11.1: Introduction
We live in a globalized world. Products are designed in one place, assembled in another from parts produced in multiple other places. These products are marketed nearly everywhere. Until a few decades ago, such a process would have been impossible. Two hundred years ago, such an idea would have been beyond comprehension. What happened to change the world in such a way. What eventually tied all the economies of the world into a global economy? Industry did. The Industrial Revolution changed the world as much as the Agricultural Revolution. Industry has made the modern lifestyle possible.
During the 2016 U.S. presidential election, something very peculiar happened. Candidates from both major parties (Donald Trump and Hillary Clinton) agreed over and over again on one thing (and only one thing). The U.S. needed to create and/or bring back manufacturing jobs. Both candidates promised, if elected, to create new, well-paid manufacturing jobs. This was an odd shift, because for the previous 40+ years Republicans typically embraced free trade that allows manufacturers to choose where and what to produce (and many chose to move operations outside of the U.S.), while Democrats claimed to be working for the interests of blue-collar, working class people, whose jobs and wages had diminished since the 1980s period of de-industrialization both in the U.S. and throughout the developed/industrialized world. In the U.S. manufacturing provided jobs to 13 million workers in 1950, rising to 20 million in 1980 but by 2017 that number was back to 12 million – similar to levels last seen in 1941. A similar story can be found in Great Britain where jobs in manufacturing in 2017 were half of what they were in 1978 and output that once was 30% of GDP accounts for only 10% in 2017. Similar stories can be found in Germany, Japan, and other ‘industrialized’ economies. You may ask yourself, “Where did all of those jobs go?” But if you think about it, you can probably come up with your own answers.
It’s important to note that even as jobs declined, manufacturing output in most industrialized countries continued to increase, so fewer people were producing more things. The first and simplest explanation for this is automation. For years, science fiction writers have warned us that the robots are coming. In the case of manufacturing technology...they’re already here! Workers today are aided by software, robots, and sophisticated tools that have simply replaced millions of workers. Working at a manufacturing facility is no longer simply a labor-intensive effort, but one that requires extensive training, knowledge, and willingness to learn new technologies all the time. The second explanation is the relocation of manufacturing from wealthy countries to poorer ones because of lower wages in the latter.
As discussed earlier in this chapter, most countries have moved away from the protectionist model of development, allowing corporations to choose for themselves the location of production. No better example of this shift can be found than Walmart, which in the 1970’s advertised that the majority of all of the products it sold were made in the USA. Thirty years later, it would be difficult for to find ANY manufactured product that was still ‘made in the USA’. A third reason for the decline in manufacturing jobs is a decrease in demand for certain types of items. Steel production in the U.S. and England dropped precipitously during the period of deindustrialization (since the 1980’s) not just due to automation or cheaper wages elsewhere, but also because demand for steel also declined. During the 20th century the U.S., Europe, and Japan required enormous amounts of steel in the construction of bridges, dams, railroad, skyscrapers, and even automobiles. Building of such items in the 21st century has slowed down, not because those countries are in decline, but because there is a limit as to how many bridges and skyscrapers are needed in any country! Demand for steel in a country diminishes as GDP per capita reaches about $20,000. Meanwhile, demand for steel will continue to rise in Japan and India for several years as they (and other industrializing countries) continue to expand cities, rail lines, and other large-scale construction projects. Such a decline in steel production does NOT mean that a country is in decline, but rather that there has been a shift in the type of manufacturing that occurs. The U.S., Germany, and Japan all continue to increase manufacturing output ( Figure 11.3 ), even as their share of global output continues to decline.
Another significant shift in manufacturing relates specifically to the geography of production and is best understood in the consideration of 2 different modes of production: 1) Fordism 2) Post-Fordism . Fordism is associated with the assembly line style of production credited to Henry Ford, who dramatically improved efficiency by instituting assembly line techniques to specialize/simplify jobs, standardize parts, reduce production errors, and keep wages high. Those techniques drove massive growth in manufacturing output throughout most of the 20th century and brought the cost of goods down to levels affordable by the masses. Nearly all of the automobile assembly plants located in and around the Great Lakes region of North America adopted the same strategies, which also provided healthy amounts of competition and new innovation for decades, as North America became the world’s leading producer of automobiles. Post-Fordism begins to take hold in the 1980’s as a new, global mode of production that seeks to relocate various components of production across multiple places, regions, and countries. Under Fordism, the entire unit would be produced locally, while Post-Fordism seeks the lowest cost location for every different component, no matter where that might be. Consider an optical, wireless mouse for a moment. The optical component may come from Korea, the rubber cord from Thailand, the plastic from Taiwan, and the patent from the U.S. Meanwhile, all of those items are most likely transported to China, where low-wage workers manually assemble the finished project and an automated packing system boxes and wraps it for shipping to all corners of the world. Global trade has been occurring for hundreds of years, dating back to the days of the Silk Road, Marco Polo, and the Dutch East India Company, but Post- Fordism, in which a single item is comprised of multiple layers of manufacturing from multiple places around the world, is a very recent innovation. The system has reconfigured the globe, such that manufacturers are constantly searching for new locations of cheap production. Consumers tend to benefit greatly from the system in that even poor middle school students in the U.S. can somehow afford to own a pocket computer (smart phone) that is more powerful than the most advanced computer system in the world from the previous generation. This is kind of a miracle. On the other hand, manufacturing jobs that once were a pathway to upward economic mobility, no longer assure people of such a decent standard of living as they once did.
