10.7: Looking for a Better Way- Improving Production and Operations
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6. How can quality-management and lean-manufacturing techniques help firms improve production and operations management?
Competing in today’s business world is challenging. To compete effectively, firms must keep production costs down. At the same time, however, it’s becoming increasingly complex to produce and deliver the high-quality goods and services customers demand. Methods to help meet these challenges include quality-management techniques, lean manufacturing, and technology and automation.
Putting Quality First
Successful businesses recognize that quality and productivity must go hand in hand. Quality goods and services meet customer expectations by providing reliable performance. Defective products waste materials and time, increasing costs. Worse, poor quality causes customer dissatisfaction, which usually results in lost sales.
A consumer measures quality by how well a product serves its purpose. From the manufacturer’s point of view, quality is the degree to which the product conforms to a set of predetermined standards. Quality control involves creating quality standards, producing goods that meet them, and measuring finished goods and services against them. It takes more than just inspecting goods at the end of the assembly line to ensure quality control, however. Quality control requires a company-wide dedication to managing and working in a way that builds excellence into every facet of operations.
Dr. W. Edwards Deming, an American management consultant, was the first to say that quality control should be a company-wide goal. His ideas were adopted by the Japanese in the 1950s but largely ignored in the United States until the 1970s. Deming believed that quality control starts with top management, who must foster a company-wide culture dedicated to producing quality.
Deming’s concept of Total Quality Management (TQM) emphasizes the use of quality principles in all aspects of a company’s production and operations. It recognizes that all employees involved with bringing a product or service to customers—marketing, purchasing, accounting, shipping, manufacturing—contribute to its quality. TQM focuses on continuous improvement , a commitment to constantly seek better ways of doing things in order to achieve greater efficiency and improve quality. Company-wide teams work together to prevent problems and systematically improve key processes instead of troubleshooting problems only as they arise. Continuous improvement continually measures performance using statistical techniques and looks for ways to apply new technologies and innovative production methods.
Another quality-control method is the Six Sigma quality program. Six Sigma is a company-wide process that focuses on measuring the number of defects that occur and systematically eliminating them in order to get as close to “zero defects” as possible. In fact, Six Sigma quality aims to have every process produce no more than 3.4 defects per million. Six Sigma focuses on designing products that not only have fewer defects but that also satisfy customer needs. A key process of Six Sigma is called DMAIC . This stands for Define, Measure, Analyze, Improve, and Control. Employees at all levels define what needs to be done to ensure quality, then measure and analyze production results using statistics to see if standards are met. They are also charged with finding ways to improve and control quality.
General Electric was one of the first companies to institute Six Sigma throughout the organization. GE employees are trained in Six Sigma concepts, and many analysts believe this has given GE a competitive manufacturing advantage. Service firms and government entities have applied Six Sigma to their quality initiatives as well.
Malcolm Baldrige National Quality Award
Named for a former secretary of commerce, the Malcolm Baldrige National Quality Award was established by the U.S. Congress in 1987 to recognize U.S. companies that offer goods and services of world-class quality. The award promotes awareness of quality and allows the business community to assess which quality control programs are most effective.
Administered by the U.S. Department of Commerce’s National Institute of Standards and Technologies (NIST), the award’s most important criterion is a firm’s effectiveness at meeting customer expectations, as well as demonstrating that it offers quality goods and services. To qualify for the award, a company must also show continuous improvement in internal operations. Company leaders and employees must be active participants in the firm’s quality program, and they must respond quickly to data and analysis.
Organizations in a wide variety of industries have won the Baldrige Award since it was first presented in 1987. In 2017, for example, the Baldrige Award winners included Bristol Tennessee Essential Services, an electricity and fiber services utility company, in the small business sector; the city of Fort Collins, Colorado, in the nonprofit sector; and Southcentral Foundation in Anchorage, Alaska, in the health care sector. 9
Worldwide Excellence: International Quality Standards
The International Organization for Standardization (ISO), located in Geneva, Switzerland, is an industry organization that has developed standards of quality that are used by businesses around the world. ISO 9000 , introduced in the 1980s, is a set of five technical standards designed to offer a uniform way of determining whether manufacturing plants and service organizations conform to sound quality procedures. To register, a company must go through an audit of its manufacturing and customer service processes, covering everything from how it designs, produces, and installs its products, to how it inspects, packages, and markets them. Over 500,000 organizations worldwide have met ISO 9000 standards.
ISO 14000 , launched after ISO 9000, was designed in response to environmental issues such as global warming and water pollution and promotes clean production processes. To meet ISO 14000 standards, a company must commit to continually improving environmental management and reducing pollution resulting from its production processes.
Lean Manufacturing Trims the Fat
Manufacturers are discovering that they can better respond to rapidly changing customer demands, while keeping inventory and production costs down, by adopting lean-manufacturing techniques. Lean manufacturing streamlines production by eliminating steps in the production process that do not add benefits customers want. In other words, non-value-added production processes are cut so that the company can concentrate its production and operations resources on items essential to satisfying customers. Toyota was a pioneer in developing these techniques, but today manufacturers in many industries have adopted the lean-manufacturing philosophy.
Another Japanese concept, just-in-time (JIT) , goes hand in hand with lean manufacturing. JIT is based on the belief that materials should arrive exactly when they are needed for production, rather than being stored on-site. Relying closely on computerized systems such as MRP, MRPII, and ERP, manufacturers determine what parts will be needed and when and then order them from suppliers so they arrive “just in time.” Under the JIT system, inventory and products are “pulled” through the production process in response to customer demand. JIT requires close teamwork between vendors and purchasing and production personnel because any delays in deliveries of supplies could bring JIT production to a halt.
Unexpected events like the September 11 terrorist attacks or the shutdown of ports due to Hurricane Harvey and the devastation and flooding caused by Hurricane Maria in Puerto Rico can cause chaos in the supply chains of manufacturers, resulting in problems for firms relying on JIT. But if employed properly, and in spite of these risks, a JIT system can greatly reduce inventory-holding costs and smooth production highs and lows.
CONCEPT CHECK
- How can managers use techniques to improve efficiency?
- Define Six Sigma.
- What was Edward Demming’s contribution to operations management?


16.2: Accounting in Business
What you’ll learn to do: define accounting, and explain its role as a form of business communication
How businesses express their financial health and stability is by presenting an “accounting” of all their financial transactions. In this section you will explore in depth what accounting is and the vital information it communicates about the business.
Learning Objectives
- Identify the users and uses of financial accounting
- Identify the users and uses of managerial accounting
What Is Accounting?
Why Do we Need Financial Information?
As you learned earlier in the course, businesses have large groups of stakeholders who have a vested interested in the continued success of the enterprise. If a business, whether for-profit or nonprofit, becomes financially insolvent and can’t pay its bills, it will be forced to close. Financial information enables a business to track its accounts and avoid insolvency.
Each business needs financial information to be able to answer questions such as the following:
- How much cash does the business need to pay its bills and employees?
- Is the business profitable, earning more income than it pays in expenses, or is it losing money and possibly in danger of closing?
- How much of a particular product or mixture of products should the business produce and sell?
- What is the cost of making the goods or providing the service?
- What are the business’s daily, monthly, and annual expenses?
- Do customers owe money to the business, and are they paying on time?
- How much money does the business owe to vendors (suppliers), banks, or other investors?
The video below gives a brief overview of many of the topics in this section. Before you review the video, consider these questions:
- What is accounting?
- What is business?
- Who are the three people that want to know the story of your business?
- What language of accounting does the government use?
- What language of accounting do investors use?
- What language of accounting do internal users employ?
Accounting Is the Language of Business
Every business organization that has economic resources, such as money, machinery, and buildings, uses accounting information. For this reason, accounting is called the language of business. Accounting also serves as the language providing financial information about not-for-profit organizations such as governments, churches, charities, fraternities, and hospitals. However, in this module we will focus on accounting for business firms.
The accounting process provides financial data for a broad range of individuals whose objectives in studying the data vary widely. Bank officials, for example, may study a company’s financial statements to evaluate the company’s ability to repay a loan. Prospective investors may compare accounting data from several companies to decide which company represents the best investment. Accounting also supplies management with significant financial data useful for decision making.
Definition of Accounting
As the video explained, accounting is “the language of business.” The American Accounting Association defines accounting as “the process of identifying, measuring, and communicating economic information to permit informed judgments and decisions by the users of the information.”
This information is primarily financial—stated in money terms. Accounting, then, is a measurement and communication process used to report on the activities of profit-seeking business organizations. As a measurement and communication process for business, accounting supplies information that permits informed judgments and decisions by users of the data.
Internal and External Users
Users of accounting information are separated into two groups, internal and external. Internal users are the people within a business organization who use accounting information. For example, the human resource department needs to have information about how profitable the business is in order to set salaries and benefits. Likewise, production managers need to know if the business is doing well enough to afford to replace worn-out machinery or pay overtime to production workers. External users are people outside the business entity that use accounting information. These external users include potential investors, the Internal Revenue Service, banks and finance companies, as well as local taxing authorities. Accounting information is valuable to both groups when it comes time to evaluate the financial consequences of various alternatives. Accountants reduce uncertainty by using professional judgment to quantify the future financial impact of taking action or delaying action. In short, although accounting information plays a significant role in reducing uncertainty within an organization, it also provides financial data for persons outside the company.
Financial accounting information appears in financial statements that are intended primarily for external use (although management also uses them for certain internal decisions). Stockholders and creditors are two of the outside parties who need financial accounting information. These outside parties decide on matters pertaining to the entire company, such as whether to increase or decrease their investment in a company or to extend credit to a company. Consequently, financial accounting information relates to the company as a whole, while managerial accounting focuses on the parts or segments of the company.
Because the external users of accounting information vary greatly, the way that financial information is presented must be consistent from year to year and company to company. In order to facilitate this, financial accountants adhere to set of rules called Generally Accepted Accounting Principles (GAAP). GAAP are a uniform set of accounting rules that allow users to compare the financial statements issued by one company to those of another company in the same industry. These principles for financial reporting are issued by an independent non-profit agency created by the Securities Exchange Commission (SEC) called the Financial Accounting Standards Board (FASB) . The FASB’s mission is “to establish and improve standards of financial accounting and reporting that foster financial reporting by nongovernmental entities that provides decision-useful information to investors and other users of financial reports.”
Tax accounting information includes financial accounting information, written and presented in the tax code of the government—namely the Internal Revenue Code. Tax accounting focuses on compliance with the tax code and presenting the profit and loss story of a business to minimize its tax liability.
Accounting is more than just reporting income to taxing authorities or providing revenue and expense information to potential investors. As the language of business, accounting is used for decision-making as well.
Managerial accounting information is for internal use and provides special information for the managers of a company. The information managers use may range from broad, long-range planning data to detailed explanations of why actual costs varied from cost estimates. The employees of a firm who perform these managerial accounting functions are often referred to as Cost Accountants. Managerial accounting is more concerned with forward looking projections and making decisions that will affect the future of the organization, than in the historical recording and compliance aspects of the financial accountants. There are no reporting guidelines such as GAAP; therefore, managerial accounting reports will vary widely in both scope and content. Also, much of the information generated by managerial accountants is confidential and not intended to be shared outside of the organization. Managerial accounting focus on range of topics from production planning to budgets for raw materials. When a company makes a decision to purchase a component part instead of manufacture it in house, that decision is based primarily on managerial accounting information. For this reason, many managerial accountants consider themselves to be provide “accounting information for decision making.”
