimately seeking to produce the optimum level of output and the lowest possible cost.
Labor Market Equilibrium
In order to find the equilibrium quantity and price of labor, economists generally make several assumptions:
- The marginal product of labor (MPL) is decreasing;
- Firms are price-takers in the goods market (cannot affect the price of output) as well as in the labor market (cannot affect the wage rate);
- The supply of labor is elastic and increases with the wage rate (upward sloping supply); and
- Firms are profit-maximizers.
The marginal revenue product of labor (MRPL) is equal to the MPL multiplied by the price of output. The MRPL represents the additional revenue that a firm can expect to gain from employing one additional unit of labor – it is the marginal benefit to the firm from labor. Under the above assumptions, the MRPL is decreasing as the quantity of labor increases, and firms can increase profit by hiring more labor if the MRPL is greater than the marginal cost of that additional unit of labor – the wage rate. Thus, firms will hire more labor when the MRPL is greater than the wage rate, and stop hiring as soon as the two values are equal. The point at which the MRPL equals the prevailing wage rate is the labor market equilibrium.
Optimal Demand for Labor : The optimal demand for labor is located where the marginal product equals the real wage rate. The curved line represents the falling marginal product of labor, the y-axis is the marginal product/wage rate, and the x-axis is the quantity of labor.
Optimizing Capital and Labor
In the long run, firms maximize profit by choosing the optimal combination of labor and capital to produce a given amount of output. It’s possible that an automobile company could manufacture 1,000 cars using only expensive, technologically advanced robots and machinery (capital) that do not require any human participation. It’s also possible that the company could produce the same number of vehicles using only employee work (labor), without any assistance from machines or technology. For most industries, however, relying solely on capital or solely on labor is more expensive than using some combination of the two.
Factory Worker : Most firms need a combination of both labor and capital in order to produce their product.
Firms use the marginal decision rule in order to decide what combination of labor, capital, and other factors of production to use in the creation of output. The marginal decision rule says that a firm will shift spending among factors of production as long as the marginal benefit of such a shift exceeds the marginal cost. Imagine that a firm must decide whether to spend an additional dollar on labor. To determine the marginal benefit of that dollar, we divide the marginal product of labor (MP L ) by it’s price (the wage rate, P L ): MP L /P L . If capital and labor are the only factors of production, then spending an additional $1 on labor while holding the total cost constant means taking $1 out of capital. The cost of that action will be the output lost from cutting back on capital, which is the ratio of the marginal product of capital (MP K ) to the price of capital (the rental rate, P K ). Thus, the cost of cutting back on capital is MP K /P K .
If the marginal benefit of additional labor, MP L /P L , exceeds the marginal cost, MP K /PK, then the firm will be better off by spending more on labor and less on capital. On the other hand, if MP K /P K is greater than MP L /P L , the firm will be better off spending more on capital and less on labor. The equilibrium – the point at which the firm is producing the maximum amount of output at a given cost – occurs where MP L /P L =MP K /P K .
The Wage Rate
The wage rate is determined by the intersection of supply of and demand for labor.
learning objectives
- Describe the factors that determine the wage rate
When labor is an input to production, firms hire workers. Firms are demand labor and workers provide it at a price called the wage rate. Colloquially, “wages” refer to just the dollar amount paid to a worker, but in economics, it refers to total compensation (i.e. it includes benefits).
The marginal benefit of hiring an additional unit of labor is called the marginal product of labor: it is the additional revenue generated from the last unit of labor. In theory, as with other inputs to production, firms will hire workers until the wage rate (marginal cost) equals the marginal revenue product of labor (marginal benefit).
Changes in Supply and Demand
In competitive markets, the demand curve for labor is the same as the marginal revenue curve. Thus, shifts in the demand for labor are a function of changes in the marginal product of labor. This can occur for a number of reasons. First of all, you can imagine that a new product or company is created that represents new demand for labor of a certain type. There are also three main factors that would shift the labor demand curve:
- Technology which affects the output of a unit of labor.
- Changes in the price of the output which affect the value of the unit of labor.
- Changes in the price of labor relative to other factors of production.
In the long run, the supply of labor is a function of the population. A decrease in the supply of labor will typically cause an increase in the wage rate. The fact that a reduction in supply tends to strengthen wages explains why unions and other professional associations have often sought to limit the number of workers in their particular industry. Physicians, for example, have a financial incentive to enforce rigorous training, licensing, and certification requirements in order to limit the number of practitioners and keep the labor supply low.
Wage Rate in the Long Run : In the long run the supply of labor is fixed and demand is downward-sloping. The wage rate is determined by their intersection.
Compensation Differentials
Some differences in wage rates across places, occupations, and demographic groups can be explained by compensation differentials.
learning objectives
- Describe nonmonetary factors that affect wage rates
According to the basic theory of the labor market, there ought to be one equilibrium wage rate that applies to all workers across industries and countries. Of course this is not the case; doctors typically make more per hour than retail clerks, and workers in the United States typically earn a higher wage than workers in India. These wage differences are called compensation differentials and can be explained by many factors, such as differences in the skills of the workers, the country or geographical area in which jobs are performed, or the characteristics of the jobs themselves.
Education Differentials
One common source of differences in wage rates is human capital. More skilled and educated workers tend to have higher wages because their marginal product of labor tends to be higher. Additionally, the differential pay for more education tends to compensate workers for the time, effort, and foregone wages from obtaining the necessary training. If all jobs paid the same rate, for example, fewer people would go through the expense and effort of law school. The compensation differential ensures that individuals are willing to invest in their own human capital.
Education Differentials : Workers seek increased compensation by attaining higher levels of education
Geographic Compensation Differentials
If a certain part of a country is a particularly attractive area to live in and if labor mobility is perfect, then more and more workers will move to that area, which in turn will increase the supply of labor and depress wages. If the attractiveness of that area compared to other areas does not change, the wage rate will be set at such a rate that workers will be indifferent between living in areas that are more attractive but with a lower wage and living in areas which are more attractive with a higher wage. In this way, a sustained equilibrium with different wage rates across different areas can occur.
Discrimination and Compensation Differentials
In the United States, minorities and women make lower wages on average than Caucasian men. Some of this is due to historical trends affecting these groups that result in less human capital or a concentration in certain lower-paying occupations. Another source of differing wage rates, however, is discrimination. Several studies have shown that, in the United States, several minority groups (including black men and women, Hispanic men and women, and white women) suffer from decreased wage earning for the same job with the same performance levels and responsibilities as white males.
Compensating Differential
Not to be confused with a compensation differential, a compensating differential is a term used in labor economics to analyze the relation between the wage rate and the unpleasantness, risk, or other undesirable attributes of a particular job. It is defined as the additional amount of income that a given worker must be offered in order to motivate them to accept a given undesirable job, relative to other jobs that worker could perform. One can also speak of the compensating differential for an especially desirable job, or one that provides special benefits, but in this case the differential would be negative: that is, a given worker would be willing to accept a lower wage for an especially desirable job, relative to other jobs.
Hazard Differential : Hazard pay is a type of compensating differential. Occupations that are dangerous, such as police work, will typically have higher pay to compensate for the risk associated with that job.
Performance and Pay
Theoretically there is a direct connection between job performance and pay, but in reality other factors often distort this relationship.
learning objectives
- Identify the relationship between performance and wages
According to economic theory, workers’ wages are equal to the marginal revenue product of their labor. If one employee is very productive he or she will have a high marginal revenue product: one additional hour of their work will produce a significant increase in output. It follows that more productive employees should have higher wages than less productive employees. Imagine if this were not true: a firm decides to pay a highly productive worker less than the marginal revenue product of his labor. Any other firm could make a profit by offering a higher salary to attract the productive employee to their company, and the worker’s wage would rise. Theoretically, therefore, there is a direct relationship between job performance and pay.
We know that this is not always the case in reality. Wages are determined not only by one’s productivity, but also by seniority, networking, ambition, and luck. It is very rare for an entry-level worker to make the same wage as an experienced member of the same profession regardless of their relative levels of productivity because the older worker has had time to receive pay raises and promotions for which the younger employee is simply not eligible. Discrimination is sometimes responsible for members of minority racial or gender groups receiving wages that are less than wages for the majority group even when productivity levels are the same. Finally, outside forces, such as unions or government regulations, can distort pay rates.
Wages and Productivity in the U.S. : On a macroeconomic level, this graph shows the disconnect, beginning around 1975, between the productivity of labor and the wage rate in the U.S. If the economic theory were correct in the real world, wages and productivity would increase together.
Linking Performance and Pay
Some of the disconnect between performance and pay can be addressed with alternate pay schemes. While a salary or hourly pay does not directly take into account the quality of work, performance-related pay compensates workers with higher levels of productivity directly. One example is commission-based pay. In this type of pay scheme, workers receive some percentage of the profit that they generate for their company. This may be paid on top of a baseline salary or may be the only form of compensation. This type of system is very common among car salespeople and insurance brokers.
Another alternative is piece-work, in which employees are paid a fixed rate for every unit produced or action performed, regardless of the time it takes. This is common in settings where it is easy to measure the output of piece work, such as when a garment worker is paid per each piece of cloth sewn or a telemarketer is paid for every call placed.