11.1.1 History of Industrialization
Industrialization was not a process that emerged, fully-formed in England in the eighteenth century. It was the result of centuries of incremental developments that were assembled and deployed in the 18th century. Early industrialization involved using water power to run giant looms that produced cloth at a very low cost. This early manufacturing didn’t use coal and belch smoke into the sky, but it initiated an industrial mindset. Costs could be reduced by relying on inanimate power (first water, then steam, then electricity), converting production to simple steps that cheap low-skilled laborers could do ( Taylorism ), getting larger and concentrated in an area (economy of scale), and cranking out large numbers of the same thing ( Fordism ). This is industry in a nutshell. The advantage of industry was that a company could sell a cheaper product, but at a greater profit.
As this mindset was applied to other goods, and then services, the world was changed forever. Places which had been producing goods for millennia suddenly (really suddenly) found themselves competing with a product that was far cheaper. Economist Joseph Schumpeter coined the term “creative destruction” to describe the process in which new industries destroy old ones. Hand production of goods for the masses began to decline precipitously. They quickly became too expensive in comparison to manufactured goods. Today, hand produced goods are often reserved for the wealthy.
A contemporary example of the industrial mode of production is fast food. Looking inside the kitchen of a fast food restaurant will reveal industrially prepared ingredients prepared just in time for sale to a customer. It is not the same process that you would use at home.
In the abstract, companies do not exist to provide jobs or even to make things. Companies exist to produce a profit. If changing the method of making a profit is necessary, then the company will do that in order to survive. If it cannot, then it will go away. For example, many companies today are highly diversified, for example Mitsubishi produces such unrelated products as cars and tuna fish. What is the connection between the two. They both produce profit.
11.1.2 Industrial Geography
How is industry related to geography? For one thing, industrial societies have more goods in them. Since the goods are cheaper, people just have more things. For another, the means of production, the factories, shipping terminals, and distribution centers are visible for anyone to see. The lifestyles of industrialized people are different. Pre-industrialized societies are not regulated by clocks, for example, people wear lass-produced similar clothing. They listen to globally marketed music. If it seems like you already read this in the chapter on pop culture, you have. Pop culture is a function of industry. Geography is concerned with places and industry changed the way the world operates. It changed the relationships between places. Places that industrialized early gained the ability to economically and politically dominate other parts of the world that had not industrialized. Something as simple as having access to cheap, mass produced guns had impacts far beyond mere trade relationships.
Over time industrial production changed from one that disrupted local economies to one that completely changed the relationship that most human beings had with their material culture, their environment, and with one another. Industry has improved standards of living and increased food production on one hand and it has despoiled environments and promoted massive inequality on the other.
Industrialization is about applying rational thought to production of goods. Specifically, this form of rational thought refers to discovering ways to reduce unnecessary labor, materials, capital (money), and time. In the same way thatfactories changed how things were produced, it also changed where things were produced. Locational criteria are used to determine where a factory even gets built.


6.14: Joints
Joints are the structures in the skeleton where 2 or more bones meet. There are several different types of joints. Some are immovable once the animal has reached maturity. Examples of these are those between the bones of the skull and the midline joint of the pelvic girdle. Some are slightly moveable like the joints between the vertebrae but most joints allow free movement and have a typical structure with a fluid filled cavity separating the articulating surfaces (surfaces that move against each other) of the two bones. This kind of joint is called a synovial joint (see diagram 6.17). The joint is held together by bundles of white fibrous tissue called ligaments and a fibrous capsule encloses the joint. The inner layers of this capsule secrete the synovial fluid that acts as a lubricant. The articulating surfaces of the bones are covered with cartilage that also reduces friction and some joints, e.g. the knee, have a pad of cartilage between the surfaces that articulate with each other.
The shape of the articulating bones in a joint and the arrangement of ligaments determine the kind of movement made by the joint. Some joints only allow a to and from gliding movement e.g. between the ankle and wrist bones; the joints at the elbow, knee and fingers are hinge joints and allow movement in two dimensions and the axis vertebra pivots on the atlas vertebra. Ball and socket joints , like those at the shoulder and hip, allow the greatest range of movement.