Bookkeeping vs. Accounting
Accounting is often confused with bookkeeping. Bookkeeping is a mechanical process that records the routine economic activities of a business. Accounting includes bookkeeping, but it goes further to analyze and interpret financial information, prepare financial statements, conduct audits, design accounting systems, prepare special business and financial studies, prepare forecasts and budgets, and provide tax services.
Importance of Accounting
You probably will find that of all the business knowledge you have acquired or will learn, the study of accounting will be the most useful. Your financial and economic decisions as a student and consumer involve accounting information. When you file income tax returns, accounting information helps determine your taxes payable.
Understanding the discipline of accounting also can influence many of your future professional decisions. You cannot escape the effects of accounting information on your personal and professional life.


7.1: Volume Properties
Two volume properties of a closed system are defined as follows: \begin{equation} \textbf{cubic expansion coefficient} \quad \alpha \defn \frac{1}{V}\Pd{V}{T}{p} \tag{7.1.1} \end{equation} \begin{equation} \textbf{isothermal compressibility} \quad \kT \defn -\frac{1}{V}\Pd{V}{p}{T} \tag{7.1.2} \end{equation}
The cubic expansion coefficient is also called the coefficient of thermal expansion and the expansivity coefficient. Other symbols for the isothermal compressibility are \(\beta\) and \(\g_T\).
These definitions show that \(\alpha\) is the fractional volume increase per unit temperature increase at constant pressure, and \(\kT\) is the fractional volume decrease per unit pressure increase at constant temperature. Both quantities are intensive properties. Most substances have positive values of \(\alpha\), and all substances have positive values of \(\kT\), because a pressure increase at constant temperature requires a volume decrease.
The cubic expansion coefficient is not always positive. \(\alpha\) is negative for liquid water below its temperature of maximum density, \(3.98\units{\(\degC\)}\). The crystalline ceramics zirconium tungstate (ZrW\(_2\)O\(_8\)) and hafnium tungstate (HfW\(_2\)O\(_8\)) have the remarkable behavior of contracting uniformly and continuously in all three dimensions when they are heated from \(0.3\K\) to about \(1050\K\); \(\alpha\) is negative throughout this very wide temperature range (T. A. Mary et al, Science , 272 , 90–92, 1996). The intermetallic compound YbGaGe has been found to have a value of \(\alpha\) that is practically zero in the range \(100\)–\(300\K\) (James R. Salvador et al, Nature , 425 , 702–705, 2003).
If an amount \(n\) of a substance is in a single phase, we can divide the numerator and denominator of the right sides of Eqs. 7.1.1 and 7.1.2 by \(n\) to obtain the alternative expressions \begin{gather} \s{ \alpha = \frac{1}{V\m}\Pd{V\m}{T}{\!p} } \tag{7.1.3} \cond{(pure substance, \(P{=}1\))} \end{gather} \begin{gather} \s{ \kT = -\frac{1}{V\m}\Pd{V\m}{p}{T} } \tag{7.1.4} \cond{(pure substance, \(P{=}1\))} \end{gather} where \(V\m\) is the molar volume. \(P\) in the conditions of validity is the number of phases. Note that only intensive properties appear in Eqs. 7.1.3 and 7.1.4; the amount of the substance is irrelevant. Figures 7.1 and 7.2 show the temperature variation of \(\alpha\) and \(\kT\) for several substances.
If we choose \(T\) and \(p\) as the independent variables of the closed system, the total differential of \(V\) is given by \begin{equation} \dif V = \Pd{V}{T}{\!p}\dif T + \Pd{V}{p}{T}\difp \tag{7.1.5} \end{equation} With the substitutions \(\pd{V}{T}{p} = \alpha V\) (from Eq. 7.1.1) and \(\pd{V}{p}{T} = -\kT V\) (from Eq. 7.1.2), the expression for the total differential of \(V\) becomes \begin{gather} \s{ \dif V = \alpha V \dif T - \kT V \difp } \tag{7.1.6} \cond{(closed system,} \nextcond{\(C{=}1\), \(P{=}1\))} \end{gather} To find how \(p\) varies with \(T\) in a closed system kept at constant volume, we set \(\dif V\) equal to zero in Eq. 7.1.6: \(0 = \alpha V\dif T - \kappa _T V\difp\), or \(\difp/\dif T = \alpha /\kappa _T\). Since \(\difp/\dif T\) under the condition of constant volume is the partial derivative \(\pd{p}{T}{V}\), we have the general relation \begin{gather} \s{ \Pd{p}{T}{V} = \frac{\alpha}{\kT} } \tag{7.1.7} \cond{(closed system,} \nextcond{\(C{=}1\), \(P{=}1\))} \end{gather}


2.3.5: Axiomatic Derivations
Axiomatic derivations are the oldest and simplest logical derivation systems. Its derivations are simply sequences of sentences. A sequence of sentences counts as a correct derivation if every sentence \(A\) in it satisfies one of the following conditions:
- \(A\) is an axiom, or
- \(A\) is an element of a given set \(\Gamma\) of sentences, or
- \(A\) is justified by a rule of inference.
To be an axiom, \(A\) has to have the form of one of a number of fixed sentence schemas. There are many sets of axiom schemas that provide a satisfactory (sound and complete) derivation system for first-order logic. Some are organized according to the connectives they govern, e.g., the schemas \[A \lif (B \lif A) \qquad B \lif (B \lor C) \qquad (B \land C) \lif B\nonumber\] are common axioms that govern \(\lif\) , \(\lor\) and \(\land\) . Some axiom systems aim at a minimal number of axioms. Depending on the connectives that are taken as primitives, it is even possible to find axiom systems that consist of a single axiom.
A rule of inference is a conditional statement that gives a sufficient condition for a sentence in a derivation to be justified. Modus ponens is one very common such rule: it says that if \(A\) and \(A \lif B\) are already justified, then \(B\) is justified. This means that a line in a derivation containing the sentence \(B\) is justified, provided that both \(A\) and \(A \lif B\) (for some sentence \(A\) ) appear in the derivation before \(B\) .
The \(\Proves\) relation based on axiomatic derivations is defined as follows: \(\Gamma \Proves A\) iff there is a derivation with the sentence \(A\) as its last formula (and \(\Gamma\) is taken as the set of sentences in that derivation which are justified by (2) above). \(A\) is a theorem if \(A\) has a derivation where \(\Gamma\) is empty, i.e., every sentence in the derivation is justfied either by (1) or (3). For instance, here is a derivation that shows that \(\Proves A \lif (B \lif (B \lor A))\) :
1.
\(B \lif (B \lor A)\)
2.
\((B \lif (B \lor A)) \lif (A \lif (B \lif (B \lor A)))\)
3.
\(A \lif (B \lif (B \lor A))\)
The sentence on line 1 is of the form of the axiom \(A \lif (A \lor B)\) (with the roles of \(A\) and \(B\) reversed). The sentence on line 2 is of the form of the axiom \(A \lif (B \lif A)\) . Thus, both lines are justified. Line 3 is justified by modus ponens: if we abbreviate it as \(D\) , then line 2 has the form \(C \lif D\) , where \(C\) is \(B \lif (B \lor A)\) , i.e., line 1.
A set \(\Gamma\) is inconsistent if \(\Gamma \Proves \lfalse\) . A complete axiom system will also prove that \(\lfalse \lif A\) for any \(A\) , and so if \(\Gamma\) is inconsistent, then \(\Gamma \Proves A\) for any \(A\) .
Systems of axiomatic derivations for logic were first given by Gottlob Frege in his 1879 Begriffsschrift , which for this reason is often considered the first work of modern logic. They were perfected in Alfred North Whitehead and Bertrand Russell’s Principia Mathematica and by David Hilbert and his students in the 1920s. They are thus often called “Frege systems” or “Hilbert systems.” They are very versatile in that it is often easy to find an axiomatic system for a logic. Because derivations have a very simple structure and only one or two inference rules, it is also relatively easy to prove things about them. However, they are very hard to use in practice, i.e., it is difficult to find and write proofs.


24.2: Microbial Diseases of the Mouth and Oral Cavity
- Explain the role of microbial activity in diseases of the mouth and oral cavity
- Compare the major characteristics of specific oral diseases and infections
Despite the presence of saliva and the mechanical forces of chewing and eating, some microbes thrive in the mouth. These microbes can cause damage to the teeth and can cause infections that have the potential to spread beyond the mouth and sometimes throughout the body.
Dental Caries
Cavities of the teeth, known clinically as dental caries, are microbial lesions that cause damage to the teeth. Over time, the lesion can grow through the outer enamel layer to infect the underlying dentin or even the innermost pulp. If dental caries are not treated, the infection can become an abscess that spreads to the deeper tissues of the teeth, near the roots, or to the bloodstream.
Tooth decay results from the metabolic activity of microbes that live on the teeth. A layer of proteins and carbohydrates forms when clean teeth come into contact with saliva. Microbes are attracted to this food source and form a biofilmcalled plaque. The most important cariogenic species in these biofilms is Streptococcus mutans . When sucrose, a disaccharide sugar from food, is broken down by bacteria in the mouth, glucose and fructose are produced. The glucose is used to make dextran, which is part of the extracellular matrix of the biofilm. Fructose is fermented, producing organic acids such as lactic acid. These acids dissolve the minerals of the tooth, including enamel, even though it is the hardest material in the body. The acids work even more quickly on exposed dentin (Figure \(\PageIndex{1}\)). Over time, the plaque biofilm can become thick and eventually calcify. When a heavy plaque deposit becomes hardened in this way, it is called tartar or dental calculus (Figure \(\PageIndex{2}\)). These substantial plaque biofilms can include a variety of bacterial species, including Streptococcus and Actinomyces species.
Some tooth decay is visible from the outside, but it is not always possible to see all decay or the extent of the decay. X-ray imaging is used to produce radiographs that can be studied to look for deeper decay and damage to the root or bone (Figure \(\PageIndex{2}\)). If not detected, the decay can reach the pulp or even spread to the bloodstream. Painful abscesses can develop.
To prevent tooth decay, prophylactic treatment and good hygiene are important. Regular tooth brushing and flossing physically removes microbes and combats microbial growth and biofilm formation. Toothpaste contains fluoride, which becomes incorporated into the hydroxyapatite of tooth enamel, protecting it against acidity caused by fermentation of mouth microbiota. Fluoride is also bacteriostatic, thus slowing enamel degradation. Antiseptic mouthwashes commonly contain plant-derived phenolics like thymol and eucalyptol and/or heavy metals like zinc chloride (see Using Chemicals to Control Microorganisms ). Phenolics tend to be stable and persistent on surfaces, and they act through denaturing proteins and disrupting membranes.