Marginal Revenue Productivity and Wages
In a perfectly competitive market, the wage rate is equal to the marginal revenue product of labor.
learning objectives
- Explain how wages are determines by marginal revenue productivity
Just as in any market, the price of labor, the wage rate, is determined by the intersection of supply and demand. When the supply of labor increases the equilibrium price falls, and when the demand for labor increases the equilibrium price rises. In the long run the supply of labor is a simple function of the size of the population, so in order to understand changes in wage rates we focus on the demand for labor.
To determine demand in the labor market we must find the marginal revenue product of labor (MRPL), which is based on the marginal productivity of labor (MPL) and the price of output. Conceptually, the MRPL represents the additional revenue that the firm can generate by adding one additional unit of labor (recall that MPL is the additional output from the additional unit of labor). Thus, MRPL is simply the product of MPL and the price of the output.
The MPL is generally decreasing: adding a 100th unit of labor will not increase output as much as adding a 99th. Since competitive industries are price takers and cannot change the price of output by changing their level of production, the MRPL curve will have the same downward slope as the MPL curve.
From the perspective of the firm, the MRPL is the marginal benefit to the firm of hiring an additional unit of labor. We know that a profit-maximizing firm will increase its factors of production until their marginal benefit is equal to the marginal cost. Therefore, firms will continue to add labor (hire workers) until the MRPL equals the wage rate. Thus, workers earn a wage equal to the marginal revenue product of their labor. For example, in a perfectly competitive market, an employee who earns $20/hour has a marginal productivity that is worth exactly $20.
Marginal Product and Wages : The graph shows that a factor of production – in our case, labor – has a fixed supply in the long run, so the wage rate is determined by the factor demand curve – in our case, the marginal revenue product of labor. The intersection of vertical supply and the downward sloping demand gives the wage rate.
Changes in Equilibrium for Shifts in Market Supply and Market Demand
A shift in the supply or demand of labor will cause a change in the market equilibrium.
learning objectives
- Discuss the factors that influence the shape and position of the labor supply curve
As in all competitive markets, the equilibrium price and quantity of labor is determined by supply and demand.
Labor Supply
Labour supply curves are derived from the ‘labor-leisure’ trade-off. More hours worked earn higher incomes but necessitate a cut in the amount of other things workers enjoy such as going to movies, hanging out with friends, or sleeping. The opportunity cost of working is leisure time and vis versa. Considering this tradeoff, workers collectively offer a set of labor to the market which economists call the supply of labor.
To see how changes in wages affect the supply of labor, suppose wages rise. This increases the cost of leisure and causes the supply of labor to rise – this is the substitution effect , which states that as the relative price of one good increases, consumption of that good will decrease. However, there is also an income effect – an increased wage means higher income, and since leisure is a normal good, the quantity of leisure demanded will go up. In general, at low wage levels the substitution effect dominates the income effect and higher wages cause an increase in the supply of labor. At high incomes, however, the negative income effect could offset the positive substitution effect and higher wage levels could actually cause labor to decrease. A worker making $800/hour who receives a raise to $1200/hour may not have much use for the extra money and may choose to work less while maintaining the same standard of living, for example. This creates a supply curve that bends backwards, initially increasing with the wage rate but later decreasing.
Backward Bending Supply : While normally hours of labor supplied will increase with the wage rate, the income effect may produce the opposite effect at high wage levels.
People supply labor in order to increase their utility —just as they demand goods and services in order to increase their utility. The supply curve for labor will shift in response to changes in the same factors that shift demand for goods and services. These include changes in preferences, changes in income, changes in population, and changes in expectations. A change in preferences that causes people to prefer more leisure, for example, will shift the supply curve to the left, creating a lower level of employment and a higher wage rate.
Labor Demand
An increase in the demand for labor will increase both the level of employment and the wage rate. We have already seen that the demand for labor is based on the marginal product of labor and the price of output. Thus, any factor that affects productivity or output prices will also shift labor demand. Some of these factors include:
- Available technology (marginal productivity of labor)
- The skills or education of the workforce (marginal productivity of labor)
- Level of physical capital (marginal productivity of labor)
- Price of physical capital (price of output)
- Price of substitute or complement goods (price of output)
- Consumer preferences (price of output)
All of the above may cause the demand for labor to shift and change the equilibrium quantity and price of labor.
Labor Union Impacts on Equilibrium
Unions are organizations of workers that seek to improve working conditions and raise the equilibrium wage rate.
learning objectives
- Examine the role of unions and collective bargaining in labor-firm relations
A labor union is an organization of workers who have banded together to achieve common goals. The primary activity of the union is to bargain with the employer on behalf of union members and negotiate labor contracts. The most common purpose of associations or unions is maintaining or improving the conditions of employment, which may include the negotiation of wages, work rules, complaint procedures, promotions, benefits, workplace safety, and policies.
In order to achieve these goals unions engage in collective bargaining: the process of negotiation between a company’s management and a labor union. When collective bargaining fails, union members may go on strike, refusing to work until a firm addresses the workers’ grievances.
Union Impacts on Equilibrium
Fundamentally, unions seek higher wages for its member workers (though, here “wages” encompases all types of compensation, not just cash paid to the workers by the employer).
The effect of unions on the labor market equilibrium can be analyzed like any other price increase. If employers (those who demand labor) have an inelastic demand for labor, the increase in wages (the price of labor) will not translate into a drop in employment (quantity of labor supplied). If, however, their demand is elastic, employers will simply respond to union demands for higher wages by hiring fewer workers.
However, the reality of unions is more complex. As an organized body, unions are also active in the political realm. They can lobby for legislation that will affect the market not only for labor, but also for the goods they produce. For example, unions may advocate for trade restrictions to protect the markets in which they work from foreign competition. By preventing domestic firms from having to compete with unrestricted foreign firms, they can ensure that consumers do not have lower cost alternatives which would drive employers who pay a higher union wage out of business.
Union Members Strike : One tool that unions may use to raise wages is to go on strike.
Key Points
- Firms will hire more labor when the marginal revenue product of labor is greater than the wage rate, and stop hiring as soon as the two values are equal.
- The point at which the MRPL equals the prevailing wage rate is the labor market equilibrium.
- The marginal decision rule says that a firm will shift spending among factors of production as long as the marginal benefit of such a shift exceeds the marginal cost.
- If the marginal benefit of additional labor, MPL/PL, exceeds the marginal cost, MPK/PK, then the firm will be better off by spending more on labor and less on capital.
- According to the marginal decision rule, equilibrium in the labor market must occur where MPL/PL=MPK/PK.
- An increase in demand or a reduction in supply will raise wages; an increase in supply or a reduction in demand will lower them.
- The demand curve depends on the marginal product of labor and the price of the good labor produces. If the demand curve shifts to the right, either because productivity or the price of output has increased, wages will be pushed up.
- In the long run the supply of labor is simply a function of the population size, but in the short run it depends on variables such as worker preferences, the skills and training a job requires, and wages available in alternative occupations.
- Although basic economic theory suggests that there ought to be one prevailing wage rate for all labor, this is not the case.
- Wage differences are called compensation differentials and can be explained by many factors, such as differences in the skills of the workers, the country or geographical area in which jobs are performed, or the characteristics of the jobs themselves.
- One common source of differences in wage rates is human capital. More skilled and educated workers tend to have higher wages because their marginal product of labor tends to be higher.
- If a certain area is a desirable place to live, the supply of labor will be higher than in other areas and wages will be lower. This is a type of geographical differential.
- Discrimination against gender or racial groups can cause compensation differentials.
- A compensating differential is the additional amount of income that a given worker must be offered in order to motivate them to accept a given undesirable job, relative to other jobs that worker could perform.
- According to economic theory, workers’ wages are equal to the marginal revenue product of their labor. If one employee is very productive he or she will have a high marginal revenue product.
- In reality, wages are determined not only by one’s productivity, but also by seniority, networking, ambition, and luck.
- Some of the disconnect between performance and pay can be addressed with alternate pay schemes.
- In the long run the supply of labor is a simple function of the size of the population, so in order to understand changes in wage rates we focus on the demand for labor.
- The marginal product of labor (MPL) is the increase in output that a firm experiences from adding one additional unit of labor.
- The marginal benefit to the firm of hiring an additional unit of labor is called the marginal revenue product of labor (MRPL). It is calculated by multiplying MPL by the price of the output.
- The MRPL represents the firm’s demand curve for labor, which means that the firm will continue to hire more labor until the MRPL is equal to the wage rate.
- The opportunity cost of leisure is the wages lost while not working; as wages rise, the cost of leisure increases.
- The substitution effect means that when wages rise, people are likely to substitute more labor for less leisure.
- However, the income effect means that as people become wealthier, their demand for normal goods such as leisure increases.
- Typically the substitution effect dominates the supply of labor at normal wage rates, but the income effect may come to dominate at higher wage rates. This creates a backward bending labor supply curve.
- The supply curve for labor will shift in response to changes in preferences, changes in income, changes in population, and changes in expectations.
- The demand curve for labor will shift in response to changes in human capital, changes in technology, changes in the price of complements or substitutes for output, and changes in consumer preferences.
- Unions ‘ primary work involves negotiating wages, work rules, complaint procedures, promotions, benefits, workplace safety and policies with company management.