Diagram 6.17 - A synovial joint


41.1: Appendix B- Common Abbreviations and Lab Values
Nurses are required to review and interpret lab values. Table B1 lists common abbreviations, and Table B2 lists common lab values for nurses to learn.
| Abbreviation | Meaning | Abbreviation | Meaning |
|---|---|---|---|
| cells/mcL or cells/µL | Cells per microliter | mg/dL | Milligrams per deciliter |
| cells/mL | Cells per milliliter (cubic centimeter) | microunits/mL | Microunits per milliliter |
| cm | Centimeter | mL | Milliliter |
| fl oz | Fluid ounce | mL/minute | Milliliters per minute |
| fL | Femtoliter | mm | Millimeter |
| g | Gram | mm 3 | Cubic millimeter |
| g/dL | Grams per deciliter | mm Hg | Millimeters of mercury |
| HPF | High-power field | mmol/L | Millimoles per liter |
| IU | International unit | ng/mL | Nanograms per milliliter |
| kg | Kilogram | oz | Ounce |
| L | Liter | pg/mL | Picograms per milliliter |
| lb or # | Pound | Tbsp | Tablespoon |
| mcg/dL or µg/dL | Micrograms per deciliter | tsp | Teaspoon |
| mEq/L | Milliequivalents per liter | units/L | Units per liter |
| mg | Milligram |
Note that the abbreviation for “micro” is sometimes shown as the Greek mu (µ, as in µL); however, that practice is considered to be error prone by the Institute for Safe Medication Practices and has been supplanted by “mc,” as in “mcg” for microgram or “mcL” for microliter.
| Arterial Blood Gases | |||||||
| Base excess/base deficit | –2 to +2 mEq/L | HCO 3 | 22–26 mEq/L | PaCO 2 | 35–45 mm Hg | ||
| PaO 2 | 75–100 mm Hg | pH | 7.35–7.45 | SaO 2 | 95%–100% | ||
| Serum Blood Tests | |||||||
| Albumin | 3.5–5.5 g/dL | Glucose (fasting) | 70–99 mg/dL | MCV | 80–98 fL | ||
| Alk phos | 30–120 units/L | Glucose tolerance test |
(Start) 70–100 mg/dL
(1 hour) <180 mg/dL (2 hours) <140 mg/dL |
Monocytes | 4000–11,000 cells/mcL | ||
| ALT | 10–40 units/L | Neutrophils | 2000–8250 cells/mcL | ||||
| Ammonia | 40–70 mcg/dL | Phosphorus | 3.0–4.5 mg/dL | ||||
| Amylase | 3–125 units/L | Platelet count | 150–450 × 10 9 /L | ||||
| AST | 10–40 units/L | HDL |
(F) >50 mg/dL
(M) >40 mg/dL |
Potassium | 3.5–5.0 mEq/L | ||
| Bicarb (HCO 3 ) | 23–28 mEq/L | Hematocrit |
(F) 37%–47%
(M) 42%–50% |
Protein | 5.5–9.0 g/dL | ||
| Bilirubin, total | 0.3–1.0 mg/dL |
PT
PTT |
11–13 seconds
25–35 seconds |
||||
| BNP | 0–100 pg/mL | Hemoglobin |
(F) 12–16 g/dL
(M) 14–18 g/ dL |
RBC |
(F) 4.2–5.4 million/mm
3
(M) 4.7–6.1 million/mm 3 |
||
| BUN | 8–20 mg/dL | ||||||
| Calcium | 8.6–10.2 mg/ dL | HgbA1c | 4.0%–5.6% | RDW | 9.0%–14.5% | ||
| Chloride | 98–106 mEq/L | INR | 0.8–1.2 seconds | SaO 2 | 95%–100% | ||
| Cholesterol | <200 mg/dL | Iron | 50–150 mcg/ dL | Sodium | 136–145 mEq/L | ||
| CK |
(F) 30–135 units/L
(M) 55–170 units/L |
Lactic acid |
(Art) <0.6 mmol/L
(Ven) 0.7–1.8 mmol/L |
Triglycerides (fasting) | <150 mg/dL | ||
| CO 2 (CMP) | 23–30 mEq/L | LDL | <130 mg/dL | Troponin T | 0–0.01 ng/mL | ||
| Creatinine |
(F) 0.5–1.10 mg/dL
(M) 0.70–1.30 mg/dL |
Lipase | 10–140 units/L | TSH | 0.4–6 mcg/mL | ||
| D-dimer | <0.5 mcg/mL | Lymphocytes | 20–40 × 10 9 /L | WBC | 5000–10,000/mL | ||
| GFR | ≥90 mL/minute | Magnesium | 1.6–2.6 mEq/L | ||||
| Miscellaneous Tests | |||||||
| Fungal (yeast) culture | Negative | ||||||
| Sputum culture | Negative | ||||||
| Urine Tests | |||||||
| Bacteria | Negative | Color | Yellow–amber | pH | 4.5–8.0 | ||
| Bilirubin | Negative | Glucose | Negative | Protein | ≤150 mg/dL | ||
| Blood | Negative | Ketones | Negative | Specific gravity | 1.002 | ||
| Casts | 0–5 HPF | Leukocytes | Negative | WBC | 0–5 HPF | ||
| Clarity | Clear | Nitrates | Negative | Yeast | Negative |
Note: The reference values provided in these tables should be used as guidelines only,as they can vary by testing laboratory, instrument type, and demographics of the healthy client population used for comparison. Always use your facility’s lab value guidelines for interpretation or testing. In the Next Generation NCLEX exams, assessment items that contain a numeric laboratory value will include the corresponding normal reference range. See NCLEX FAQs .


10.12: Introduction to Types of Compensation
What you’ll learn to do: Discuss various types of compensation
Prior to developing a compensation policy, human resource professionals should develop an understanding of the nature of rewards. In this section, we’ll discuss the connection between compensation and motivation, differentiate between intrinsic and extrinsic rewards, and discuss types of compensation, factors affecting compensation, and methods of developing a pay structure.