Regular dental cleanings allow for the detection of decay at early stages and the removal of tartar. They may also help to draw attention to other concerns, such as damage to the enamel from acidic drinks. Reducing sugar consumption may help prevent damage that results from the microbial fermentation of sugars. Additionally, sugarless candies or gum with sugar alcohols (such as xylitol) can reduce the production of acids because these are fermented to nonacidic compounds (although excess consumption may lead to gastrointestinal distress). Fluoride treatment or ingesting fluoridated water strengthens the minerals in teeth and reduces the incidence of dental caries.
If caries develop, prompt treatment prevents worsening. Smaller areas of decay can be drilled to remove affected tissue and then filled. If the pulp is affected, then a root canal may be needed to completely remove the infected tissues to avoid continued spread of the infection, which could lead to painful abscesses.
- Name some ways that microbes contribute to tooth decay.
- What is the most important cariogenic species of bacteria?
Periodontal Disease
In addition to damage to the teeth themselves, the surrounding structures can be affected by microbes. Periodontal disease is the result of infections that lead to inflammation and tissue damage in the structures surrounding the teeth. The progression from mild to severe periodontal disease is generally reversible and preventable with good oral hygiene.
Inflammation of the gums that can lead to irritation and bleeding is called gingivitis. When plaque accumulates on the teeth, bacteria colonize the gingival space. As this space becomes increasingly blocked, the environment becomes anaerobic. This allows a wide variety of microbes to colonize, including Porphyromonas , Streptococcus , and Actinomyces . The bacterial products, which include lipopolysaccharide (LPS), proteases, lipoteichoic acids, and others, cause inflammation and gum damage (Figure \(\PageIndex{3}\)). It is possible that methanogenic archaeans (including Methanobrevibacter oralis and other Methanobrevibacter species) also contribute to disease progression as some species have been identified in patients with periodontal disease, but this has proven difficult to study. 1 2 3 Gingivitis is diagnosed by visual inspection, including measuring pockets in the gums, and X-rays, and is usually treated using good dental hygiene and professional dental cleaning, with antibiotics reserved for severe cases.
Over time, chronic gingivitis can develop into the more serious condition of periodontitis (Figure \(\PageIndex{4}\)). When this happens, the gums recede and expose parts of the tooth below the crown. This newly exposed area is relatively unprotected, so bacteria can grow on it and spread underneath the enamel of the crown and cause cavities. Bacteria in the gingival space can also erode the cementum, which helps to hold the teeth in place. If not treated, erosion of cementum can lead to the movement or loss of teeth. The bones of the jaw can even erode if the infection spreads. This condition can be associated with bleeding and halitosis (bad breath). Cleaning and appropriate dental hygiene may be sufficient to treat periodontitis. However, in cases of severe periodontitis, an antibiotic may be given. Antibiotics may be given in pill form or applied directly to the gum (local treatment). Antibiotics given can include tetracycline, doxycycline, macrolides or β-lactams. Because periodontitis can be caused by a mix of microbes, a combination of antibiotics may be given.
Trench Mouth
When certain bacteria, such as Prevotella intermedia , Fusobacterium species, and Treponema vicentii , are involved and periodontal disease progresses, acute necrotizing ulcerative gingivitis or trench mouth, also called Vincent's disease, can develop. This is severe periodontitis characterized by erosion of the gums, ulcers, substantial pain with chewing, and halitosis (Figure \(\PageIndex{5}\)) that can be diagnosed by visual examination and X-rays. In countries with good medical and dental care, it is most common in individuals with weakened immune systems, such as patients with AIDS. In addition to cleaning and pain medication, patients may be prescribed antibiotics such as amoxicillin, amoxicillin clavulanate, clindamycin, or doxycycline.
How does gingivitis progress to periodontitis?
Good oral health promotes good overall health, and the reverse is also true. Poor oral health can lead to difficulty eating, which can cause malnutrition. Painful or loose teeth can also cause a person to avoid certain foods or eat less. Malnutrition due to dental problems is of greatest concern for the elderly, for whom it can worsen other health conditions and contribute to mortality. Individuals who have serious illnesses, especially AIDS, are also at increased risk of malnutrition from dental problems.
Additionally, poor oral health can contribute to the development of disease. Increased bacterial growth in the mouth can cause inflammation and infection in other parts of the body. For example, Streptococcus in the mouth, the main contributor to biofilms on teeth, tartar, and dental caries, can spread throughout the body when there is damage to the tissues inside the mouth, as can happen during dental work. S. mutans produces a surface adhesin known as P1, which binds to salivary agglutinin on the surface of the tooth. P1 can also bind to extracellular matrix proteins including fibronectin and collagen. When Streptococcus enters the bloodstream as a result of tooth brushing or dental cleaning, it causes inflammation that can lead to the accumulation of plaque in the arteries and contribute to the development of atherosclerosis, a condition associated with cardiovascular disease, heart attack, and stroke. In some cases, bacteria that spread through the blood vessels can lodge in the heart and cause endocarditis (an example of a focal infection).
Oral Infections
As noted earlier, normal oral microbiota can cause dental and periodontal infections. However, there are number of other infections that can manifest in the oral cavity when other microbes are present.
Herpetic Gingivostomatitis
As described in Viral Infections of the Skin and Eyes , infections by herpes simplex virus type 1 (HSV-1) frequently manifest as oral herpes, also called acute herpes labialis and characterized by cold sores on the lips, mouth, or gums. HSV-1 can also cause acute herpetic gingivostomatitis, a condition that results in ulcers of the mucous membranes inside the mouth (Figure \(\PageIndex{6}\)). Herpetic gingivostomatitis is normally self-limiting except in immunocompromised patients. Like oral herpes, the infection is generally diagnosed through clinical examination, but cultures or biopsies may be obtained if other signs or symptoms suggest the possibility of a different causative agent. If treatment is needed, mouthwashes or antiviral medications such as acyclovir, famciclovir, or valacyclovir may be used.
Oral Thrush
The yeast Candida is part of the normal human microbiota, but overgrowths, especially of Candida albicans , can lead to infections in several parts of the body. When Candida infection develops in the oral cavity, it is called oral thrush. Oral thrush is most common in infants because they do not yet have well developed immune systems and have not acquired the robust normal microbiota that keeps Candida in check in adults. Oral thrush is also common in immunodeficient patients and is a common infection in patients with AIDS.
Oral thrush is characterized by the appearance of white patches and pseudomembranes in the mouth (Figure \(\PageIndex{7}\)) and can be associated with bleeding. The infection may be treated topically with nystatin or clotrimazole oral suspensions, although systemic treatment is sometimes needed. In serious cases, systemic azoles such as fluconazole or itraconazole (for strains resistant to fluconazole), may be used. Amphotericin B can also be used if the infection is severe or if the Candida species is azole-resistant.
Mumps
The viral disease mumps is an infection of the parotid glands, the largest of the three pairs of salivary glands ( Figure 24.1.2 ). The causative agent is mumps virus (MuV), a paramyxovirus with an envelope that has hemagglutinin and neuraminidase spikes. A fusion protein located on the surface of the envelope helps to fuse the viral envelope to the host cell plasma membrane.
Mumps virus is transmitted through respiratory droplets or through contact with contaminated saliva, making it quite contagious so that it can lead easily to epidemics. It causes fever, muscle pain, headache, pain with chewing, loss of appetite, fatigue, and weakness. There is swelling of the salivary glands and associated pain (Figure \(\PageIndex{8}\)). The virus can enter the bloodstream (viremia), allowing it to spread to the organs and the central nervous system. The infection ranges from subclinical cases to cases with serious complications, such as encephalitis, meningitis, and deafness. Inflammation of the pancreas, testes, ovaries, and breasts may also occur and cause permanent damage to those organs; despite these complications, a mumps infection rarely cause sterility.
Mumps can be recognized based on clinical signs and symptoms, and a diagnosis can be confirmed with laboratory testing. The virus can be identified using culture or molecular techniques such as RT-PCR. Serologic tests are also available, especially enzyme immunoassays that detect antibodies. There is no specific treatment for mumps, so supportive therapies are used. The most effective way to avoid infection is through vaccination. Although mumps used to be a common childhood disease, it is now rare in the United States due to vaccination with the measles, mumps, and rubella (MMR) vaccine.
Compare and contrast the signs and symptoms of herpetic gingivostomatitis, oral thrush, and mumps.
Infections of the mouth and oral cavity can be caused by a variety of pathogens, including bacteria, viruses, and fungi. Many of these infections only affect the mouth, but some can spread and become systemic infections. Figure \(\PageIndex{9}\) summarizes the main characteristics of common oral infections.
Key Concepts and Summary
- Dental caries , tartar , and gingivitis are caused by overgrowth of oral bacteria, usually Streptococcus and Actinomyces species, as a result of insufficient dental hygiene.
- Gingivitis can worsen, allowing Porphyromonas , Streptococcus , and Actinomyces species to spread and cause periodontitis . When Prevotella intermedia , Fusobacterium species, and Treponema vicentii are involved, it can lead to acute necrotizing ulcerative gingivitis .
- The herpes simplex virus type 1 can cause lesions of the mouth and throat called herpetic gingivostomatitis.
- Other infections of the mouth include oral thrush , a fungal infection caused by overgrowth of Candida yeast, and mumps , a viral infection of the salivary glands caused by the mumps virus, a paramyxovirus.
Footnotes
- 1 Hans-Peter Horz and Georg Conrads. “Methanogenic Archaea and Oral Infections—Ways to Unravel the Black Box.” Journal of Oral Microbiology 3(2011). doi: 10.3402/jom.v3i0.5940.
- 2 Hiroshi Maeda, Kimito Hirai, Junji Mineshiba, Tadashi Yamamoto, Susumu Kokeguchi, and Shogo Takashiba. “Medical Microbiological Approach to Archaea in Oral Infectious Diseases.” Japanese Dental Science Review 49: 2, p. 72–78.
- 3 Paul W. Lepp, Mary M. Brinig, Cleber C. Ouverney, Katherine Palm, Gary C. Armitage, and David A. Relman. “Methanogenic Archaea and Human Periodontal Disease.” Proceedings of the National Academy of Sciences of the United States of America 101 (2003): 16, pp. 6176–6181. doi: 10.1073/pnas.0308766101.


15.2: Simple Harmonic Motion
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Learning Objectives
- Define the terms period and frequency
- List the characteristics of simple harmonic motion
- Explain the concept of phase shift
- Write the equations of motion for the system of a mass and spring undergoing simple harmonic motion
- Describe the motion of a mass oscillating on a vertical spring
When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time (Figure \(\PageIndex{1}\)). The string vibrates around an equilibrium position, and one oscillation is completed when the string starts from the initial position, travels to one of the extreme positions, then to the other extreme position, and returns to its initial position. We define periodic motion to be any motion that repeats itself at regular time intervals, such as exhibited by the guitar string or by a child swinging on a swing. In this section, we study the basic characteristics of oscillations and their mathematical description.
Period and Frequency in Oscillations
In the absence of friction, the time to complete one oscillation remains constant and is called the period (T) . Its units are usually seconds, but may be any convenient unit of time. The word ‘period’ refers to the time for some event whether repetitive or not, but in this chapter, we shall deal primarily in periodic motion, which is by definition repetitive.