- If the labor market is a competitive one in which wages are determined by demand and supply, increasing the wage requires either increasing the demand for labor or reducing the supply.
- Increasing demand for labor requires increasing the marginal product of labor or raising the price of the good produced by labor.
- Unions can restrict the supply of labor in two ways: slowing the growth of the labor force and promoting policies that make it difficult for workers to enter a particular craft.
Key Terms
- marginal product : The extra output that can be produced by using one more unit of the input.
- marginal revenue product : The change in total revenue earned by a firm that results from employing one more unit of labor.
- capital : Already-produced durable goods available for use as a factor of production, such as steam shovels (equipment) and office buildings (structures).
- Union : an organization of workers who have banded together to achieve common goals
- differential : a qualitative or quantitative difference between similar or comparable things
- discrimination : Distinct treatment of an individual or group to their disadvantage; treatment or consideration based on class or category rather than individual merit; partiality; prejudice; bigotry.
- commission : A fee charged by an agent or broker for carrying out a transaction
- piece work : Work that a worker is paid for according to the number of units produced, rather than the number of hours worked.
- marginal benefit : The extra benefit received from a small increase in the consumption of a good or service. It is calculated as the increase in total benefit divided by the increase in consumption.
- marginal revenue product : The change in total revenue earned by a firm that results from employing one more unit of labor.
- normal good : A good for which demand increases when income increases and falls when income decreases but price remains constant.
- Opportunity cost : The cost of any activity measured in terms of the value of the next best alternative forgone (that is not chosen).
- collective bargaining : A method of negotiation in which employees negotiate as a group with their employers.
- strike : A work stoppage (or otherwise concerted stoppage of an activity) as a form of protest.
- minimum wage : The lowest rate at which an employer can legally pay an employee; usually expressed as pay per hour.
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- Opportunity cost. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/Opportunity+cost. License : CC BY-SA: Attribution-ShareAlike
- Fator. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Fator.jpg. License : CC BY-SA: Attribution-ShareAlike
- All sizes | Solar wafer manufacturing | Flickr - Photo Sharing!. Provided by : Flickr. Located at : www.flickr.com/photos/oregond...n/photostream/. License : CC BY: Attribution
- Factor compensation. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Fa...mpensation.jpg. License : CC BY-SA: Attribution-ShareAlike
- Police Poland 2 AB. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Po...oland_2_AB.jpg. License : Public Domain: No Known Copyright
- Johns Hopkins University on Fotopedia. Provided by : Fotopedia. Located at : http://www.fotopedia.com/wiki/Johns_...ckr-4647211575 . License : CC BY: Attribution
- US productivity and real wages. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...real_wages.jpg. License : Public Domain: No Known Copyright
- Factor compensation. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Fa...mpensation.jpg. License : CC BY-SA: Attribution-ShareAlike
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- collective bargaining. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/collective_bargaining. License : CC BY-SA: Attribution-ShareAlike
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- Trade union. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Trade_union. License : CC BY-SA: Attribution-ShareAlike
- strike. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/strike. License : CC BY-SA: Attribution-ShareAlike
- minimum wage. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/minimum_wage. License : CC BY-SA: Attribution-ShareAlike
- Fator. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Fator.jpg. License : CC BY-SA: Attribution-ShareAlike
- All sizes | Solar wafer manufacturing | Flickr - Photo Sharing!. Provided by : Flickr. Located at : http://www.flickr.com/photos/oregond...n/photostream/ . License : CC BY: Attribution
- Factor compensation. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Fa...mpensation.jpg. License : CC BY-SA: Attribution-ShareAlike
- Police Poland 2 AB. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Po...oland_2_AB.jpg. License : Public Domain: No Known Copyright
- Johns Hopkins University on Fotopedia. Provided by : Fotopedia. Located at : http://www.fotopedia.com/wiki/Johns_...ckr-4647211575 . License : CC BY: Attribution
- US productivity and real wages. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...real_wages.jpg. License : Public Domain: No Known Copyright
- Factor compensation. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Fa...mpensation.jpg. License : CC BY-SA: Attribution-ShareAlike
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2.1: Prelude to The Kinetic Theory of Gases
As we discussed in the preceding chapter, the study of heat and temperature is part of an area of physics known as thermodynamics, in which we require a system to be macroscopic, that is, to consist of a huge number (such as \(10^{23}\)) of molecules. We begin by considering some macroscopic properties of gases: volume, pressure, and temperature. The simple model of a hypothetical “ideal gas” describes these properties of a gas very accurately under many conditions. We move from the ideal gas model to a more widely applicable approximation, called the Van der Waals model.
Gases are literally all around us—the air that we breathe is a mixture of gases. Other gases include those that make breads and cakes soft, those that make drinks fizzy, and those that burn to heat many homes. Engines and refrigerators depend on the behaviors of gases, as we will see in later chapters.
To understand gases even better, we must also look at them on the microscopic scale of molecules. In gases, the molecules interact weakly, so the microscopic behavior of gases is relatively simple, and they serve as a good introduction to systems of many molecules. The molecular model of gases is called the kinetic theory of gases and is one of the classic examples of a molecular model that explains everyday behavior.


5.2: Introduction to Classical Composers
We’re going to take a different approach to our study of Classical composers than we did with earlier periods. Rather than study a chronological progression of composers who contributed to the evolution of Classical style, we’ll examine in greater depth the lives and music of the composers whose work represents the highest achievements of the era. This section contains the readings on the three composers we study from the Classical era: Haydn, Mozart, and Beethoven.
This section includes the following pages:
- Slide Show: Classical Composers
- Franz Joseph Haydn
- Wolfgang Amadeus Mozart
- Ludwig van Beethoven
- Symphony No. 5 in C minor
Contributors and Attributions
CC licensed content, Original
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Provided by : Lumen Learning. License : CC BY: Attribution


13.2: Solvable Groups
A subnormal series of a group \(G\) is a finite sequence of subgroups
\[ G = H_n \supset H_{n-1} \supset \cdots \supset H_1 \supset H_0 = \{ e \}\text{,} \nonumber \]
where \(H_i\) is a normal subgroup of \(H_{i+1}\text{.}\) If each subgroup \(H_i\) is normal in \(G\text{,}\) then the series is called a normal series . The length of a subnormal or normal series is the number of proper inclusions.
Example \(13.11\)
Any series of subgroups of an abelian group is a normal series.
Solution
Consider the following series of groups:
\begin{gather*} {\mathbb Z} \supset 9{\mathbb Z} \supset 45{\mathbb Z} \supset 180{\mathbb Z} \supset \{0\},\\ {\mathbb Z}_{24} \supset \langle 2 \rangle \supset \langle 6 \rangle \supset \langle 12 \rangle \supset \{0\}\text{.} \end{gather*}
Example \(13.12\)
A subnormal series need not be a normal series. Consider the following subnormal series of the group \(D_4\text{:}\)
\[ D_4 \supset \{ (1), (1 \, 2)(3 \, 4), (1 \, 3)(2 \, 4), (1 \, 4)(2 \, 3) \} \supset \{ (1), (1 \, 2)(3 \, 4) \} \supset \{ (1) \}\text{.} \nonumber \]
Solution
The subgroup \(\{ (1), (1 \, 2)(3 \, 4) \}\) is not normal in \(D_4\text{;}\) consequently, this series is not a normal series.
A subnormal (normal) series \(\{ K_j \}\) is a refinement of a subnormal (normal) series \(\{ H_i \}\) if \(\{ H_i \} \subset \{ K_j \}\text{.}\) That is, each \(H_i\) is one of the \(K_j\text{.}\)
Example \(13.13\)
The series
\[ {\mathbb Z} \supset 3{\mathbb Z} \supset 9{\mathbb Z} \supset 45{\mathbb Z} \supset 90{\mathbb Z} \supset 180{\mathbb Z} \supset \{0\} \nonumber \]
Solution
is a refinement of the series
\[ {\mathbb Z} \supset 9{\mathbb Z} \supset 45{\mathbb Z} \supset 180{\mathbb Z} \supset \{0\}\text{.} \nonumber \]
The best way to study a subnormal or normal series of subgroups, \(\{ H_i \}\) of \(G\text{,}\) is actually to study the factor groups \(H_{i+1}/H_i\text{.}\) We say that two subnormal (normal) series \(\{H_i \}\) and \(\{ K_j \}\) of a group \(G\) are isomorphic if there is a one-to-one correspondence between the collections of factor groups \(\{H_{i+1}/H_i \}\) and \(\{ K_{j+1}/ K_j \}\text{.}\)
Example \(13.14\)
The two normal series
\begin{gather*} {\mathbb Z}_{60} \supset \langle 3 \rangle \supset \langle 15 \rangle \supset \{ 0 \}\\ {\mathbb Z}_{60} \supset \langle 4 \rangle \supset \langle 20 \rangle \supset \{ 0 \} \end{gather*}
of the group \({\mathbb Z}_{60}\) are isomorphic
Solution
since
\begin{gather*} {\mathbb Z}_{60} / \langle 3 \rangle \cong \langle 20 \rangle / \{ 0 \} \cong {\mathbb Z}_{3}\\ \langle 3 \rangle / \langle 15 \rangle \cong \langle 4 \rangle / \langle 20 \rangle \cong {\mathbb Z}_{5}\\ \langle 15 \rangle / \{ 0 \} \cong {\mathbb Z}_{60} / \langle 4 \rangle \cong {\mathbb Z}_4\text{.} \end{gather*}
A subnormal series \(\{ H_i \}\) of a group \(G\) is a composition series if all the factor groups are simple; that is, if none of the factor groups of the series contains a normal subgroup. A normal series \(\{ H_i \}\) of \(G\) is a principal series if all the factor groups are simple.