Contributors and Attributions
CC licensed content, Shared previously
- Authored by : kschneider2991. Provided by : Pixabay. Located at : pixabay.com/photos/money-money-tower-coins-euro-2180330/. License : CC0: No Rights Reserved . License Terms : Pixabay License


5.4: Hierarchical Regression Explanation, Assumptions, Interpretation, and Write Up
At the end of this section you should be able to answer the following questions:
- Explain how hierarchical regression differs from multiple regression.
- Discuss where you would use “control variables” in a hierarchical regression analyses.
Hierarchical Regression Explanation and Assumptions
Hierarchical regression is a type of regression model in which the predictors are entered in blocks. Each block represents one step (or model). The order (or which predictor goes into which block) to enter predictors into the model is decided by the researcher, but should always be based on theory.
The first block entered into a hierarchical regression can include “control variables,” which are variables that we want to hold constant. In a sense, researchers want to account for the variability of the control variables by removing it before analysing the relationship between the predictors and the outcome.
The example research question is “what is the effect of perceived stress on physical illness, after controlling for age and gender?”. To answer this research question, we will need two blocks. One with age and gender, then the next block including perceived stress.
It is important to note that the assumptions for hierarchical regression are the same as those covered for simple or basic multiple regression. You may wish to go back to the section on multiple regression assumptions if you can’t remember the assumptions or want to check them out before progressing through the chapter.
Hierarchical Regression Interpretation
PowerPoint: Hierarchical Regression
For this example, please click on the link for Chapter Five – Hierarchical Regression below. You will find 4 slides that we will be referring to for the rest of this section.
For this test, the statistical program used was Jamovi, which is freely available to use. The first two slides show the steps to get produce the results. The third slide shows the output with any highlighting. You might want to think about what you have already learned, to see if you can work out the important elements of this output.
Slide 2 shows the overall model statistics. The first model, with only age and gender, can be seen circled in red. This model is obviously significant. The second model (circled in green) includes age, gender, and perceived stress. As you can see, the F statistic is larger for the second model. However, does this mean it is significantly larger?
To answer this question, we will need to look at the model change statistics on Slide 3. The R value for model 1 can be seen here circled in red as .202. This model explains approximately 4% of the variance in physical illness. The R value for model 2 is circled in green, and explains a more sizeable part of the variance, about 25%.
The significance of the change in the model can be seen in blue on Slide 3. The information you are looking at is the R squared change, the F statistic change, and the statistical significance of this change.
On Slide 4, you can examine the role of each individual independent variable on the dependant variable. For model one, as circled in red, age and gender are both significantly associated with physical illness. In this case, age is negatively associated (i.e. the younger you are, the more likely you are to be healthy), and gender is positively associated (in this case being female is more likely to result in more physical illness). For model 2, gender is still positively associated and now perceived stress is also positively associated. However, age is no longer significantly associated with physical illness following the introduction of perceived stress.
Hierarchical Regression Write Up
An example write up of a hierarchal regression analysis is seen below:
In order to test the predictions, a hierarchical multiple regression was conducted, with two blocks of variables. The first block included age and gender (0 = male, 1 = female) as the predictors, with difficulties in physical illness as the dependant variable. In block two, levels of perceived stress was also included as the predictor variable, with difficulties in perceived stress as the dependant variable.
Overall, the results showed that the first model was significant F (2,364) = 7.75, p = .001, R 2 =.04. Both age and gender were significantly associated with perceived life stress ( b =-0.14, t = -2.78, p = .006, and b =.14, t = 2.70, p = .007, respectively). The second model ( F (3,363) = 39.61, p < .001, R 2 =.25), which included physical illness ( b =0.47, t = 9.96, p < .001) showed significant improvement from the first model ∆ F (1,363) = 99.13, p < .001, ∆R 2 =.21, , Overall, when age and location of participants were included in the model, the variables explained 8.6% of the variance, with the final model, including physical illness accounted for 24.7% of the variance, with model one and two representing a small, and large effect size, respectively.


4.2: §26. Latin Adjectivesst and 2nd Declension Type
- 8343
The basic Latin adjective that meant “big” or “great” was a word with the base magn -; the ending that followed this base depended on a variety of factors, including the gender of the noun to which the adjective was linked. A “big page” was a Magna Charta ; a man known as “Charles the Great”—we call him Charlemagne —had the Latin name of Carolus Magnus ; a “big work” (the main achievement of one’s life, perhaps) can be described in Latin as a magnum opus . These are the three genders of the standard Latin adjective: magnus (m.), magna , (f.), and magnum (n.). If it modifies a feminine noun, an adjective of this type will use first-declension endings; for example, when the adjective firmus , firma , firmum is combined with the feminine noun terra , it creates the phrase terra firma (“firm earth”). If attached to a masculine or neuter noun, an adjective of this type will use second-declension endings, as seen in the examples Caro lus Magnus and magnum opus above. [1] For obvious reasons, then, this very common type is described as an adjective of the first and second declensions, or a 1ST AND 2ND DECLENSION ADJECTIVE.