A concept closely related to period is the frequency of an event. Frequency (f) is defined to be the number of events per unit time. For periodic motion, frequency is the number of oscillations per unit time. The relationship between frequency and period is
\[f = \frac{1}{T} \ldotp \label{15.1}\]
The SI unit for frequency is the hertz (Hz) and is defined as one cycle per second :
\[1\; Hz = 1\; cycle/sec\; or\; 1\; Hz = \frac{1}{s} = 1\; s^{-1} \ldotp\]
A cycle is one complete oscillation
Example \(\PageIndex{1}\): Determining the Frequency of Medical Ultrasound
Ultrasound machines are used by medical professionals to make images for examining internal organs of the body. An ultrasound machine emits high-frequency sound waves, which reflect off the organs, and a computer receives the waves, using them to create a picture. We can use the formulas presented in this module to determine the frequency, based on what we know about oscillations. Consider a medical imaging device that produces ultrasound by oscillating with a period of 0.400 \(\mu\)s. What is the frequency of this oscillation?
Strategy
The period (T) is given and we are asked to find frequency (f).
Solution
Substitute 0.400 µs for T in f = \(\frac{1}{T}\):
\[f = \frac{1}{T} = \frac{1}{0.400 \times 10^{-6}\; s} \ldotp \nonumber\]
Solve to find
\[f = 2.50 \times 10^{6}\; Hz \ldotp \nonumber\]
Significance
This frequency of sound is much higher than the highest frequency that humans can hear (the range of human hearing is 20 Hz to 20,000 Hz); therefore, it is called ultrasound. Appropriate oscillations at this frequency generate ultrasound used for noninvasive medical diagnoses, such as observations of a fetus in the womb.
Characteristics of Simple Harmonic Motion
A very common type of periodic motion is called simple harmonic motion (SHM) . A system that oscillates with SHM is called a simple harmonic oscillator .
Simple Harmonic Motion
In simple harmonic motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and acts in the opposite direction of the displacement.
A good example of SHM is an object with mass \(m\) attached to a spring on a frictionless surface, as shown in Figure \(\PageIndex{2}\). The object oscillates around the equilibrium position, and the net force on the object is equal to the force provided by the spring. This force obeys Hooke’s law F s = −kx, as discussed in a previous chapter.
If the net force can be described by Hooke’s law and there is no damping (slowing down due to friction or other nonconservative forces), then a simple harmonic oscillator oscillates with equal displacement on either side of the equilibrium position, as shown for an object on a spring in Figure \(\PageIndex{2}\). The maximum displacement from equilibrium is called the amplitude (A) . The units for amplitude and displacement are the same but depend on the type of oscillation. For the object on the spring, the units of amplitude and displacement are meters.
What is so significant about SHM? For one thing, the period \(T\) and frequency \(f\) of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, oscillates with the same frequency whether plucked gently or hard.
Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is. A very stiff object has a large force constant (k) , which causes the system to have a smaller period. For example, you can adjust a diving board’s stiffness—the stiffer it is, the faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one. In fact, the mass m and the force constant k are the only factors that affect the period and frequency of SHM. To derive an equation for the period and the frequency, we must first define and analyze the equations of motion. Note that the force constant is sometimes referred to as the spring constant .
Equations of SHM
Consider a block attached to a spring on a frictionless table (Figure \(\PageIndex{3}\)). The equilibrium position (the position where the spring is neither stretched nor compressed) is marked as x = 0 . At the equilibrium position, the net force is zero.
Work is done on the block to pull it out to a position of x = + A, and it is then released from rest. The maximum x-position (A) is called the amplitude of the motion. The block begins to oscillate in SHM between x = + A and x = −A, where A is the amplitude of the motion and T is the period of the oscillation. The period is the time for one oscillation. Figure \(\PageIndex{4}\) shows the motion of the block as it completes one and a half oscillations after release.
Figure \(\PageIndex{4}\) shows a plot of the position of the block versus time. When the position is plotted versus time, it is clear that the data can be modeled by a cosine function with an amplitude \(A\) and a period \(T\). The cosine function cos\(\theta\) repeats every multiple of 2\(\pi\), whereas the motion of the block repeats every period T. However, the function \(\cos \left(\dfrac{2 \pi}{T} t \right)\) repeats every integer multiple of the period. The maximum of the cosine function is one, so it is necessary to multiply the cosine function by the amplitude A.
\[x(t) = A \cos \left(\dfrac{2 \pi}{T} t \right) = A \cos (\omega t) \ldotp \label{15.2}\]
Recall from the chapter on rotation that the angular frequency equals \(\omega = \frac{d \theta}{dt}\). In this case, the period is constant, so the angular frequency is defined as 2\(\pi\) divided by the period, \(\omega = \frac{2 \pi}{T}\).
The equation for the position as a function of time \(x(t) = A\cos( \omega t)\) is good for modeling data, where the position of the block at the initial time t = 0.00 s is at the amplitude A and the initial velocity is zero. Often when taking experimental data, the position of the mass at the initial time t = 0.00 s is not equal to the amplitude and the initial velocity is not zero. Consider 10 seconds of data collected by a student in lab, shown in Figure \(\PageIndex{6}\).
The data in Figure \(\PageIndex{6}\) can still be modeled with a periodic function, like a cosine function, but the function is shifted to the right. This shift is known as a phase shift and is usually represented by the Greek letter phi (\(\phi\)). The equation of the position as a function of time for a block on a spring becomes
\[x(t) = A \cos (\omega t + \phi) \ldotp\]
This is the generalized equation for SHM where t is the time measured in seconds, \(\omega\) is the angular frequency with units of inverse seconds, A is the amplitude measured in meters or centimeters, and \(\phi\) is the phase shift measured in radians (Figure \(\PageIndex{7}\)). It should be noted that because sine and cosine functions differ only by a phase shift, this motion could be modeled using either the cosine or sine function.
The velocity of the mass on a spring, oscillating in SHM, can be found by taking the derivative of the position equation:
\[v(t) = \frac{dx}{dt} = \frac{d}{dt} (A \cos (\omega t + \phi)) = -A \omega \sin(\omega t + \varphi) = -v_{max} \sin (\omega t + \phi) \ldotp\]
Because the sine function oscillates between –1 and +1, the maximum velocity is the amplitude times the angular frequency, v max = A\(\omega\). The maximum velocity occurs at the equilibrium position (x = 0) when the mass is moving toward x = + A. The maximum velocity in the negative direction is attained at the equilibrium position (x = 0) when the mass is moving toward x = −A and is equal to −v max .
The acceleration of the mass on the spring can be found by taking the time derivative of the velocity:
\[a(t) = \frac{dv}{dt} = \frac{d}{dt} (-A \omega \sin (\omega t + \phi)) = -A \omega^{2} \cos (\omega t + \varphi) = -a_{max} \cos (\omega t + \phi) \ldotp\]
The maximum acceleration is a max = A\(\omega^{2}\). The maximum acceleration occurs at the position (x = −A), and the acceleration at the position (x = −A) and is equal to −a max .
Summary of Equations of Motion for SHM
In summary, the oscillatory motion of a block on a spring can be modeled with the following equations of motion:
\[ \begin{align} x(t) &= A \cos (\omega t + \phi) \label{15.3} \\[4pt] v(t) &= -v_{max} \sin (\omega t + \phi) \label{15.4} \\[4pt] a(t) &= -a_{max} \cos (\omega t + \phi) \label{15.5} \end{align}\]
with
\[ \begin{align} x_{max} &= A \label{15.6} \\[4pt] v_{max} &= A \omega \label{15.7} \\[4pt] a_{max} &= A \omega^{2} \ldotp \label{15.8} \end{align}\]
Here, \(A\) is the amplitude of the motion, \(T\) is the period, \(\phi\) is the phase shift, and \(\omega = \frac{2 \pi}{T}\) = 2\(\pi\)f is the angular frequency of the motion of the block.
Example 15.2: Determining the Equations of Motion for a Block and a Spring
A 2.00-kg block is placed on a frictionless surface. A spring with a force constant of k = 32.00 N/m is attached to the block, and the opposite end of the spring is attached to the wall. The spring can be compressed or extended. The equilibrium position is marked as x = 0.00 m. Work is done on the block, pulling it out to x = + 0.02 m. The block is released from rest and oscillates between x = + 0.02 m and x = −0.02 m. The period of the motion is 1.57 s. Determine the equations of motion.
Strategy
We first find the angular frequency. The phase shift is zero, \(\phi\) = 0.00 rad, because the block is released from rest at x = A = + 0.02 m. Once the angular frequency is found, we can determine the maximum velocity and maximum acceleration.
Solution
The angular frequency can be found and used to find the maximum velocity and maximum acceleration:
\[\begin{split} \omega & = \frac{2 \pi}{1.57\; s} = 4.00\; s^{-1}; \\ v_{max} & = A \omega = (0.02\; m)(4.00\; s^{-1}) = 0.08\; m/s; \\ a_{max} & = A \omega^{2} = (0.02; m)(4.00\; s^{-1})^{2} = 0.32\; m/s^{2} \ldotp \end{split}\]
All that is left is to fill in the equations of motion:
\[\begin{split} x(t) & = a \cos (\omega t + \phi) = (0.02\; m) \cos (4.00\; s^{-1} t); \\ v(t) & = -v_{max} \sin (\omega t + \phi) = (-0.8\; m/s) \sin (4.00\; s^{-1} t); \\ a(t) & = -a_{max} \cos (\omega t + \phi) = (-0.32\; m/s^{2}) \cos (4.00\; s^{-1} t) \ldotp \end{split}\]
Significance
The position, velocity, and acceleration can be found for any time. It is important to remember that when using these equations, your calculator must be in radians mode.
The Period and Frequency of a Mass on a Spring
One interesting characteristic of the SHM of an object attached to a spring is that the angular frequency, and therefore the period and frequency of the motion, depend on only the mass and the force constant, and not on other factors such as the amplitude of the motion. We can use the equations of motion and Newton’s second law (\(\vec{F}_{net} = m \vec{a}\)) to find equations for the angular frequency, frequency, and period.
Consider the block on a spring on a frictionless surface. There are three forces on the mass: the weight, the normal force, and the force due to the spring. The only two forces that act perpendicular to the surface are the weight and the normal force, which have equal magnitudes and opposite directions, and thus sum to zero. The only force that acts parallel to the surface is the force due to the spring, so the net force must be equal to the force of the spring:
\[\begin{split} F_{x} & = -kx; \\ ma & = -kx; \\ m \frac{d^{2} x}{dt^{2}} & = -kx; \\ \frac{d^{2} x}{dt^{2}} & = - \frac{k}{m} x \ldotp \end{split}\]
Substituting the equations of motion for x and a gives us
\[-A \omega^{2} \cos (\omega t + \phi) = - \frac{k}{m} A \cos (\omega t +\phi) \ldotp\]
Cancelling out like terms and solving for the angular frequency yields
\[\omega = \sqrt{\frac{k}{m}} \ldotp \label{15.9}\]
The angular frequency depends only on the force constant and the mass, and not the amplitude. The angular frequency is defined as \(\omega = \frac{2 \pi}{T}\), which yields an equation for the period of the motion:
\[T = 2 \pi \sqrt{\frac{m}{k}} \ldotp \label{15.10}\]
The period also depends only on the mass and the force constant. The greater the mass, the longer the period. The stiffer the spring, the shorter the period. The frequency is
\[f = \frac{1}{T} = \frac{1}{2 \pi} \sqrt{\frac{k}{m}} \ldotp \label{15.11}\]
Vertical Motion and a Horizontal Spring
When a spring is hung vertically and a block is attached and set in motion, the block oscillates in SHM. In this case, there is no normal force, and the net effect of the force of gravity is to change the equilibrium position. Consider Figure \(\PageIndex{8}\). Two forces act on the block: the weight and the force of the spring. The weight is constant and the force of the spring changes as the length of the spring changes.