Example \(13.15\)
The group \({\mathbb Z}_{60}\) has a composition series
\[ {\mathbb Z}_{60} \supset \langle 3 \rangle \supset \langle 15 \rangle \supset \langle 30 \rangle \supset \{ 0 \} \nonumber \]
with factor groups
\begin{align*} {\mathbb Z}_{60} / \langle 3 \rangle & \cong {\mathbb Z}_{3}\\ \langle 3 \rangle / \langle 15 \rangle & \cong {\mathbb Z}_{5}\\ \langle 15 \rangle / \langle 30 \rangle & \cong {\mathbb Z}_{2}\\ \langle 30 \rangle / \{ 0 \} & \cong {\mathbb Z}_2\text{.} \end{align*}
Solution
Since \({\mathbb Z}_{60}\) is an abelian group, this series is automatically a principal series. Notice that a composition series need not be unique. The series
\[ {\mathbb Z}_{60} \supset \langle 2 \rangle \supset \langle 4 \rangle \supset \langle 20 \rangle \supset \{ 0 \} \nonumber \]
is also a composition series.
Example \(13.16\)
For \(n \geq 5\text{,}\) the series
\[ S_n \supset A_n \supset \{ (1) \} \nonumber \]
Solution
is a composition series for \(S_n\) since \(S_n / A_n \cong {\mathbb Z}_2\) and \(A_n\) is simple.
Example \(13.17\)
Not every group has a composition series or a principal series. Suppose that
\[ \{ 0 \} = H_0 \subset H_1 \subset \cdots \subset H_{n-1} \subset H_n = {\mathbb Z} \nonumber \]
is a subnormal series for the integers under addition.
Solution
Then \(H_1\) must be of the form \(k {\mathbb Z}\) for some \(k \in {\mathbb N}\text{.}\) In this case \(H_1 / H_0 \cong k {\mathbb Z}\) is an infinite cyclic group with many nontrivial proper normal subgroups.
Although composition series need not be unique as in the case of \({\mathbb Z}_{60}\text{,}\) it turns out that any two composition series are related. The factor groups of the two composition series for \({\mathbb Z}_{60}\) are \({\mathbb Z}_2\text{,}\) \({\mathbb Z}_2\text{,}\) \({\mathbb Z}_3\text{,}\) and \({\mathbb Z}_5\text{;}\) that is, the two composition series are isomorphic. The Jordan-Hölder Theorem says that this is always the case.
Theorem \(13.18\). Jordan-Hölder
Any two composition series of \(G\) are isomorphic.
- Proof
-
We shall employ mathematical induction on the length of the composition series. If the length of a composition series is 1, then \(G\) must be a simple group. In this case any two composition series are isomorphic.
Suppose now that the theorem is true for all groups having a composition series of length \(k\text{,}\) where \(1 \leq k \lt n\text{.}\) Let
\begin{gather*} G = H_n \supset H_{n-1} \supset \cdots \supset H_1 \supset H_0 = \{ e \}\\ G = K_m \supset K_{m-1} \supset \cdots \supset K_1 \supset K_0 = \{ e \} \end{gather*}
be two composition series for \(G\text{.}\) We can form two new subnormal series for \(G\) since \(H_i \cap K_{m-1}\) is normal in \(H_{i+1} \cap K_{m-1}\) and \(K_j \cap H_{n-1}\) is normal in \(K_{j+1} \cap H_{n-1}\text{:}\)
\begin{gather*} G = H_n \supset H_{n-1} \supset H_{n-1} \cap K_{m-1} \supset \cdots \supset H_0 \cap K_{m-1} = \{ e \}\\ G = K_m \supset K_{m-1} \supset K_{m-1} \cap H_{n-1} \supset \cdots \supset K_0 \cap H_{n-1} = \{ e \}\text{.} \end{gather*}
Since \(H_i \cap K_{m-1}\) is normal in \(H_{i+1} \cap K_{m-1}\text{,}\) the Second Isomorphism Theorem (Theorem \(11.12\)) implies that
\begin{align*} (H_{i+1} \cap K_{m-1}) / (H_i \cap K_{m-1}) & = (H_{i+1} \cap K_{m-1}) / (H_i \cap ( H_{i+1} \cap K_{m-1} ))\\ & \cong H_i (H_{i+1} \cap K_{m-1})/ H_i\text{,} \end{align*}
where \(H_i\) is normal in \(H_i (H_{i+1} \cap K_{m-1})\text{.}\) Since \(\{ H_i \}\) is a composition series, \(H_{i+1} / H_i\) must be simple; consequently, \(H_i (H_{i+1} \cap K_{m-1})/ H_i\) is either \(H_{i+1}/H_i\) or \(H_i/H_i\text{.}\) That is, \(H_i (H_{i+1} \cap K_{m-1})\) must be either \(H_i\) or \(H_{i+1}\text{.}\) Removing any nonproper inclusions from the series
\[ H_{n-1} \supset H_{n-1} \cap K_{m-1} \supset \cdots \supset H_0 \cap K_{m-1} = \{ e \}\text{,} \nonumber \]
we have a composition series for \(H_{n-1}\text{.}\) Our induction hypothesis says that this series must be equivalent to the composition series
\[ H_{n-1} \supset \cdots \supset H_1 \supset H_0 = \{ e \}\text{.} \nonumber \]
Hence, the composition series
\[ G = H_n \supset H_{n-1} \supset \cdots \supset H_1 \supset H_0 = \{ e \} \nonumber \]
and
\[ G = H_n \supset H_{n-1} \supset H_{n-1} \cap K_{m-1} \supset \cdots \supset H_0 \cap K_{m-1} = \{ e \} \nonumber \]
are equivalent. If \(H_{n-1} = K_{m-1}\text{,}\) then the composition series \(\{H_i \}\) and \(\{ K_j \}\) are equivalent and we are done; otherwise, \(H_{n-1} K_{m-1}\) is a normal subgroup of \(G\) properly containing \(H_{n-1}\text{.}\) In this case \(H_{n-1} K_{m-1} = G\) and we can apply the Second Isomorphism Theorem once again; that is,
\[ K_{m-1} / (K_{m-1} \cap H_{n-1}) \cong (H_{n-1} K_{m-1}) / H_{n-1} = G/H_{n-1}\text{.} \nonumber \]
Therefore,
\[ G = H_n \supset H_{n-1} \supset H_{n-1} \cap K_{m-1} \supset \cdots \supset H_0 \cap K_{m-1} = \{ e \} \nonumber \]
and
\[ G = K_m \supset K_{m-1} \supset K_{m-1} \cap H_{n-1} \supset \cdots \supset K_0 \cap H_{n-1} = \{ e \} \nonumber \]
are equivalent and the proof of the theorem is complete.
A group \(G\) is solvable if it has a subnormal series \(\{ H_i \}\) such that all of the factor groups \(H_{i+1} / H_i\) are abelian. Solvable groups will play a fundamental role when we study Galois theory and the solution of polynomial equations.
Example \(13.19\)
The group \(S_4\) is solvable since
\[ S_4 \supset A_4 \supset \{ (1), (1 \, 2)(3 \, 4), (1 \, 3)(2 \, 4), (1 \, 4)(2 \, 3) \} \supset \{ (1) \} \nonumber \]
has abelian factor groups;
Solution
however, for \(n \geq 5\) the series
\[ S_n \supset A_n \supset \{ (1) \} \nonumber \]
is a composition series for \(S_n\) with a nonabelian factor group. Therefore, \(S_n\) is not a solvable group for \(n \geq 5\text{.}\)


18.6: Skills Needed for MTI
- No matter what method is used, what skills do you need to successfully manage technology and innovation?
There are a number of skills needed to successfully manage technology and innovation in the organization. No matter what organization you are a part of, there are two skills the organization must develop to be successful—the ability to manage learning and knowledge processes, and the ability to analyze and forecast future trends. Individual skills that are critical to the organization’s success include leadership/followership and creative thinking.
Organizational skills involve how the firm puts people and resources together to create value—building capabilities. With the right capabilities, the organization can develop a competitive advantage. In the world of technology and innovation, the management of learning and knowledge processes is critical. The organization needs to have systems in place that allow it to collect data that can be analyzed to form information. The information needs to be used to gain knowledge and insight. At each step, learning takes place. Organizational learning is the acquisition of knowledge through the collection of data that is analyzed to gather information, which is then transferred and shared through communication among members of the organization. This communication process provides the foundation for knowledge acquisition and enhancement within the firm. There are two types of knowledge that must be managed: explicit knowledge (codified or written down as rules or guidelines) and tacit knowledge (which emerges from the experience of an individual). Tacit knowledge can become explicit at some point if the expert is able to codify the knowledge for others. However, it is not always possible to codify tacit knowledge. For example, Henry Bessemer was sued by the patent purchasers who could not get his steel-making process to work. In the end, Bessemer set up his own steel company because he knew how to gauge when to add and subtract heat based on the impurities in the iron ore, even though he could not convey it to his patent users. Bessemer’s company became one of the largest in the world and changed the face of steelmaking. After the introduction of the Bessemer process, steel and wrought iron became similarly priced, and some users, primarily railroads, turned to steel. 7 The insights and experiences that are gained from the gathering of data and converting that data into information are key to successful MTI. Organizational knowledge is the sharing and utilization of the learning that takes place in the firm.