For the purposes of this course, we seldom have to worry about questions of Latin gender; and we can blissfully ignore all problems of adjective-noun agreement, a topic that creates some anguish for students of Latin grammar. It will be enough for us to know that most Latin adjectives belong to the same class as magnus , magna , magnum and firmus , firma , firmum. For the sake of brevity, we shall usually refer to words of this type simply as magnus or firmus , assuming the existence of feminine and neuter forms to match the masculine. IF A LATIN ADJECTIVE IS LISTED WITH THE ENDING – us , YOU CAN BE CERTAIN THAT IT BELONGS TO THE 1ST ADN 2ND DECLENSION TYPE. As you would expect from the parallel nouns that we studied in chapter 2, the base of these adjectives can be determined by removing the – us ending.
It is often possible to guess the meaning of a Latin adjective from obvious English clues. We have already met firmus (“firm,” “steadfast”), whose base provides the synonymous English derivative. The same principle applies to Latin adjectives like justus , vastus , solidus , timidus , validus , and rotundus . In meaning, these words may not exactly match their English derivatives, but the differences aren’t worth worrying about. Occasionally, however, semantic changes have occurred—the Latin word crispus , for example, meant “curly-haired.” If you know that Latin curtus meant “clipped” or “shortened,” you may have a better feeling for the connotations of the English word curt . Once in a while, appearances can be deceiving: Latin longus is not the origin of English long , which is a cognate word of Germanic descent.
| aequus | even, equal | multus | much, many |
| bonus | good | pius | dutiful, good |
| justus | upright, just | planus | level, flat |
| longus | long | sanus | healthy, sound |
| magnus | big, great | solus | alone, only, sole |
| malus | bad, evil | vacuus | empty |
| medius | middle | verus | true |
For hundreds of years, students of Latin have been learning 1st and 2nd declension adjective forms from the model of bonus , bona , bonum ( bonus – a – um ), one of the commonest words in Latin. It is the origin of the corresponding adjective in the Romance Languages—French bon , Italian buono , and Spanish bueno . Its main derivatives in English, however, are two nouns— bonus and boon . It is not uncommon for adjectives to evolve into nouns in this fashion. In English, as in Latin, we can refer to “the good” as an abstract concept; the Latin phrase summum bonum (“the highest good”) is sometimes used in English. Similarly, we can talk about a “happy medium , ” using another Latin adjective as a noun. A medium may also be a physical substance “in the middle,” or a means of mass communication (plural, media) , or a spiritualist who attempts contact with the dead. A magnum can be a large (two-quart) bottle—probably of champagne. Another example of the evolution of neuter adjective into noun can be seen in vacuum , an “empty space.” [2] This same Latin adjective has given us the English adjective vacuous.
You will surely spot the connection between aequus and English equal , equity , etc. Don’t be puzzled by the change in spelling: it is a regular development for the Latin diphthong ae to evolve into English e. Problems can arise, however, when this change blurs the contrast between two distinct Latin words. In English spelling terms, there is no difference between the Latin adjective aequus and the Latin noun equus (“horse”), since both of them appear in our language with the base equ-. If Julius Caesar should be suddenly brought back to life today, he might think that the mysterious English word equator referred to a horseback rider, or that equidistant meant that you had a cheap seat at the Kentucky Derby.
Solus and sanus , with their derivatives sole and sane , illustrate a fairly common type of morphological change, where the English word is the Latin base plus silent – e (see §14, 3.a). Further examples are amplus , curvus , and pronus (with its opposite, supinus ). The development of Latin sanus to English sane shows also a semantic change: the meaning has been specialized, since Latin sanus involved both physical and mental health. The poet Juvenal said that all of us should pray for a healthy mind in a healthy body— mens sana in corpore sano.
Quite a few Latin adjectives ending in – ius and – uus have entered English as adjectives in – ious and – uous . Vacuus > vacuous is one case in point. Further examples include varius > various , pius > pious (and its opposite im-pius [3] > impious) , spurius (originally, “illegitimate”) > spurious , noxius (“harmful”) > noxious , in-nocuus > innocuous , and strenuus > strenuous .
As in the case of 2nd declension nouns in – us (see §13), there is a small subtype of 1st and 2nd declension adjectives with – er in the masculine singular:
|
liber, libera, liberum
|
free |
|
miser, misera, miserum
|
wretched, miserable |
|
pulc
her, pulchra, pulchrum
|
beautiful |
|
sacer, sacra, sacrum
|
sacred |
|
integer, integra, integrum
|
untouched, whole |
Some of these, like liber and miser , keep the vowel – e – in their base (and therefore in their English derivatives). Others drop the – e -, having bases like pulchr -, sacr -, and integr -. Notice that miser and integer are two English noun derivatives from this list. English sinister is the masculine form of a Latin adjective that meant “left-handed”; throughout human history, southpaws have always been abused linguistically.