When the block reaches the equilibrium position, as seen in Figure \(\PageIndex{8}\), the force of the spring equals the weight of the block, F net = F s − mg = 0, where
\[-k (- \Delta y) = mg \ldotp\]
From the figure, the change in the position is \( \Delta y = y_{0}-y_{1} \) and since \(-k (- \Delta y) = mg\), we have
\[k (y_{0} - y_{1}) - mg = 0 \ldotp\]
If the block is displaced and released, it will oscillate around the new equilibrium position. As shown in Figure \(\PageIndex{9}\), if the position of the block is recorded as a function of time, the recording is a periodic function. If the block is displaced to a position y, the net force becomes F net = k(y 0 - y) − mg. But we found that at the equilibrium position, mg = k\(\Delta\)y = ky 0 − ky 1 . Substituting for the weight in the equation yields
\[F_{net} =ky_{0} - ky - (ky_{0} - ky_{1}) = k (y_{1} - y) \ldotp\]
Recall that y 1 is just the equilibrium position and any position can be set to be the point y = 0.00 m. So let’s set y 1 to y = 0.00 m. The net force then becomes
\[\begin{split}F_{net} & = -ky; \\ m \frac{d^{2} y}{dt^{2}} & = -ky \ldotp \end{split}\]
This is just what we found previously for a horizontally sliding mass on a spring. The constant force of gravity only served to shift the equilibrium location of the mass. Therefore, the solution should be the same form as for a block on a horizontal spring, y(t) = Acos(\(\omega\)t + \(\phi\)). The equations for the velocity and the acceleration also have the same form as for the horizontal case. Note that the inclusion of the phase shift means that the motion can actually be modeled using either a cosine or a sine function, since these two functions only differ by a phase shift.


6.4: Chapter 3
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- Chippewa Valley Technical College via OpenRN
Critical Thinking Activities
You can review additional information regarding these answers in the corresponding section in which the Critical Thinking activities appear.
Critical Thinking Activity Section 3.2a
Patient education regarding the importance of adhering to the prescribed medication regimen is vital to help prevent drug resistance. During patient education, the nurse should emphasize the need to complete the full course of medication, in the dosages and frequencies prescribed, to treat the infection and prevent the dangers of drug resistance. In addition to patient education, another solution used to prevent drug resistance in high-risk medications is called directly observed therapy (DOT). DOT is the supervised administration of medications to patients. Patients are required to visit a health-care facility to receive their medications or a health-care professional administers medications in the patients’ homes or other designated location. DOT has been implemented worldwide for the treatment of tuberculosis (TB), and research has been shown it to be effective in treating infections successfully and preventing additional drug resistance.
Critical Thinking Activity Section 3.5a
The administration of penicillin should be postponed for four hours because citrus juice can impede absorption of drugs like penicillin. The remaining doses of penicillin for the day should be rescheduled based on the time the breakfast dose was actually administered. Additionally, the patient should be educated about avoiding citrus juice while taking penicillin, and the dietary department should be notified to remove citrus juice from the meal choices.
Critical Thinking Activity Section 3.6a
The changes in the patient’s renal labs demonstrate decreased renal function. The prescribing provider should be notified prior to administering additional doses of cefazolin because the medication or the dosage will likely need to be revised based on the patient’s response.
Critical Thinking Activity Section 3.7a
The nurse should check the progress notes in the electronic medical record to determine if anything is documented about John’s allergies and the decision to use imipenem. If nothing is documented, then the nurse should notify the prescribing provider of the patient’s allergies to penicillin to confirm the appropriateness of this medication for John, document the provider’s response in the medical record, and provide this information in the end-of-shift handoff report.
Critical Thinking Activity Section 3.8a
Monobactams are narrow-spectrum antibacterial medications used primarily to treat gram-negative bacteria like Pseudomonas aeruginosa. However, MRSA is a gram-positive bacteria, so aztreonam will not be effective in fighting this infection. The nurse should notify the prescribing provider of the results of the new culture report before administering the azotreonam.
Critical Thinking Activity Section 3.9a
The nurse should review the other medications the patient is taking. Trimethoprim-Sulfamethoxazole has many significant drug interactions, including oral diabetics. This medication may increase hypoglycemic effects requiring closer monitoring of blood sugars. Additionally, the patient’s renal status should be verified before administration of trimethoprim-sulfamethoxazole because dose adjustment may be required.
Critical Thinking Activity Section 3.10a
The nurse should immediately stop the medication and notify the provider regarding the new onset of tendon pain because this symptom indicates an adverse reaction of levofloxacin may be occurring.
Critical Thinking Activity Section 3.11a
The nurse should notify the provider of the patient’s change condition because it may indicate an adverse effect of liver damage is occurring.
Critical Thinking Activity Section 3.12a
The nurse should not administer the medication until the trough levels have been drawn. The nurse should phone the lab and check on the status of the laboratory trough level.
Critical Thinking Activity Section 3.13a
The patient is under the age of six and is at risk for the adverse effect of teeth discoloration. The nurse should advocate for this patient by notifying the prescribing provider of this concern and requesting an alternate medication.
Critical Thinking Activity Section 3.14a
Oseltamivir should be administered within the first 24-48 hours of the onset of influenza symptoms. The patient may have already passed the window for maximum therapeutic effectiveness of oseltamivir. The provider should be notified regarding the onset of symptoms to clarify the prescription.
Critical Thinking Activity Section 3.15a
If there are no signs of improvement from the prescribed medication therapy, the nurse should notify the provider.
Critical Thinking Activity Section 3.16a
In order to prevent malaria, the CDC recommends patients should take antimalarial medications for four weeks after leaving the infected area. The nurse should provide additional patient education to the patient regarding this recommendation and evaluate for patient understanding.
Critical Thinking Activity Section 3.17a
Metronidazole is commonly used to treat C-diff. The medication must be given by mouth for the indication of a gastrointestinal infection like C-diff.
Critical Thinking Activity Section 3.18a
The nurse should provide education regarding the use of the medication, as well as ways to prevent re-infection. Methods to prevent reinfection include using proper handwashing, washing all fruits and vegetables, and wearing shoes in the barn or where animals and their feces are present.
Critical Thinking Activity Section 3.19a
The nurse should explain that directly observed therapy (DOT) means the administration of this medication will be supervised to ensure all doses are taken as prescribed to be sure the infection is treated properly and drug resistance does not develop. The patient will be required to visit a health-care facility to receive their medications or a health-care professional will administer the medication in the patient’s home or other designated location.
Critical Thinking Activity Section 3.20a
The nurse should not administer the vancomycin until after the trough level is drawn. The nurse should call the lab to request prioritization of completing the trough level.


8.4: Arresting Juveniles
Whenever a juvenile is arrested for an act of juvenile delinquency, he must immediately be advised of his legal rights. [ 18 U.S.C.A. § 5033 (West 1985).] The Attorney General (United States Attorney) shall be notified. Id. The juvenile's parents, guardian or custodian must also be immediately notified of his arrest as well as his rights and of the nature of the alleged offense. This requirement is not invoked when a juvenile is arrested and placed into administrative detention, but rather is initiated by the juvenile's placement into custody after the filing of an information alleging delinquent conduct. [ United States v. Juvenile Male , 74 F.3d 526, 530 (4th Cir. 1996 ).] Notification made after a statement has been given or made without spelling out the juvenile's right to notify a responsible adult cannot satisfy the statutory mandate. [ United States v. Nash , 620 F. Supp. 1439, 1442-43 (S.D.N.Y. 1985). ]
Figure 8.3 USMS-Omaha-2 by the Omaha Police Department .is used under a CC BY 2.0 license.
If the juvenile is an alien, a reasonable effort must be made to reach his parents, and if not feasible, prompt notice to his country's Consulate should be made. [ United States v. Doe , 862 F. 2d 776, 780 (9th Cir. 1988). ]
Figure 8.4 James G. Bowles Juvenile Hall is operated 24 hours a day, 365 days a year by the Kern County Probation Department as a secure detention facility for minors. JGB Juvenile Hall by Tabitha Raber is used under a CC BY 4.0 license.
Regarding cases of juvenile delinquency, the United States Attorney can proceed in different ways. The case can be referred to state authorities, attempts to adjudicate the juvenile as a delinquent can be made, or the United States Attorney can move to transfer the juvenile for criminal prosecution as an adult.


2.3: John Winthrop (1588–1649)
John Winthrop was born into a prosperous family in Groton, England, and followed the path of many such prosperous gentlemen by studying at Cambridge University. Though he practiced law at the Inner Temple, he soon shifted paths when he became a Puritan, devoted to purifying the Anglican Church from within and eschewing lingering Catholic practices and rituals. When Charles I ascended the throne, Puritans such as Winthrop faced being ruled by a monarch with clear and expressed sympathies for Catholicism. To avoid losing his earthly possessions to the throne, Winthrop joined a group of Puritans who obtained permission from the king to leave England for America. They gained a charter from the Council for New England and formed themselves as “The Company of Massachusetts Bay in New England,” free to found a religious colony beyond the king’s rule. Their colony would in time become New England’s chief colony.
Image \(\PageIndex{1}\): John Winthrop
In 1629, Winthrop was chosen governor, a position he would hold for twenty years. The initial group of colonists left England on April 8, 1630, sailing on the Arbella . Either before embarkation or early in the voyage itself, Winthrop gave his sermon A Model of Christian Charity which envisaged a harmonious Puritan community that would serve as guide and model for future emigrants. Preparing the colonists to face adversity and temptation, the sermon also prepared for their future society’s being built on and guided by Christian principles. As governor of the colony, Winthrop himself modeled these principles through his steadfast morality and selfless concern for others.
A Model of Christian Charity speaks plainly and clearly of an earthly life in the wilderness guiding towards God’s heavenly city, the new Jerusalem.
Image \(\PageIndex{2}\): Massachusetts Bay Colony Seal


1.2: Vector Multiplication
Now we know how to do some math with vectors, and the question arises, “If we can add and subtract vectors, can we also multiply them?” The answer is yes and no. It turns out that there is not one unique way to define a product of two vectors, but instead there are two…
Scalar Product
Soon we will be looking at how we can describe the effect that a force pushing on an object has on its speed as it moves from one position to another. The force is a vector, because it has a magnitude (the amount of the push) and a direction (the way the push is directed). And the movement of the object is also a vector (tail is at the object’s starting point, and head is at its ending point). It will turn out that this effect is describable mathematically as the product of the amount of force and the amount of movement. This is simple to compute if the push is along the direction of movement, but what if it is not? It turns out that only the amount of push that acts in the direction of the movement will affect the object's speed.