The ability to forecast the future is another key organizational skill in the management of technology and innovation. This involves scanning the environment for trends and possible areas of value-creation opportunities. It also involves understanding the risk involved with newness in the firm and the risk involved in not seeking newness—both can cause the firm to lose value. Any method of forecasting comes with limitations. These include:
- Forecasting methods, by definition, are uncertain in their outcomes. Usually the firm is trying to develop scenarios concerning best, worst, and most likely outcomes. With this information, risk can be assessed.
- Forecasts are imperfect—the firm cannot predict all potential influences in the competitive marketplace. Bessemer knew he had a better process, but he did not predict the problems he had licensing his patent.
- Forecasts are at best an educated guess. Many forecast techniques rely on statistical analysis, but the numbers used in the analysis are forecasts themselves or rely on patterns of behavior continuing in the marketplace.
- With all the issues with forecasting, a company that produces excellent forecasts will most likely formulate better strategy and capture more value.
The knowledge-management system of the firm can help the ability of the firm to forecast. Experience and learning about industry and general environment trends can help individuals and teams forecast more accurately.
Individuals within the firm also need to have certain skills to enhance the management of technology and innovation processes. These skills include a balance of leadership and followership and the ability to think creatively.
Most individuals in the organization understand what leadership is. For MTI, it is important that the right person be in the leadership position when needed. For example, in new product development, the leader during the design phase is likely to be an engineer, the leader in the prototype development phase may be an engineer or a production person, and as the product is introduced to the marketplace the leader may be a marketing person. It is necessary for these individuals to communicate, and they may all be on a project team that is under the direction/coordination of a dedicated project manager. However, the leadership on the project shifts within the creation-to-market process. While leadership is critical, so is followership. Followership is the mirror image of leadership. Most will never have taken a class in followership. You cannot have leaders without followers. There is a skill set for leadership and a skill set for followership. It is the actions of followers that determine the success of a leader. The success of organizations is more the result of good followership than of great leadership. Leadership is influencing others, and followership is seeking or accepting influence. In the case of new product development outlined above, each of the individuals were leaders during some point in the project and each were followers during the project. Individuals spend a lot of time seeking and learning about leadership, but followership is also critical to organizational success. Innovative companies are often lead by a combination of two individuals who lead and follow each other. For example, Microsoft was founded by Bill Gates and Paul Allen. The names of firms started or built by two people are common: Sears & Roebuck, Proctor & Gamble, Marks, and Spencer. The characteristics of a good follower include:
- They are truthful.
- They are supportive. Don’t blame your boss for an unpopular decision or policy. “I know this is an unpopular decision, but…” Absent person example of trust. [I call it confessing the sins of the boss in the hallway after the meeting.]
- They give the boss the benefit of their knowledge and experience. Your job is to make the organization successful.
- They take the initiative to solve problems by providing solutions, not just issues.
- They keep the leader Informed. The higher a manager is in an organization, the more people are less inclined to talk openly with them. Great followers provide the good, the bad, and the ugly of information, knowledge, and experience.
- What is organizational learning?
- What are the differences between leadership and followership?
- What forecasting techniques are used in the management of technology and innovation?


4.2: Pleasure
What does the term pleasure mean to you? It’s a good thing right? Embracing embodied living and being present in our mind and bodies in order to tap into our senses. Pleasure, a feeling of happy satisfaction and enjoyment, heals. It is something we live for whether we know it or not. Every human seeks pleasure in one way or another, so why is so much of it taboo in our culture? Let’s unpack this taboo a little bit and then put it away so we can get back to what’s important, pleasure! As we discuss pleasure in general, it is important to note that while all people seek pleasure, we all have varying interest in sexual activity either with others or alone at different points throughout our lives. Sometimes the term Asexual can be used to describe a lack of interest in sexual activity and people who are asexual may self-identify as Ace .  All people also have desires and seek pleasure; it just may be expressed in ways that differ from how you as an individual define what is pleasurable. The desire for pleasure is universal. Whether it be embodied or in our minds, a connection to our sensual selves helps us tap into this vital piece of our humanity. Sensuality, the notion of being highly tuned into your senses, is an important way to discover what it is you as an individual like. Paying attention to your senses rather than letting standards or norms guide our expressions of gender or sexuality will allow greater satisfaction. Pleasure is deeply connected, but not limited to our sexuality, and sexual health is an important part of overall well-being. The World Health Organization (WHO) considers our sexual health part of our overall quality of life (The World Health Organization Quality of Life (WHOQOL), 2012). Whether it be in the form of accessibility to sex education, reproductive rights and health care, or the right to openly express our sexual identity exactly as it is this very moment all add to our quality of life. Having desires and fulfilling them is part of our comprehensive wellness.
In terms of our sexual gratification, there is a spectrum of what people do to feel good. Whether it be auto-erotic (sexual pleasure with one’s self), or with another person or persons, there is no right way to do it and what feels good to one person may not to another. As long as all parties consent to it, have fun. In many ways consent is complicated rather than a simple yes or no binary as many perceive. The ability to consent replicates the larger social structure. Think about how age, gender, cultural background, body type, race, experiences with trauma and social class afford us different levels of privilege. Consent becomes complicated by all of this. It is complicated by coercion, power, and privilege of those who may or may not consent to something authenticity as a result so a one size fits all definition of consent is difficult. The following clip describes consent in a humorous way. Tea Consent . If consent is to be authentically achieved, there must be open, honest communication and a sense of trust. Consent is ongoing and needs to be revisited. A yes now may be a no in 20 minutes so keep talking. The goal is for consent to be what is known as yes means yes or enthusiastic consent. Flipping this script helps add clarity to what people actually want. Remember, communication is lubrication so communicate about what you like, what you don’t like, and what you want to try. Please also try to let go of societal messaging about how sexuality should look, sound or feel. Getting caught up in cultural standards of beauty or how the act of sex is supposed to look can make us feel like we don’t get to enjoy ourselves until something changes (we lose 5lbs, or learn how to come in a certain position, etc.). The work of Sex Educator, Emily Nagoski influences the ways in which we in this text hope to frame sexuality and self-acceptance. Her 2015 book, Come As You Are, debunks many of the societal messaging around sexuality, gender and offers an empowering alternative.
We all grew up hearing contradictory messages about sex, and so now many of us experience ambivalence about it. That’s normal. The more aware you are of those contradictory messages, the more choice you have about whether to believe them (Nagoski, 2015, p 202).
You are deserving of pleasure right now! You get to define what that pleasure looks like and make choices that you want to make, instead of what you think society values. “When you embrace your sexuality precisely as it is right now, that’s the context that creates the greatest potential for ecstatic pleasure” (Nagoski, 2015 p. 8).
A person who identifies as asexual might be commonly referred to as “Ace” or “Aces”. Asexuality is not defined as abstinence as a result of a bad relationship, celibacy, fear of intimacy, or other commonly referenced stereotypes. Rather, asexuality can be many things, but people who identify as asexual can still fall in love, choose to masturbate, choose to engage in sexual activities, have a spouse or partners(s)! It is important to note that love does not always equal sex, and that sexuality exists on a spectrum. Asexuality, or for that matter, any other sexual identity, is not fixed, nor is it expressed the same for every person who identifies as asexual (Chen, 2020). Asexuality as an orientation is not a single way of orienting towards sex, it is not a catch all, and it can be misinterpreted or misused, so it is best to not apply labels to others, and instead, allow expression and labels to be up to individuals if they chose. For a deeper dive into understanding Asexuality, here are three resources, Yasmin Benoit, Asexuals Need Media Representation , The Amazing Aces . and Season 8, Episode 15 of the Gender Reveal Podcast : Gender Reveal Podcast with Ev'Yan Whitney on the superpower of being an asexual sex educator , which offers first person insight from an asexual sex educator.
A Guide to Fulfillment: The Kama Sutra
None more famously referenced when talking about pleasure and sexuality is the ancient Sanskrit text, The Kama Sutra . While its actual date of publication is up for debate, it is likely somewhere between 200-400 BCE (Doniger, 2003 p. i). It is a guide to life, in which they include sections regarding sexuality, eroticism, and emotional fulfillment.  Its focus is broader, written as a spiritual manual to the art of living well, the nature of love, finding a life partner, maintaining one's love life, and other aspects pertaining to pleasure-oriented faculties of human life.