-
The phrase
magnum opus
may help you remember that
opus
is a neuter noun. It also shows that a Latin adjective is not likely to rhyme with its noun if the noun belongs to a different declension. ↵
-
We often use this noun adjectivally in expressions like “
vacuum
cleaner” and “
vacuum
bottle.” English is a language where words can jump casually from one grammatical function (part of speech) to another without any change in form. The word
home
is obviously a noun (“my old
home
”), but we use it as an adverb in “I’m going
home
,” and as an adjective in “
home
town” or “
home
run.” In highly inflected languages like Latin, words normally can’t switch functions without some change in form. ↵
-
This is our first example of the negative
PREFIX in-
, of which we’ll see much more later. ↵


2.2.5: The Theory of Sets
Almost all of mathematics can be developed in the theory of sets. Developing mathematics in this theory involves a number of things. First, it requires a set of axioms for the relation \(\in\) . A number of different axiom systems have been developed, sometimes with conflicting properties of \(\in\) . The axiom system known as \(\Log{ZFC}\) , Zermelo-Fraenkel set theory with the axiom of choice stands out: it is by far the most widely used and studied, because it turns out that its axioms suffice to prove almost all the things mathematicians expect to be able to prove. But before that can be established, it first is necessary to make clear how we can even express all the things mathematicians would like to express. For starters, the language contains no constant symbols or function symbols, so it seems at first glance unclear that we can talk about particular sets (such as \(\emptyset\) or \(\Nat\) ), can talk about operations on sets (such as \(X \cup Y\) and \(\Pow{X}\) ), let alone other constructions which involve things other than sets, such as relations and functions.
To begin with, “is an element of” is not the only relation we are interested in: “is a subset of” seems almost as important. But we can define “is a subset of” in terms of “is an element of.” To do this, we have to find a formula \(A(x, y)\) in the language of set theory which is satisfied by a pair of sets \(\tuple{X, Y}\) iff \(X \subseteq Y\) . But \(X\) is a subset of \(Y\) just in case all elements of \(X\) are also elements of \(Y\) . So we can define \(\subseteq\) by the formula \[\lforall{z}{(z \in x \lif z \in y)}\nonumber\] Now, whenever we want to use the relation \(\subseteq\) in a formula, we could instead use that formula (with \(x\) and \(y\) suitably replaced, and the bound variable \(z\) renamed if necessary). For instance, extensionality of sets means that if any sets \(x\) and \(y\) are contained in each other, then \(x\) and \(y\) must be the same set. This can be expressed by \(\lforall{x}{\lforall{y}{((x \subseteq y \land y \subseteq x) \lif x = y)}}\) , or, if we replace \(\subseteq\) by the above definition, by \[\lforall{x}{\lforall{y}{((\lforall{z}{(z \in x \lif z \in y)} \land \lforall{z}{(z \in y \lif z \in x)}) \lif x = y)}}.\nonumber\] This is in fact one of the axioms of \(\Log{ZFC}\) , the “axiom of extensionality.”
There is no constant symbol for \(\emptyset\) , but we can express “ \(x\) is empty” by \(\lnot \lexists{y}{y \in x}\) . Then “ \(\emptyset\) exists” becomes the sentence \(\lexists{x}{\lnot \lexists{y}{y \in x}}\) . This is another axiom of \(\Log{ZFC}\) . (Note that the axiom of extensionality implies that there is only one empty set.) Whenever we want to talk about \(\emptyset\) in the language of set theory, we would write this as “there is a set that’s empty and …” As an example, to express the fact that \(\emptyset\) is a subset of every set, we could write \[\lexists{x}{(\lnot\lexists{y}{y \in x} \land \lforall{z}{x \subseteq z})}\nonumber\] where, of course, \(x \subseteq z\) would in turn have to be replaced by its definition.
To talk about operations on sets, such has \(X \cup Y\) and \(\Pow{X}\) , we have to use a similar trick. There are no function symbols in the language of set theory, but we can express the functional relations \(X \cup Y = Z\) and \(\Pow{X} = Y\) by \[\begin{aligned} & \lforall{u}{((u \in x \lor u \in y) \liff u \in z)}\\ & \lforall{u}{(u \subseteq x \liff u \in y )}\end{aligned}\nonumber\] since the elements of \(X \cup Y\) are exactly the sets that are either elements of \(X\) or elements of \(Y\) , and the elements of \(\Pow{X}\) are exactly the subsets of \(X\) . However, this doesn’t allow us to use \(x \cup y\) or \(\Pow{x}\) as if they were terms: we can only use the entire formulas that define the relations \(X \cup Y = Z\) and \(\Pow{X} = Y\) . In fact, we do not know that these relations are ever satisfied, i.e., we do not know that unions and power sets always exist. For instance, the sentence \(\lforall{x}{\lexists{y}{\Pow{x} = y}}\) is another axiom of \(\Log{ZFC}\) (the power set axiom).