We therefore would like to introduce the notion of the projection of one vector onto another. The best description of this is, "the amount of a given vector that points along the other vector." This could be imagined as the “shadow” one vector casts upon another vector:
Figure 1.2.1 – Projecting One Vector Onto Another
So if we want to multiply the length of a vector by the amount of a second vector that is projected onto it we get:
\[ ( \text{projection of } \overrightarrow A \text{ onto } \overrightarrow B )( \text{magnitude of } \overrightarrow B ) = (A \cos \theta) (B) = AB \cos \theta\]
This is the first of the two types of vector multiplication, and it is called a scalar product , because the result of the product is a scalar. We usually write the product with a dot (giving its alternative name of dot product ):
Exercise
The vector \(\overrightarrow A\) has a magnitude of 120 units, and when projected onto \(\overrightarrow B\), the projected portion has a value of 105 units. Suppose that \(\overrightarrow B\) is now projected onto \(\overrightarrow A\), and the projected length is 49 units. Find the magnitude of \(\overrightarrow B\).
- Solution
-
The factor that determines the length of the projection is \(\cos \theta\). The angle between the two vectors is the same regardless of which vector is projected, so the factor is the same in both directions. The projection of \(\overrightarrow A\) onto \(\overrightarrow B\) is 7/8 of the magnitude of \(\overrightarrow A\), so the magnitude of \(\overrightarrow B\) must be 8/7 of its projection, which is 56 units. Note that when the projection of one vector is multiplied by the magnitude of the other, the same product results regardless of which way the projection occurs. That is, the scalar product is the same in either order (i.e. it is commutative).
Note that a scalar product of a vector with itself is the square of the magnitude of that vector:
\[\overrightarrow A \cdot \overrightarrow A = A^2 \cos 0 = A^2\]
It should be immediately clear what the scalar products of the unit vectors are. They have unit length, so a scalar product of a unit vector with itself is just 1.
\[\widehat i \cdot \widehat i = \widehat j \cdot \widehat j = \widehat k \cdot \widehat k = 1\]
They are also mutually orthogonal , so the scalar products with each other are zero:
\[\widehat i \cdot \widehat j = \widehat j \cdot \widehat k = \widehat k \cdot \widehat i = 0\]
This gives us an alternative way to look at components, which are projections of a vector onto the coordinate axes. Since the unit vectors point along the \(x\), \(y\), and \(z\) directions, the components of a vector can be expressed as a dot product. For example:
\[ \begin{align} \overrightarrow A \cdot \widehat i &= (A_x \widehat i + A_y \widehat j + A_z \widehat k) \cdot \widehat i \nonumber\\[5pt] &= A_x \cancelto{1}{\widehat i \cdot \widehat i } + A_y \cancelto{0} {\widehat j \cdot \widehat i} +A_z \cancelto{0}{ \widehat k \cdot \widehat i} \end{align}\]
Unit vectors also show us an easy way to take the scalar product of two vectors whose components we know. Start with two vectors written in component form:
\[\overrightarrow A = A_x\widehat i + A_y \widehat j\nonumber\]
\[\overrightarrow B = B_x\widehat i + B_y \widehat j\nonumber\]
then just do "normal algebra," distributing the dot product as you would with normal multiplication:
If we didn’t have this simple result, think about what we would have to do: We would need to calculate the angles each vector makes with (say) the \(x\)-axis. Then from those two angles, we need to figure out the angles between the two vectors. Then we would need to compute the magnitudes of the two vectors. Finally, with the magnitudes of the vectors and the angle between the vectors, we could finally plug into our scalar product equation.
Alert
With two different ways to compute a scalar product, it should be clear that the simplest method to use will depend upon what information is provided. If you are given (or can easily ascertain) the magnitudes of the vectors and the angle between them, then use Equation 1.2.2 , but if you are given (or can easily ascertain) the components of the vectors, use Equation 1.2.7 .
Exercise
The two vectors given below are perpendicular to each other. Find the unknown \(z\)-component.
\[\overrightarrow A = +5 \widehat i - 4 \widehat j - \widehat k \;\;\;\;\;\; \overrightarrow B = +2 \widehat i + 3 \widehat j + B_z \widehat k \nonumber\]
- Solution
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The scalar product of two vectors is proportional to the cosine of the angle between them. This means that if they are orthogonal, the scalar product is zero. The dot product is easy to compute when given the components, so we do so and solve for \(B_z\):
\[0 = \overrightarrow A \cdot \overrightarrow B = \left( +5 \right) \left( +2 \right) + \left( -4 \right) \left( +3 \right) \ + \left( -1 \right) \left( B_z \right) \;\;\; \Rightarrow \;\;\; B_z = -2 \nonumber\]
The scalar product of two vectors in terms of column vectors works exactly how you would expect – simply multiply the similar components and sum all the products.
Vector Product
As mentioned earlier, there are actually two ways to define products of vectors. If the scalar product involves the amount of one vector that is parallel to the other vector, then it should not be surprising that our other product involves the amount of a vector that is perpendicular to the other vector.
Figure 1.2.2 – Portion of One Vector Perpendicular to Another
If we take a product like before, multiplying this perpendicular piece by the magnitude of the other vector, we get an expression similar to what we got for the scalar product, this time with a sine function rather than a cosine. For reasons that will be clear soon, this type of product is referred to as a vector product . Because this is distinct from the scalar product, we use a different mathematical notation as well – a cross rather than a dot (giving it an alternative name of cross product ). This has a simple (though not entirely useful, at least not in physics) geometric interpretation in terms of the parallelogram defined by the two vectors:
Figure 1.2.3 – Constructing a Vector Product (Magnitude)
But there is another even bigger difference between the vector and scalar products. While the projection always lands parallel to the second vector, the perpendicular part implies an orientation, since the perpendicular part can point in multiple directions. Any quantity that has an orientation has the potential to be a vector, and in fact we will define a vector that results from this type of product as follows: If we follow the perimeter of the parallelogram above in the direction given by the two vectors, we get a clockwise orientation [ Would we get the same orientation if the product was in the opposite order: \(\overrightarrow B\) \(\times\) \(\overrightarrow A\)? ]. We turn this circulation direction into a vector direction (which points in a specific direction in space) using a convention called the right-hand-rule :
Point the fingers of your right hand in the direction of the first vector in the product, then orient your hand such that those fingers curl naturally into the direction of the second vector in the product. As your fingers curl, your extended thumb points in a direction that is perpendicular to both vectors in the product. This is the direction of the vector that results from the cross product.
If we perform the cross product with the vectors in the opposite order, our fingers curl in the opposite direction, which makes our thumb point in the opposite direction in space. This means that the cross product has an anticommutative property:
\[ \overrightarrow A \times \overrightarrow B = -\overrightarrow B \times \overrightarrow A\]
A cross product of any vector with itself gives zero, since the part of the first vector that is perpendicular to the second vector is zero:
\[ | \overrightarrow A \times \overrightarrow A | = A^2 \sin 0 = 0\]
As with the scalar product, the vector product can be easily expressed with components and unit vectors. The vector products of the unit vectors with themselves are zero. Each of the unit vectors is at right angles with the other two unit vectors, so the magnitude of the cross product of two unit vectors is also a unit vector (since the sine of the angle between them is 1).
Convention: Right-handed Coordinate Systems
We will always choose a right-handed coordinate system, which means that using the right-hand-rule on the +\(x\) to +\(y\) axis yields the +\(z\) axis.
In terms of the unit vectors, we therefore have:
\[ \widehat i \times \widehat i = \widehat j \times \widehat j = \widehat k \times \widehat k = 0\]
and
This allows us to do cross products purely mathematically (without resorting to the right-hand-rule) when we know the components, as we did for the scalar product. Again start with two vectors in component form:
\[\overrightarrow A = A_x\widehat i + A_y \widehat j\nonumber\]
\[\overrightarrow B = B_x\widehat i + B_y \widehat j\nonumber\]
then, as in the case of the scalar product, just do "normal algebra" (apart from the cross product being anti-commutative) by distributing the cross product, and applying the unit vector cross products above:
\[ \begin{align} \overrightarrow A \times \overrightarrow B &= (A_x \widehat i + A_y \widehat j) \times (B_x \widehat i + B_y \widehat j) \\[5pt] &= (A_x B_x) \cancelto{0}{\widehat i \times \widehat i} + (A_yB_x) \cancelto{- \widehat k} {\widehat j \times \widehat i} + (A_x B_y ) \cancelto{+\widehat k}{ \widehat i \times \widehat j} + (A_y B_y) \cancelto{0}{\widehat j \times \widehat j} \\[5pt] &= (A_x B_y - A_yB_x ) \widehat k \end{align}\]
It is not obvious right now how we will use the dot and cross product in physics, but it is coming, so it’s a good idea to get a firm grasp on these important tools now.
Exercise
Using the two vectors \(\overrightarrow A\) and \(\overrightarrow B\) from the previous exercise...
- compute the vector product \(\overrightarrow A\times\overrightarrow B\), and
- use the result of the vector product to confirm that \(\overrightarrow A\) and \(\overrightarrow B\) are perpendicular to each other.
- Solution
-
a. The vector product is a straightforward computation that only requires the distributive property and Equations 1.2.11 and 1.2.12:
\[\begin{array}{l}\overrightarrow A\times\overrightarrow B && = && \left(+5 \widehat i - 4 \widehat j - \widehat k\right)\times\left(+2 \widehat i + 3\widehat j - 2\widehat k\right)\\ && = && +10 \cancelto{0}{\left(\widehat i \times \widehat i\right)}+15 \left(\widehat i \times \widehat j \right) - 10 \left(\widehat i \times \widehat k \right) - 8 \left(\widehat j \times \widehat i \right) -12 \cancelto{0}{\left(\widehat j \times \widehat j\right)} + 8 \left(\widehat j \times \widehat k \right) - 2\left(\widehat k \times \widehat i \right) - 3 \left(\widehat k \times \widehat j \right) + 2\cancelto{0}{\left(\widehat k\times \widehat k\right)} \\ && = && +11 \widehat i +8\widehat j +23\widehat k \end{array}\nonumber\]
b. We can compute the magnitudes of the three vectors \(\overrightarrow A\), \(\overrightarrow B\), and \(\overrightarrow C=\overrightarrow A \times \overrightarrow B\), and use them to find the angle:
\[\left. \begin{array}{l} \left|\overrightarrow A\right| = \sqrt{(+5)^2+(-4)^2+(-1)^2} = \sqrt{42} \\ \left|\overrightarrow B\right| = \sqrt{(+2)^2+(+3)^2+(-2)^2} = \sqrt{17} \\ \left|\overrightarrow C\right| = \sqrt{(+11)^2+(+8)^2+(23)^2} = \sqrt{714}\end{array} \right\} \;\;\;\left|\overrightarrow C\right|=\left|\overrightarrow A\right|\left|\overrightarrow B\right|\sin\theta\;\;\;\Rightarrow\;\;\;\sin\theta=\frac{\sqrt{714}}{\sqrt{42}\sqrt{17}}=1\;\;\;\Rightarrow\;\;\;\theta=90^o\nonumber\]


10.12.1: The Social Construction of Sexuality
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Excerpt from: Coleman, K. Alienation through Social Construction: A Call for the Re-humanization of Sexuality. Journal of Positive Sexuality, Vol. 1, June 2015, page 25-30.