The Erotic
“I want to live the rest of my life, however long or short, with as much sweetness as I can decently manage, loving all the people I love, and doing as much as I can of the work I still have to do. I am going to write fire until it comes out of my ears, my eyes, my nose holes--everywhere. Until it's every breath I breathe.  - Audre Lorde
Audre Lorde (1934-1992), represented many things to many people. Black, poet, womanist, feminist, writer and scholar, and activist whose work included pleasure. Her use of the term Erotic was meant to describe a state of being alive, a feeling one gets when they are tapping into something that provides pleasure. In some cases, the erotic has sexual connotations, but it can also refer to a really good feeling a person experiences when tapping into their joy, like feeling the warm sun on their skin or indulging in something they really like. Lorde pointed out that if we were to really feel deeply and sensually, we would begin to understand what joy we are capable of feeling and live our lives in accordance. Can you imagine?
Stigma: How Culture Constrains Sexual Liberation
It is not hard to understand why we are socialized to not seek pleasure. Religion plays a role in decentering pleasure. Each nation worldwide has a religious foundation that shapes the values and beliefs held. This foundation is the basis for culture. Those raised with religion in their homes may be taught the notion of sin. From an early age, through various agents of socialization like, family, schools, the media and the church, we learn that there is a great deal of stigma surrounding sexual pleasure. As mentioned early on in this book, nations that offer comprehensive and sex-positive sex education from a young age, built intentionally into K-12 curriculum, have lower unintended pregnancy rates and overall comfort levels about communicating about sex. In the United States especially, where White Anglo-Saxon Protestant (WASP) ethics shape culture, this is not the case. Instead, the focus is more about denying pleasure and shaming sexuality. Lust , a strong passion or longing for something especially related to sexual desire, is one of the seven deadly sins written about in the Christian Bible. Masturbation is labeled a sinful act, and when young children are “caught” exploring their bodies, they are often shamed for it. Rather than seeing the moment as an opportunity to educate our kids about their bodies, parents usually mishandle the opportunity, and kids come away from the experience feeling like sexual pleasure is something they should hide and be ashamed of. This cultural message about sexual pleasure gets internalized at a very young age, and we quickly learn that sex is something that we are not supposed to talk about and depending on our gender identity, we are also not supposed to ask for it, desire it, or be in charge of it, unless we are trying to procreate. Those gendered female and non-binary are not usually allowed to initiate sex, want sex, or control the act of sex. Our bodies are seen as something to be ashamed of. Medieval anatomists called female external genitalia the pudendum which is derived from the Latin, pudere, meaning to make ashamed (Nagoski, 2015, p 18). Hysteria , another Latin term commonly used to describe exaggerated or uncontrollable emotion or excitement, literally translates to uterus. And then we have hedonism, the belief that pleasure is the most important thing in life which is derived from the Greek word, delight. Society often juxtaposes hedonism , the pursuit of pleasure or sensual self-indulgence, with discipline and inhibiting one’s desires, and devalues the notion that as humans we deserve the right to follow our desires.
In U.S. culture hard work, material gain, and goal achievement are values that supersede personal desire. Pleasure is not quantifiable in the capitalist sense, and so it becomes more of an extra than something we espouse as a necessity. As a result, we receive mixed messages about sex, pleasure and fulfillment. Knowing that there is a cultural hang up with accessing pleasure can help us navigate the messy waters of sexuality and society and inform our decisions. Understanding this contradiction gives us the ability to question and debunk myths around pleasure, self-care and overall well-being. Hang-ups about our bodies or what we like sexually are entirely shaped by culture, and the sooner we move past self-criticism into self-love, the better. The ability to enjoy your sexuality in this very moment rests on what you are telling yourself about yourself.
You, too, are healthy and normal at the start of your sexual development, as you grow, and as you bear the fruits of living with confidence and joy inside your body. You are healthy when you need lots of sun, and you’re healthy when you enjoy some shade. That’s the true story. We are all the same. We are all different. We are all normal (Nagoski, 2015 pp. 7).
Sometimes the confines of culture clouds our ability to love ourselves exactly how we are. We may also experience trauma related to our sexuality that needs to be processed in order to get to a place of radical self-acceptance.  We all deserve the right to feel good.  A world full of people who took care of their pleasure as a regular part of their overall wellness would look very different than the one we currently live in. Undoing all the cultural “no’s” we've been told is not easy, and those “no’s” may have prevented us from learning more about our amazing pleasure seeking selves. A really good resource for questions about sexuality and tapping into desire is, Meet Lindsey Doe! - Welcome to Sexplanations - 1 . Check it out!
As you read that last section, what is coming up for you? Are you buying it? Do you deserve to feel pleasure? If not, stop and ask yourself why. Quickly rule out that “why”, and move along to what it is you desire. Tap into the Erotic. Make a list of 5 things that bring you joy. Any kind of joy. Put it somewhere you can see. Each day try to give yourself at least one of those things. More is always better so extra credit if you can give yourself all five.
Pleasure and Social Justice
What does agency over one's body mean exactly? It seems pretty straightforward and is something we imagine all humans have a right to. Historical legacies of colonization, slavery, imprisonment, sexism and homophobia diminish agency for those impacted by these injustices. The impacts of these are still there for many, and so it begs the question, what does pleasure look like if you are navigating a world that constantly has a target on your back?
These impacts are felt more severely for already marginalized groups particularly because of this historical legacy. For Black women and girls in particular, their physical bodies have always been used as a way to maintain a racial hierarchy long after emancipation and the end of Jim Crow segregation.  A 2017 Georgetown University study found that,
Black girls as young as 5 years old are already seen as less innocent and in need of less support than white girls of the same age. This presumption leads teachers and other authority figures to treat Black girls as older than they actually are and more harshly than white female students, with the disparity being particularly wide for 10-to 14-year-olds (Epstein et al, 2017).
From a young age then, these types of inappropriate attributions leave Black women and girls at a disadvantage. Based on nothing more than ignorance, this myth removes agency and inappropriately sexualizes young children. As indicated in this study, a perception of an individual's sexuality is something that has farther reaching consequences than access to pleasure. Sometimes, as in the case of Black women and girls, their bodies are weaponized against them, diminishing overall life satisfaction and opportunity.
Those who create these falsehoods and myths and present them as facts are unfortunately often the same experts we rely on to help guide us in our understanding of human sexuality. Prominent sex educators make assertions about sexuality, health, and normalcy from a very specific frame. Overwhelmingly, sex researchers tend to be cis gendered, white, heterosexual, and male, and may or may not have an understanding of the diversity inherent in human sexuality. Yet these are the people defining cultural standards of what are normative sexual attitudes or behavior. Not everyone identifies with these standards. Even as this book is being written, it is important to note that it is situated in a specific time and place. Although here we undertake a cultural critique of the narrow definition of “normal” sexuality, this too will shift as time passes. As the field of sex educators becomes more diverse and representative of the many demographics present in the culture, the understanding will widen to be more inclusive and more accurate.
I’ve spent the majority of my adult life participating in hookup culture, and until recently, I did not experience any issues with regard to pleasure, sexual arousal and sexual responses. When I reached my mid-twenties, I developed a realization that I did not enjoy sex as much as I did in my early years as an adult, and it became more difficult to become aroused. Hookup apps like Grindr, Tinder, and other web-based services can allow users to experience sexual freedom, however, can be addictive, and (as it relates to me) might alter the user’s ability to engage in intimacy due to the anonymous nature of sex apps.
 As I became more familiar with Grindr, I began using the app for what most people believe to be its purpose, which is sex. After a few years, I began dating men and realized I was experiencing difficulties reaching orgasms with the people I dated. Once I would break up with my partner, I would quickly get back on the Grindr train and would be able to easily experience orgasms. After a few cycles of dating people, breaking up with them, and heading back to the world of anonymous sex, I realized that the use of hookup apps distorted the ways in which I was able to experience sexual pleasure.


11.8: Review and Additional Resources
Review
Review and assess your learning. Start with the "Important Terms and Concepts" to ensure you know the terminology related to the topic of the chapter and concepts discussed. Move on to the "Review Questions" to answer critical thinking questions about concepts and processes discussed in the chapter. Finally, test your overall understanding by taking the "Self-assessment quiz".
-
- Parent material
- the material that soil develops from, and may be rock that has decomposed in place, or material that has been deposited by wind, water, or ice
-
- Soil forming factors
- parent material, climate, topography, organisms, and time
-
- Soil horizon
- different colored layers; form as a result of the four horizon development processes, additions, transformation, translocation, and removal
-
- O horizon
- primarily composed of organic matter; filled with nutrients and has strong soil moisture retention
-
- A horizon
- the beginning of the true mineral soil; organic material mixed with inorganic products of weathering; typically dark colored due to the presence organic matter
-
- E horizon
- generally light-colored with eluviation being the dominant process; Leaching is active in this horizon
-
- B horizon
- accumulation of fine material leads to the creation of a dense layer
-
- C horizon
- the soil parent material, either created in situ or transported into its present location
-
- Soil profile
- All the horizons taken together
-
- Humus
- dark brown, nutrient-rich material made from decomposed leaves and plant roots.
-
- Material addition
- materials that are transported into the location where a soil is forming
-
- Material transformation
- occur by chemical and biological processes acting on them
-
- Material translocation
- involves the movement of soil-forming materials through the developing soil profile
-
- Material removal
- the materials are completely removed from the soil profile
-
- pH scale
- how hydrogen ion concentration in the soil is measured
-
- Bulk density
- the mass per unit volume including the pore space
-
- Soil structure
- the way soil particles aggregate together into peds.