Now what about talk of ordered pairs or functions? Here we have to explain how we can think of ordered pairs and functions as special kinds of sets. One way to define the ordered pair \(\tuple{x, y}\) is as the set \(\{\{x\}, \{x, y\}\}\) . But like before, we cannot introduce a function symbol that names this set; we can only define the relation \(\tuple{x, y} = z\) , i.e., \(\{\{x\}, \{x, y\}\} = z\) : \[\lforall{u}{(u \in z \liff (\lforall{v}{(v \in u \liff v = x)} \lor \lforall{v}{(v \in u \liff (v = x \lor v = y))}))}\nonumber\] This says that the elements \(u\) of \(z\) are exactly those sets which either have \(x\) as its only element or have \(x\) and \(y\) as its only elements (in other words, those sets that are either identical to \(\{x\}\) or identical to \(\{x, y\}\) ). Once we have this, we can say further things, e.g., that \(X \times Y = Z\) : \[\lforall{z}{(z \in Z \liff \lexists{x}{\lexists{y}{(x \in X \land y \in Y \land \tuple{x, y} = z)}})}\nonumber\]
A function \(f \colon X \to Y\) can be thought of as the relation \(f(x) = y\) , i.e., as the set of pairs \(\Setabs{\tuple{x,y}}{f(x) = y}\) . We can then say that a set \(f\) is a function from \(X\) to \(Y\) if (a) it is a relation \(\subseteq X \times Y\) , (b) it is total, i.e., for all \(x \in X\) there is some \(y \in Y\) such that \(\tuple{x, y} \in f\) and (c) it is functional, i.e., whenever \(\tuple{x, y}, \tuple{x, y'} \in f\) , \(y = y'\) (because values of functions must be unique). So “ \(f\) is a function from \(X\) to \(Y\) ” can be written as: \[\begin{aligned} \lforall{u}{(u \in f \lif {}} & \lexists{x}{\lexists{y}{(x \in X \land y \in Y \land \tuple{x, y} = u)}}) \land {}\\ \lforall{x}{(x \in X \lif {}} & (\lexists{y}{(y \in Y \land \mathrm{maps}(f, x, y))} \land {}\\ & (\lforall{y}{\lforall{y'}{((\mathrm{maps}(f, x, y) \land \mathrm{maps}(f, x, y')) \lif y = y')}}))\end{aligned}\nonumber\] where \(\mathrm{maps}(f, x, y)\) abbreviates \(\lexists{v}{(v \in f \land \tuple{x, y} = v)}\) (this formula expresses “ \(f(x) = y\) ”).
It is now also not hard to express that \(f\colon X \to Y\) is injective, for instance: \[\begin{gathered} f \colon X \to Y \land \lforall{x}{\lforall{x'}{((x \in X \land x' \in X \land {}}} \\ \lexists{y}{(\mathrm{maps}(f, x, y) \land \mathrm{maps}(f, x', y))}) \lif x = x')\end{gathered}\nonumber\] A function \(f\colon X \to Y\) is injective iff, whenever \(f\) maps \(x, x' \in X\) to a single \(y\) , \(x = x'\) . If we abbreviate this formula as \(\mathrm{inj}(f, X, Y)\) , we’re already in a position to state in the language of set theory something as non-trivial as Cantor’s theorem: there is no injective function from \(\Pow{X}\) to \(X\) : \[\lforall{X}{\lforall{Y}{(\Pow{X} = Y \lif \lnot\lexists{f}{\mathrm{inj}(f, Y, X)})}}\nonumber\]
One might think that set theory requires another axiom that guarantees the existence of a set for every defining property. If \(A(x)\) is a formula of set theory with the variable \(x\) free, we can consider the sentence \[\lexists{y}{\lforall{x}{(x \in y \liff A(x))}}.\nonumber\] This sentence states that there is a set \(y\) whose elements are all and only those \(x\) that satisfy \(A(x)\) . This schema is called the “comprehension principle.” It looks very useful; unfortunately it is inconsistent. Take \(A(x) \ident \lnot x \in x\) , then the comprehension principle states \[\lexists{y}{\lforall{x}{(x \in y \liff x \notin x)}},\nonumber\] i.e., it states the existence of a set of all sets that are not elements of themselves. No such set can exist—this is Russell’s Paradox. \(\Log{ZFC}\) , in fact, contains a restricted—and consistent—version of this principle, the separation principle: \[\lforall{z}{\lexists{y}{\lforall{x}{(x \in y \liff (x \in z \land A(x))}}}.\nonumber\]
Show that the comprehension principle is inconsistent by giving a derivation that shows \[\lexists{y}{\lforall{x}{(x \in y \liff x \notin x)}} \Proves \lfalse.\nonumber\] It may help to first show \((A \lif \lnot A) \land (\lnot A \lif A) \Proves \lfalse\) .


1.2: Thermodynamic Variables
The system is defined and studied using parameters that are called variables. These variables are quantities that we can measure, such as pressure and temperature. However, don’t be surprised if, on some occasions, you encounter some variable that is a little harder to measure directly, such as entropy. The variables depend only on the current state of the system, and therefore they define it. If I know the values of all the “relevant variables” of a system, I know the state of the system. The relationship between the variables is described by mathematical functions called state functions, while the “relevant variables” are called natural variables.
What are the “relevant variables” of a system? The answer to this question depends on the system, and it is not always straightforward. The simplest case is the case of an ideal gas, for which the natural variables are those that enter the ideal gas law and the corresponding equation:
\[ PV=nRT \label{1.2.1} \]
Therefore, the natural variables for an ideal gas are the pressure P, the volume V, the number of moles n, and the temperature T, with R being the ideal gas constant. Recalling from the general chemistry courses, R is a universal dimensional constant which has the values of R = 8.31 kJ/mol in SI units.