Sexuality is a topic that comes with a great deal of controversy. The debate over whether sexuality is ascribed or achieved is an old and a loaded one. Often the same people (newscasters, elected officials, religious leaders, etc.) generating discussion on the topic misuse the proper terminology surrounding the subject, resulting in misleading and fallacious constructs being deployed and socially reinforced.  So to argue whether one is born “gay” or “straight” is a loaded debate without the possibility of a solution because: People cannot be “gay” or “straight”; rather, only behaviors can be categorized in these binary constructs.  Thus, “gay” or “straight” can only be applied as descriptors to individual sexual actions rather than to people as a categorical approach to identity.
The American Sociological Association (ASA), American Medical Association (AMA), American Psychological Association, and American Pediatric Association all recognize sexuality as being experienced by the actor on a continuum and based on a personal sense of identity reflective of sexual attractions. The ASA, AMA, and American Psychological Association recognize that while there is no absolute consensus as to what determines one’s sexual orientation, most people experience little or no sense of choice pertaining to their orientations, leading researchers to conclude, historically, that sexual orientation is biologically determined.  In other words, the social actor has a choice whether to exercise behaviors indicative of current social definition of heterosexuality, homosexuality, bisexuality, or asexuality.  So, while a social actor might have strong sexual urges for someone of the same sex category, they may never act on it. Conversely, someone having sexual attraction for others of the different sex may exercise behaviors indicative of current cultural definitions descriptive of homosexuality.   For example, we learn (explicitly from parents, teachers, politicians, religious leaders, and other significant figures in our lives, or implicitly from images, themes, or messages in popular culture) how to have sex, with whom to have sex, with what motivations to have sex.  Sexual behaviors—focusing only on behaviors that are products of consensus from both (or all) parties—are behaviors (like all others) that are learned through socialization. They develop and progress as we develop and progress. Consequently, sexual behaviors are not always in response to sexual desire because of two central explanations:
- Motivations for sexual behavior vary; and
- The current social construction of normative sexual behavior
is reflective of ultra-conservative (prudish) ideals and saturated with religious underpinnings—or at least the most current culturally valued behaviors are. Nevertheless sexual orientation remains inherent in individuals and, thus, unchanging.
Common terms pertaining to sexuality (sexual orientation, desire, and behavior) are constantly presented in the media as being interchangeable. However, they are not. Subsequently, much of the population is left uneducated (or inaccurately educated) due mainly to this misrepresentation in popular culture and media and to the lack of passable education in the public school system on this topic. Currently, there is no curriculum mandate for teaching human sexuality (vastly different from “sex ed”). This is in combination with the content of sex education most often resulting in the over-emphasis given to abstinence-only education. 117 Such restriction on students’ access to fair and adequate education on human sexuality only adds to the distortion of sexuality commonly presented as “normative” in popular culture.
The inconsistency surrounding the use of the mentioned terminology (sexual orientation, sexual desire, and sexual behavior), the lack of education in our public schools, and limited interpretations of sexuality presented in the media have contributed to a poorly informed public. The incapacity to recognize the differences between these terms outside of the individual only enhances the risk of not being able to identify them correctly within the individual’s experiences and the inability for one to fully understand their own complex sexuality. At risk is our ability to understand sexuality as one part of the human experience, instead of focusing on categories created in an attempt to indicate one’s full social identity. The ability to separately define behaviors from orientation will allow actors to recognize sexuality as a continuum within the human experience with room for biological variation. After all, biology loves variation; it is we humans who tend to struggle with it.
114
American Psychological Association. (2011). Sexual orientation. Washington, DC: Author. Retrieved from:
www.apa.org/topics/lgbt/orientation.aspx
115
Ibid
116
Katz, J. (2007). The invention of heterosexuality. Chicago, IL: University of Chicago Press.
117
Landry, D. J., Darroch, J., E., Singh, S., Higgins, J., & Donovan, P. (2003). Factors associated with the content of sex
education in U.S. public secondary schools. Perspectives on Sexual and Reproductive Health, 35, 261-269.


8.2: Focusing on Normal Subgroups
We first provide a theorem that will help us in identifying when a subgroup of a group is normal. First, we provide a definition.
Definition
Let \(H\) be a subgroup of \(G\) and \(a,b\) in \(G\text{.}\) We define
\begin{equation*} aHb=\{ahb\,:h\in H\}. \end{equation*}
Theorem \(\PageIndex{1}\)
Let \(H\) be a subgroup of a group \(G\text{.}\) Then the following are equivalent:
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\(H\) is normal in \(G\text{;}\)
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\(aHa^{-1}=H\) for all \(a\in G\text{;}\)
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\(aHa^{-1}\subseteq H\) for all \(a\in G\text{.}\)
- Proof
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The equivalence of Statements 1 and 2 is clear, as is the fact that Statement 2 implies Statement 3. So it suffices to show that Statement 3 implies Statement 2. Well, assume that \(aHa^{−1}⊆H\) for all \(a∈G\), and let \(b∈G\). We want to show that \(bHb^{−1}=H\). Since Statement 3 holds, we clearly have \(bHb^{−1}⊆H\). But, again using Statement 3, we also have \(b^{−1}Hb⊆H\); multiplying both sides of this equation on the left by \(b\) and on the right by \(b^{−1}\), we obtain \(H⊆bHb^{−1}\). Hence, since \(bHb^{−1}⊆H\) and \(H⊆bHb^{−1}\), those two sets are equal. Thus, Statement 2 is proven.
We now consider some examples and nonexamples of normal subgroups of groups.
Example \(\PageIndex{1}\)
- As previously mentioned, the trivial and improper subgroups of any group \(G\) are normal in \(G\text{.}\)
- As previously mentioned, if group \(G\) is abelian then each of its subgroups is normal in \(G\text{.}\)
- Suppose \(H\leq G\) has \((G:H)=2\text{.}\) Then \(H \unlhd G\text{.}\) The proof of this is left as an exercise for the reader.
- Example \(7.2.1\) shows that subgroup \(H=\langle (12)\rangle\) isn't normal in \(S_3\) (for example, \((13)H\neq H(13)\text{.}\) But \(\langle (123)\rangle\) must be normal in \(S_3\) since \((S_3:\langle (123)\rangle )=6/3=2.\)
- \(\langle r\rangle\) is normal in \(D_n\) since \((D_n:\langle r\rangle )=2\text{.}\)
- \(\langle f\rangle\) isn't normal in \(D_4\text{:}\) for instance,
\begin{equation*} r\langle f\rangle r^{-1}=\{e,rfr^3\}=\{e, fr^3r^3\}=\{e,fr^2\}\not\subseteq \langle f\rangle . \end{equation*}
We consider two other very significant examples. First, we revisit the idea of the center of a group, first introduced in Example \(2.8.9\).
Definition: Center
Let \(G\) be a group. We let
\begin{equation*} Z(G)=\{z\in G\,:\, az=za \text{ for all } a\in G\}. \end{equation*}
\(Z(G)\) is called the center of \(G\text{.}\)
(The Z stands for “zentrum,” the German word for “center.”)
Theorem \(\PageIndex{2}\)
Let \(G\) be a group. Then \(Z(G)\) is a subgroup of \(G\text{.}\)
- Proof
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Clearly, \(e∈Z(G)\). Next, let \(z,w∈Z(G)\). Then for all \(a∈G\),
\(a(zw)=(az)w=(za)w=z(aw)=z(wa)=(zw)a\),
so \(zw∈Z(G)\). Finally, \(az=za\) so, multiplying both sides on the left and right by \(z^{−1}\), we have \(z^{−1}a=az^{−1}\); thus, \(z^{−1}∈Z(G)\). Hence, \(Z(G)≤G\).
Theorem \(\PageIndex{3}\)
Let \(G\) be a group and let \(H\) be a subgroup of \(G\) with \(H\subseteq Z(G)\text{.}\) Then \(H\unlhd G\text{.}\) In particular, \(Z(G)\) is itself a normal subgroup of \(G\text{.}\)
- Proof
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Let \(a∈G\). Then since every element of \(Z(G)\) commutes with every element of \(G\), every element of \(H\) commutes with every element of \(G\); so we must have \(aH=Ha\). Thus, \(H⊴G\).
The next definition is profoundly important for us.
Definition: Kernel
Let \(G\) and \(G'\) be groups and let \(\phi\) a homomorphism from \(G\) to \(G'\text{.}\) Letting \(e'\) be the identity element of \(G'\text{,}\) we define the kernel of \(\phi\), \(\text{Ker} \;\phi\text{,}\) by
\begin{equation*} \text{Ker}\; \phi = \{k\in G\,:\,\phi(k)=e'\}. \end{equation*}
Theorem \(\PageIndex{4}\)
Let \(G\) and \(G'\) be groups and let \(\phi\) be a homomorphism from \(G\) to \(G'\text{.}\) Then \(\text{Ker}\; \phi\) is a normal subgroup of \(G\text{.}\)
- Proof
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Let \(K=\text{Ker}\;ϕ\). We first show that \(K\) is a subgroup of \(G\). Clearly, the identity element of \(G\) is in \(K\), so \(K≠∅\). Next, let \(k,m∈K\). Then, letting \(e′\) denote the identity element of \(G′\), we have
\(ϕ(km^{−1})=ϕ(k)ϕ(m)^{−1}=e′(e′)^{−1}=e′\),
so \(km^{−1}∈K\). Thus, by the Two-Step Subgroup Test, we have that \(K\) is a subgroup of \(G\).
Next, let \(k∈K\) and let \(a∈G\). Then
\(ϕ(aka^{−1})=ϕ(a)ϕ(k)ϕ(a)^{−1}=ϕ(a)e′ϕ(a)^{−1}=ϕ(a)ϕ(a)^{−1}=e′\).
So \(aka^{−1}∈K\). Thus, \(K⊴G\).
One slick way, therefore, of showing that a particular set is a normal subgroup of a group \(G\) is by showing it's the kernel of a homomorphism from \(G\) to another group.
Example \(\PageIndex{2}\)
Let \(n\in \mathbb{Z}^+\text{.}\) Here is a rather elegant proof of the fact that \(SL(n,\mathbb{R})\) is a normal subgroup of \(GL(n,\mathbb{R})\text{:}\) Define \(\phi: GL(n,\mathbb{R}) \to \mathbb{R}^*\) by \(\phi(A)=\det A\text{.}\) Clearly, \(\phi\) is a homomorphism, and since the identity element of \(\mathbb{R}^*\) is \(1\),
\begin{equation*} \text{Ker} \;\phi=\{A\in GL(n,\mathbb{R})\,:\,\det A= 1\}=SL(n,\mathbb{R}). \end{equation*}
So \(SL(n,\mathbb{R})\unlhd GL(n,\mathbb{R})\text{.}\)


2.3: Equivalence Relations
Let \(X\) be a set and \(R\) a relation on \(X\) . We say \(R\) is an equivalence relation if
(1) \(R\) is reflexive
(2) \(R\) is symmetric
(3) \(R\) is transitive.