-
- Soil texture
- refers to the relative proportion of sand, silt and clay size particles in a sample of soil
-
- Ped
- soil structure forms; come in a variety of shapes depending on the texture, composition, and environment
-
- Granular structure
- crumb structures, look like cookie crumbs. They tend to form an open structure that allows water and air to penetrate the soil
-
- Platy structure
- looks like stacks of dinner plates overlaying one another. Platy structure tends to impede the downward movement of water and plant roots through the soil
-
- Physical weathering
- aka mechanical weathering; the breakdown of large pieces of earth material into smaller ones
-
- Chemical weathering
- breaks down earth material by chemical alteration
-
- Laterization
- occurs in the hot, rainy tropics where chemical weathering proceeds rapidly
-
- Podzolization
- The E horizon is heavily leached and basically composed a of light colored layer of sand. The upper portion of the B horizon is stained reddish color from the accumulation of sesquioxides. The profile gets lighter in color as depth increases.
-
- Calcification
- Soil tends to be rich in organic matter and high in soluble bases. The B horizon of the soil is enriched with calcium carbonate
-
- Gleization
- occurs in regions of high rainfall and low-lying areas that may be naturally waterlogged. Bacterial activity is slowed, inhibiting the decomposition of dead vegetation. Peat is found in the upper portion of the soil. the soil has a black to bluish - gray color.
-
- Salinization
- occurs in warm and dry locations where soluble salts precipitate from water and accumulate in the soil. Salt may also accumulate in soils from sea spray.
-
- Oxisol
- Chemical weathering in the presence of warm temperatures combined with heavy rainfall creates a soil rich in iron and aluminum oxides. A rich diversity of decomposers, rapid uptake by vegetation, and heavy precipitation quickly removes nutrients from the soil. What is left is a nutrient poor soil, not well-suited for agriculture. Cleared of vegetation, the exposed surface is easily eroded.
-
- Ultisol
- Highly weathered soils, they are often red/yellow in color. Found in the moister portions of the Humid Subtropical climate, they have a illuvial clay layer
-
- Alfisol
- Eluviation is moderate and base status is fairly high; well-developed and contain a subsurface layer of clay. Having a favorable moisture balance and good fertility, they are very productive soils for agriculture.
-
- Spodosol
- Surface litter breaks down in the presence of water to form a weak organic acid. Acidic soil water removes base ions in solution to create an acidic soil. Easily dissolved materials are leached from surface layers leaving behind the most resistant material creating an ashy-gray near-surface layer. Layers at depth are stained with iron and aluminum oxides.
-
- Mollisol
- well-known for their dark brown to black organic rich surface layers; have a granular structure and soft consistency when dry. Rich in calcium and others nutrients, and generally posses high moisture retention. Calcium nodules are found near the base of the soil as calcium carbonate precipitates out of soil water.
-
- Entisol
- soils lacking horizons because their parent material has only recently accumulated. Also form where the parent material is quartz sand, in which horizons do not easily form.
-
- Inceptisol
- soils just starting to show horizon development
-
- Histosol
- have a very high content of organic matter in the dark upper layer of the profile. Form in places where organic matter is slow to decompose and thus accumulates over time such as bogs and swamps.
-
- Aridisol
- typically light in color as there is little organic matter. A negative moisture balance in these soils inhibits eluviation. Calcification and salinization are acting in these soils. Soil horizons are weakly developed and sodium is often high in concentration.
-
- Andisol
- Soils developing in parent material containing at least fifty percent volcanic ash; support a dense natural cover in moist climates.
-
- Vertisol
- Dark black soils rich in expandable clay minerals. The clay readily swells upon wetting and shrinks when dried. Surface fragments fall into the cracks and are "swallowed" when the soil swells upon wetting. The soil then develops an "inverted profile" with organic material that is typically located near the surface of the profile is now found at depth.
Which soil orders lack noticeable horizons?
- Answer
-
Entisols lack noticeable horizons as parent material has not undergone significant weathering processes. On the other hand, oxisols may be so strongly weathered that visually distinguishing horizons can be difficult.
Which soil order has the highest organic content?
- Answer
-
Histosols are noted for high organic content. Peat can be mined from histosols.
Describe the various soil forming processes.
- Answer
-
Soil forming processes are what determine the type of soil that forms. Podzolization - Cool, humid environments; needle leaf forest cover common. Leads to the development of soils that are acidic, have an ash-grey E-horizon. Spodosols. Laterization - warm, humid environments; broadleaf evergreen forest cover. Creates highly oxidized soils that are red/yellow in color. Oxisols; Ultisols. Calcification - warm, dry environments; grass cover. Creates soils rich in calcium carbonate; high base status. Salinization - warm, dry environments. Creates soils containing soluble salts.
Which vegetation biome is associated with mollisols?
- Answer
-
Grassland
In which soil order might you find a subsurface layer of calcium carbonate?
- Answer
-
Mollisol
Which soil order forms an "inverted soil profile"?
- Answer
-
Vertisol
Which soil orders are associated with midlatitude temperate forests?
- Answer
-
Alfisol
What impact do broadleaf deciduous trees have on the nutrient status of soils?
- Answer
-
Broadleaf deciduous forest increase the nutrient base status of soil. A constant cycling of nutrients between the soil and tree occurs through decomposition of leaf litter and root uptake.
How does soil particle size influence field capacity, infiltration, and permeability?
- Answer
-
Fine textured soils (small particle size) have more total pore space per unit volume than does coarse textured soils (large particle size). Fine textured soils have high field capacity than coarse textured soils. Coarse textured soil has larger, better connected pore space yielding greater infiltration and permeability compared to fine textured soil.
What is soil texture and why is it important?
- Answer
-
Soil texture is the relative proportions of sand, silt, and clay size particles. Soil texture effects such processes as infiltration, permeability, and field capacity.
-
The zone of illuviation is commonly the
- O horizon
- A horizon
- B horizon
- C horizon
-
A soil texture with 30% Clay, 30 % silt, and 40% Sand is classified as a
- loam
- clay loam
- silty clay loam
- sandy clay
-
Which of the following soil structures would impede the infiltration of water the most?
- Platy
- Blocky
- Crumb
- Columnar
-
Which of the following soil orders is most associated with broadleaf deciduous temperate forests?
- Spodosol
- Aridisol
- Mollisol
- Alfisol
-
Soil profiles are likely to be the least developed
- at the top of a hill
- at the middle of a hill slope
- a slight distance ahead of a hill slope
- none of the above as hill slope site has nothing to do with soil development
-
Which of the following soil horizon processes creates the acidification of soils by the break down of organic matter?
- Additions
- Removals
- Transformation
- Transfer
-
Which of the following is not a soil forming factor?
- Time
- Organisms
- Parent material
- Season
-
Which of the following soil forming processes is likely occurring at letter L
- Calcification
- Laterization
- Podzolization
- Acidification
-
Which soil would I most likely find at location E
- Spodosol
- Ultisol
- Mollisol
- Aridisol
-
Which soil would I most likely find at location C
- Spodosol
- Ultisol
- Mollisol
- Aridisol
- Answer
-
- C
- B
- A
- D
- B
- C
- D
- A
- B
- A
Additional Resources
Use these resources to further explore the world of geography
Connections : "How to green the world's deserts and reverse climate change | Allan Savory" TED Talk


7.1: Homomorphisms
Let \(G_1\) and \(G_2\) be groups. Recall that \(\phi:G_1\to G_2\) is an isomorphism if and only if \(\phi\)
- is one-to-one,
- is onto, and
- satisfies the homomorphic property.
We say that \(G_1\) is isomorphic to \(G_2\) and write \(G_1\cong G_2\) if such a \(\phi\) exists. Loosely speaking, two groups are isomorphic if they have the “same structure." What if we drop the one-to-one and onto requirement?
Let \((G_1,*)\) and \((G_2,\odot)\) be groups. A function \(\phi:G_1\to G_2\) is a homomorphism if and only if \(\phi\) satisfies the homomorphic property: \[\phi(x*y)=\phi(x)\odot\phi(y)\] for all \(x,y\in G_1\) . At the risk of introducing ambiguity, we will usually omit making explicit reference to the binary operations and write the homomorphic property as \[\phi(xy)=\phi(x)\phi(y).\]
Group homomorphisms are analogous to linear transformations on vector spaces that one encounters in linear algebra.
Figure \(\PageIndex{1}\) captures a visual representation of the homomorphic property. We encountered this same representation in Figure \(\PageIndex{1}\). If \(\phi(x)=x'\) , \(\phi(y)=y'\) , and \(\phi(z)=z'\) while \(z'=x'\odot y'\) , then the only way \(G_2\) may respect the structure of \(G_1\) is for \[\phi(x*y)=\phi(z)=z'=x'\odot y'=\phi(x)\odot \phi(y).\]
Define \(\phi:\mathbb{Z}_3\to D_3\) via \(\phi(k)=r^k\) . Prove that \(\phi\) is a homomorphism and then determine whether \(\phi\) is one-to-one or onto. Also, try to draw a picture of the homomorphism in terms of Cayley diagrams.
Let \(G\) and \(H\) be groups. Prove that the function \(\phi:G\times H\to G\) given by \(\phi(g,h)=g\) is a homomorphism. This function is an example of a projection map .