Define a relation \(R\) on \(\mathbb{R}\) by \(x R y\) if and only if \(x^{2}=\) \(y^{2}\) . Then \(R\) is an equivalence relation.
Let \(R\) be a relation defined on \(\mathbb{Z} \times \mathbb{Z}\) as follows. If \(a, b, c, d \in \mathbb{Z}\) , \[(a, b) R(c, d) \text { if and only if } a+d=b+c \text {. }\] Then \(R\) is an equivalence relation. Indeed, let us check the three properties.
Reflexive: By (2.14), we have \((a, b) R(a, b)\) if \(a+b=a+b\) , which clearly holds. Symmetric: Suppose \((a, b) R(c, d)\) , so \(a+d=b+c\) . To see if \((c, d) R(a, b)\) , we must check whether \(c+b=d+a\) ; but this holds by the commutativity of addition.
Transitive: Suppose \((a, b) R(c, d)\) and \((c, d) R(e, f)\) . We must check that \((a, b) R(e, f)\) , in other words that \[a+f=b+e .\] We have \(a+d=b+c\) and \(c+f=d+e\) , and adding these two equations we get \[a+d+c+f=b+c+d+e .\] Cancelling \(c+d\) from each side of (2.16), we get (2.15) as desired.
Let \(R\) be a relation on \(X=\mathbb{Z} \times \mathbb{N}^{+}\) defined by \[(a, b) R(c, d) \text { if and only if } a d=b c \text {. }\] Then \(R\) is an equivalence relation on \(X\) . (Prove this; Exercise 2.4).
Let \(f: X \rightarrow Y\) . Define a relation \(R_{f}\) on \(X\) by \[x R_{f} y \text { if and only if } f(x)=f(y) .\] Then \(R_{f}\) is an equivalence relation. We check the conditions for an equivalence relation:
\(R_{f}\) is clearly reflexive, since, for any \(x \in X\) , \[f(x)=f(x) .\] \(R_{f}\) is symmetric since, for any \(x \in X\) and \(y \in X\) , \[f(x)=f(y) \text { if and only if } f(y)=f(x) .\] To show \(R_{f}\) is transitive, let \(x, y, z \in X\) . If \(f(x)=f(y)\) and \(f(y)=\) \(f(z)\) then \(f(x)=f(z)\) .
Equivalence relations have three of the key properties of identity. They allow us to relate objects in a set that we wish to consider as "the same" in a given context. This allows us to focus on which differences between mathematical objects are relevant to the discussion at hand, and which are not. For this reason, a common symbol for an equivalence relation is \(\sim\) .
Let \(R\) be an equivalence relation on a set \(X\) . If \(x \in X\) then the equivalence class of \(x\) modulo \(R\) , denoted by \([x]_{R}\) , is \[[x]_{R}=\{y \in X \mid x R y\} .\] If \(y \in[x]_{R}\) we call \(y\) a representative element of \([x]_{R}\) . The set of all equivalence classes \(\left\{[x]_{R} \mid x \in X\right\}\) is written \(X / R\) . It is called the quotient space of \(X\) by \(R\) .
We may use \([x]\) for the equivalence class of \(x\) , provided that the equivalence relation is clear.
Notation. Equivalence mod \(R, \equiv_{R}, \sim\) Let \(R\) be an equivalence relation on a set \(X\) . We may express that \(x R y\) by writing \[x \equiv y \bmod R\] or \[x \equiv_{R} y\] or \[x \sim y .\] Proposition 2.19. Suppose that \(\sim\) is an equivalence relation on \(X\) . Let \(x, y \in X\) . If \(x \sim y\) , then \[[x]=[y] .\] If \(x\) is not equivalent to \(y(x \nsim y)\) , then \[[x] \cap[y]=\emptyset .\] Proof. (i) Assume \(x \sim y\) . Let us show that \([x] \subseteq[y]\) . Let \(z \in[x]\) . This means that \(x \sim z\) . Since \(\sim\) is symmetric, and \(x \sim y\) , we have \(y \sim x\) . As \(y \sim x\) and \(x \sim z\) , by transitivity of \(\sim\) we get that \(y \sim z\) . Therefore \(z \in[y]\) . Since \(z\) is an arbitrary element of \([x]\) , we have shown that \([x] \subseteq[y]\) . As \(y \sim x\) , the same argument with \(x\) and \(y\) swapped gives \([y] \subseteq[x]\) , and therefore \([x]=[y]\) .
(ii) Now assume that \(x\) and \(y\) are not equivalent. We must show that there is no \(z\) such that \(z \in[x]\) and \(z \in[y]\) . We will argue by contradiction. Suppose there were such a \(z\) . Then we would have \[x \sim z \quad \text { and } \quad y \sim z .\] By symmetry, we have also that \(z \sim y\) , and by transitivity, we then have that \(x \sim y\) . This contradicts the assumption that \(x\) is not equivalent to \(y\) . So if \(x\) and \(y\) are not equivalent, no \(z\) can exist that is simultaneously in both \([x]\) and \([y]\) . Therefore \([x]\) and \([y]\) are disjoint sets, as required.
So what have we shown? We have not shown that any particular relation is an equivalence relation. Rather we have shown that any equivalence relation on a set partitions the set into disjoint equivalence classes.
As we shall see throughout this book, and you will see throughout your mathematical studies, this is a surprisingly powerful tool.
Definition. Pairwise disjoint Let \(\left\{X_{\alpha} \mid \alpha \in A\right\}\) be a family of sets. The family is pairwise disjoint if for any \(\alpha, \beta \in A, \alpha \neq \beta\) , \[X_{\alpha} \cap X_{\beta}=\emptyset .\] DEFINITION. Partition Let \(Y\) be a set and \(\mathcal{F}=\left\{X_{\alpha} \mid \alpha \in A\right\}\) be a family of non-empty sets. The collection \(\mathcal{F}\) is a partition of \(Y\) if \(\mathcal{F}\) is pairwise disjoint and \[Y=\bigcup_{\alpha \in A} X_{\alpha} .\] Given an equivalence relation \(\sim\) on a set \(X\) , the equivalence classes with respect to \(\sim\) give a partition of \(X\) . Conversely, partitions give rise to equivalence relations.
(i) Let \(X\) be a set, and \(\sim\) an equivalence relation on \(X\) . Then \(X / \sim\) is a partition of \(X\) . (ii) Conversely, let \(\left\{X_{\alpha} \mid \alpha \in A\right\}\) be a partition of \(X\) . Let \(\sim\) be the relation on \(X\) defined by \(x \sim y\) whenever \(x\) and \(y\) are members of the same set in the partition. Then \(\sim\) is an equivalence relation.
PROOF. Part (i) of the theorem is Proposition \(2.19\) restated, and we gave the proof above. To prove the converse, we must show that the relation \(\sim\) defined as in part (ii) of the theorem is an equivalence.
Reflexivity: Let \(x \in X\) . Then \(x\) is in some \(X_{\alpha_{0}}\) , as the union of all these sets is all of \(X\) . Therefore \(x \sim x\) .
Symmetry: Suppose \(x \sim y\) . Then there is some \(X_{\alpha_{0}}\) such that \(x \in X_{\alpha_{0}}\) and \(y \in X_{\alpha_{0}}\) . This implies that \(y \sim x\) .
Transitivity: Suppose \(x \sim y\) and \(y \sim z\) . Then there are sets \(X_{\alpha_{0}}\) and \(X_{\alpha_{1}}\) such that both \(x\) and \(y\) are in \(X_{\alpha_{0}}\) , and both \(y\) and \(z\) are in \(X_{\alpha_{1}}\) . But since the sets \(X_{\alpha}\) form a partition, and \(y\) is in both \(X_{\alpha_{0}}\) and \(X_{\alpha_{1}}\) , we must have that \(X_{\alpha_{0}}=X_{\alpha_{1}}\) . This implies that \(x\) and \(z\) are in the same member of the partition, and so \(x \sim z\) .


10.4: Benefit Trends
- Discuss benefit trends
In an article for SHRM, Compensation & Benefits Manager/Editor Stephen Miller notes the following benefit trends: [1]
- Employers will get increasingly family-friendly. Human resource consulting firm Mercer’s national practice leader Simon Camaj notes that “state and local laws, the war for talent and changing social expectations will lead more employers to implement or enhance paid-parental-leave programs and consider adding paid caregiver leave for family-related absences.”
- Voluntary benefits will become more personal. Mercer’s voluntary benefits leader Tim Weber sees benefit offerings becoming more personal, with employers using “data-driven approaches to tailor [benefit] information and guidance, all wrapped in an engaging employee experience.”
- Emotional health will move to the forefront. Benefits specialists expect increased access to behavioral health services. President and CEO of the nonprofit National Business Group on Health (NBGH) Brian Marcotte anticipates employers improving access to services to address a range of emotional, behavioral and mental health issues, including stress and anxiety, addiction, depression and other mental disorders.
- Benefits technology will simplify employees’ decision-making. Business technology provider Businessolver’s CEO Jon Shanahan expects benefit automation to help employees “learn, understand, and navigate the benefits process based on their specific needs.” He also anticipates process improvements, noting that “HR managers say they lose 14 hours a week due to lack of automation.”
In an article titled “Is Your Benefits Package Ready for the Future?” HR Morning writer Rachel Mucha cites research from Aon’s Benefits and Trends Survey 2019 that found 37% of employers are changing or plan to change their benefits package in 2020 to better accommodate a multi-generational workplace. [2]
The concept of a basic one size fits all benefits package is obsolete. Companies are not only asking employees what perks or benefits they want, they’re listening.
Mucha identifies the following 3 trends:
- Millennials are disrupting the benefits game—health savings account (HSA), student debt repayment family benefits and flexible schedules all rate high with millennials.
- Employers get innovative to rein in high healthcare costs—Employers are providing a range of wellness benefits, disease management programs and health screenings to combat chronic conditions and keep employees healthy.
- Voluntary benefits are driving retention—According to SHRM’s The Evolution of Benefits report, 72% of organizations increased their benefits offerings to retain employees in the last 12 months.
As Businessolver CEO Shanahan notes, “”In 2019 [and beyond], employees will demand more from their benefits packages, and employers will need to deliver and meet multigenerational employees’ needs to stay competitive in the job recruitment market.” [3]
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Miller, Stephen. "
6 Big Benefits Trends for 2019
." Society for Human Resources Management. January 03, 2019. Accessed August 22, 2019. ↵
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Mucha, Rachel. "
A 2020 Vision: Is Your Benefits Package Ready for the Future?
" HR Morning. February 13, 2019. Accessed August 22, 2019. ↵
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Miller, Stephen. "6 Big Benefits Trends for 2019." ↵
Contributors and Attributions
- Time on the couch together. Authored by : Alexander Dummer. Provided by : Unsplash. Located at : https://unsplash.com/photos/UH-xs-FizTk . License : CC0: No Rights Reserved . License Terms : Unsplash License