There is always at least one homomorphism between two groups.
Let \(G_1\) and \(G_2\) be groups. Define \(\phi:G_1\to G_2\) via \(\phi(g)=e_2\) (where \(e_2\) is the identity of \(G_2\) ). Then \(\phi\) is a homomorphism. This function is often referred to as the trivial homomorphism or the \(0\) -map .
Back in Section 3.3 , we encountered several theorems about isomorphisms. However, at the end of that section we remarked that some of those theorems did not require that the function be one-to-one and onto. We collect those results here for convenience.
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism.
- If \(e_1\) and \(e_2\) are the identity elements of \(G_1\) and \(G_2\) , respectively, then \(\phi(e_1)=e_2\) .
- For all \(g\in G_1\) , we have \(\phi(g^{-1})=[\phi(g)]^{-1}\) .
- If \(H\leq G_1\) , then \(\phi(H)\leq G_2\) , where \[\phi(H):=\{y\in G_2\mid \text{there exists } h\in H\text{ such that }\phi(h)=y\}.\] Note that \(\phi(H)\) is called the image of \(H\) . A special case is when \(H=G_1\) . Notice that \(\phi\) is onto exactly when \(\phi(G_1)=G_2\) .
The following theorem is a consequence of Lagrange’s Theorem.
Let \(G_1\) and \(G_2\) be groups such that \(G_2\) is finite and let \(H\leq G_1\) . If \(\phi:G_1\to G_2\) is a homomorphism, then \(|\phi(H)|\) divides \(|G_2|\) .
The next theorem tells us that under a homomorphism, the order of the image of an element must divide the order of the pre-image of that element.
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. If \(g\in G_1\) such that \(|g|\) is finite, then \(|\phi(g)|\) divides \(|g|\) .
Every homomorphism has an important subset of the domain associated with it.
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. The kernel of \(\phi\) is defined via \[\ker(\phi):=\{g\in G_1\mid \phi(g)=e_2\}.\]
The kernel of a homomorphism is analogous to the null space of a linear transformation of vector spaces.
Identify the kernel and image for the homomorphism given in Problem \(\PageIndex{1}\).
What is the kernel of a trivial homomorphism (see Theorem \(\PageIndex{1}\)).
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. Then \(\ker(\phi)\trianglelefteq G_1\) .
Let \(G\) be a group and let \(H\trianglelefteq G\) . Then the map \(\gamma:G\to G/H\) given by \(\gamma(g)=gH\) is a homomorphism with \(\ker(\gamma)=H\) . This map is called the canonical projection map .
The upshot of Theorems \(\PageIndex{5}\) and \(\PageIndex{6}\) is that kernels of homomorphisms are always normal and every normal subgroup is the kernel of some homomorphism. It turns out that the kernel can tell us whether \(\phi\) is one-to-one.
The next theorem tells us that two elements in the domain of a group homomorphism map to the same element in the codomain if and only if they are in the same coset of the kernel.
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. Then \(\phi(a) = \phi(b)\) if and only if \(a\in b\ker(\phi)\) .
One consequence of Theorem \(\PageIndex{7}\) is that if the kernel of a homomorphism has order \(k\) , then the homomorphism is \(k\) -to-1. That is, every element in the range has exactly \(k\) elements from the domain that map to it. In particular, each of these collections of \(k\) elements corresponds to a coset of the kernel.
Suppose \(\phi:\mathbb{Z}_{20}\to\mathbb{Z}_{20}\) is a group homomorphism such that \(\ker(\phi)=\{0,5,10,15\}\) . If \(\phi(13)=8\) , determine all elements that \(\phi\) maps to 8.
The next result is a special case of Theorem \(\PageIndex{7}\).
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. Then \(\phi\) is one-to-one if and only if \(\ker(\phi)=\{e_1\}\) , where \(e_1\) is the identity in \(G_1\) .
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. Given a generating set for \(G_1\) , the homomorphism \(\phi\) is uniquely determined by its action on the generating set for \(G_1\) . In particular, if you have a word for a group element written in terms of the generators, just apply the homomorphic property to the word to find the image of the corresponding group element.
Suppose \(\phi: Q_8\to V_{4}\) is a group homomorphism satisfying \(\phi(i)=h\) and \(\phi(j)=v\) .
- Find \(\phi(1)\) , \(\phi(-1)\) , \(\phi(k)\) , \(\phi(-i)\) , \(\phi(-j)\) , and \(\phi(-k)\) .
- Find \(\ker(\phi)\) .
- What well-known group is \(Q_8/\ker(\phi)\) isomorphic to?
Find a non-trivial homomorphism from \(\mathbb{Z}_{10}\) to \(\mathbb{Z}_6\) .
Find all non-trivial homomorphisms from \(\mathbb{Z}_3\) to \(\mathbb{Z}_6\) .
Prove that the only homomorphism from \(D_3\) to \(\mathbb{Z}_3\) is the trivial homomorphism.
Let \(F\) be the set of all functions from \(\mathbb{R}\) to \(\mathbb{R}\) and let \(D\) be the subset of differentiable functions on \(\mathbb{R}\) . It turns out that \(F\) is a group under addition of functions and \(D\) is a subgroup of \(F\) (you do not need to prove this). Define \(\phi:D\to F\) via \(\phi(f)=f'\) (where \(f'\) is the derivative of \(f\) ). Prove that \(\phi\) is a homomorphism. You may recall facts from calculus without proving them. Is \(\phi\) one-to-one? Onto?


7.9: References
Association of Ontario Midwives. (2018). Vaccinations in pregnancy. Retrieved from https://www.ontariomidwives.ca/vaccinations-pregnancy
Canadian Nurses Association. (2015). Framework for registered nurse prescribing in Canada. Retrieved from https://www.cna-aiic.ca/~/media/cna/page-content/pdf-en/cna-rn-prescribing-framework_e.pdf?la=en
College and Association of Registered Nurses of Alberta. (2019). Registered nurse prescribing Schedule 1 drugs and ordering diagnostic tests: Requirements and standards. Retrieved from https://www.nurses.ab.ca/docs/default-source/document-library/standards/registered-nurse-prescribing-schedule-1-drugs-and-ordering-diagnostic-tests-requirements-and-standards.pdf?sfvrsn=67eba52b_4
College of Midwives of Ontario. (2014, revised 2018). Prescribing and administering drugs. Retrieved from http://www.cmo.on.ca/wp-content/uploads/2015/07/16.Prescribing-and-Administering-Drugs.pdf
College of Nurses of Ontario. (2018a). Directives . Retrieved from https://www.cno.org/globalassets/docs/prac/41019_medicaldirectives.pdf
College of Nurses of Ontario. (2018b, December 10). Draft competencies for RN prescribing. Retrieved from http://www.cno.org/globalassets/1-whatiscno/legislation/draft-competencies-for-regcirculation.pdf
College of Nurses of Ontario. (2019a). FAQs: RN prescribing. Retrieved from http://www.cno.org/en/trending-topics/journey-to-rn-prescribing/qas-rn-prescribing/
College of Nurses of Ontario. (2019b). Journey to RN prescribing. Retrieved from http://www.cno.org/en/trending-topic...n-prescribing/
Government of Ontario. (2017, May 17). Stronger, Healthier Ontario Act (Budget Measures), 2017 [Media release]. Retrieved from https://news.ontario.ca/mof/en/2017/05/stronger-healthier-ontario-act-budget-measures-2017.html
Hoskins, E. (2015, November 4). [Letter from Minister of Health and Long-Term Care to Chair of the Health Professions Regulatory Advisory Council.] Ontario: Ontario Ministry of Health and Long-Term Care. Retrieved from https://hprac.org/en/resources/Minister-Letter-November-4-2015-FINAL-AODA.pdf
International Council of Nurses. (2013). Adult and childhood immunisation: 2013 revised edition . Retrieved from https://www.icn.ch/sites/default/files/inline-files/2013_adult_and_childhood_immunisation.pdf
International Council of Nurses. (2018). The role of nurses in immunisation: A snapshot from OECD countries. Retrieved from https://www.icn.ch/sites/default/files/inline-files/IMMUNISATION_Report%20%28002%29.pdf
Jones, A. (2019, April 18). Ontario may let pharmacists prescribe treatments for minor issues like pink eye. Toronto CityNews . Retrieved from https://toronto.citynews.ca/2019/04/...s-health-care/
Ladd, E., & Schober, M. (2018). Nurse prescribing from the global vantage point: The intersection between role and policy. Policy, Politics, & Nursing Practice, 19 (1-2), 40-49. https://doi.org/10.1177/1527154418797726
Nursing Act, 1991 , SO 1991, c 32. Retrieved from https://www.ontario.ca/laws/statute/91n32
Ontario College of Pharmacists. (2016, December 2). New authority for vaccinations (effective Dec 15). Retrieved from http://www.ocpinfo.com/library/news/new-authority-vaccinations-effective-december-15/
Ontario College of Pharmacists. (n.d.). Administering injections. Retrieved from http://www.ocpinfo.com/practice-educ...ng-injections/ .
Public Health Agency of Canada. (2008). Immunization competencies for health professionals. Retrieved from https://www.canada.ca/en/public-health/services/publications/healthy-living/immunization-competencies-health-professionals.html#ess