We will use the ideal gas equation and its variables as an example to discuss variables and functions in this chapter. We will analyze more complicated cases in the next chapters. Variables can be classified according to numerous criteria, each with its advantages and disadvantages. A typical classification is:
- Physical variables (\(P\), \(V\), \(T\) in the ideal gas law): independent of the chemical composition of the system.
- Chemical variables (\(n\) in the ideal gas law): dependent on the chemical composition of the system.
Another useful classification is:
- Intensive variables (\(P\), \(T\) in the ideal gas law): independent of the physical size (extension) of the system.
- Extensive variables (\(V\), \(n\) in the ideal gas law): dependent on the physical size (extension) of the system.
When we deal with thermodynamic systems, it is more convenient to work with intensive variables. Luckily, it is relatively easy to convert extensive variables into intensive ones by just taking the ratio between the two of them. For an ideal gas, by taking the ratio between V and n, we obtained the intensive variable called molar volume:
\[ \overline{V}=\dfrac{V}{n}. \label{1.2.2} \]
We can then recast Equation \ref{1.2.1} as:
\[ P\overline{V}=RT, \label{1.2.3} \]
which is the preferred equation that we will use for the remainder of this course. The ideal gas equation connects the 3 variables pressure, molar volume, and temperature, reducing the number of independent variables to just 2. In other words, once 2 of the 3 variables are known, the other one can be easily obtained using these simple relations:
\[ P(T,\overline{V})=\dfrac{RT}{\overline{V}}, \label{1.2.4} \]
\[ \overline{V}(T,P)=\dfrac{RT}{P}, \label{1.2.5} \]
\[ T(P,\overline{V})=\dfrac{P\overline{V}}{R}. \label{1.2.6} \]
These equations define three state functions, each one expressed in terms of two independent natural variables. For example, Equation \ref{1.2.4} defines the state function called “pressure”, expressed as a function of temperature and molar volume. Similarly, Equation \ref{1.2.5} defines the “molar volume” as a function of temperature and pressure, and Equation \ref{1.2.6} defines the “temperature” as a function of pressure and molar volume. When we know the natural variables that define a state function, we can express the function using its total differential, for example for the pressure \(P(T, \overline{V})\):
\[ dP=\left( \dfrac{\partial P}{\partial T} \right)dT + \left( \dfrac{\partial P}{\partial \overline{V}} \right)d\overline{V} \label{1.2.7} \]
Recalling Schwartz’s theorem, the mixed partial second derivatives that can be obtained from Equation \ref{1.2.7} are the same:
\[ \dfrac{\partial^2 P}{\partial T \partial \overline{V}}=\dfrac{\partial}{\partial \overline{V}}\dfrac{\partial P}{\partial T}=\dfrac{\partial}{\partial T}\dfrac{\partial P}{\partial \overline{V}}=\dfrac{\partial^2 P}{\partial \overline{V} \partial T} \label{1.2.8} \]
Which can be easily verified considering that:
\[ \dfrac{\partial}{\partial \overline{V}} \dfrac{\partial P}{\partial T} = \dfrac{\partial}{\partial \overline{V}} \left(\dfrac{R}{\overline{V}}\right) = -\dfrac{R}{\overline{V}^2} \label{1.2.9} \]
and
\[ \dfrac{\partial}{\partial T} \dfrac{\partial P}{\partial \overline{V}} = \dfrac{\partial}{\partial T} \left(\dfrac{-RT}{\overline{V}^2}\right) = -\dfrac{R}{\overline{V}^2} \label{1.2.10} \]
While for the ideal gas law, all the variables are “well-behaved” and always satisfy Schwartz’s theorem, we will encounter some variable for which Schwartz’s theorem does not hold. Mathematically, if the Schwartz’s theorem is violated (i.e., if the mixed second derivatives are not equal), then the corresponding function cannot be integrated, hence it is not a state function. The differential of a function that cannot be integrated cannot be defined exactly. Thus, these functions are called path functions; that is, they depend on the path rather than the state. The most typical examples of path functions that we will encounter in the next chapters are heat (\(Q\)) and work (\(W\)). For these functions, we cannot define exact differentials \(dQ\) and \(dW\), and we must introduce a new notation to define their “inexact differentials” \(đ Q\) and \(đ W\).
We will return to exact and inexact differential when we discuss heat and work, but for this chapter, it is crucial to notice the difference between a state function and a path function. A typical example to understand the difference between state and path function is to consider the distance between two geographical locations. Let’s, for example, consider the distance between New York City and Los Angeles. If we fly straight from one city to the other, there are roughly 4,000 km between them. This “air distance” depends exclusively on the geographical location of the two cities. It stays constant regardless of the method of transportation that I have accessibility to travel between them. Since the cities’ positions depend uniquely on their latitudes and longitudes, the “air distance” is a state function, i.e., it is uniquely defined from a simple relationship between measurable variables. However, the “air distance” is not the distance that I will practically have to drive when I go from NYC to LA. Such “travel distance” depends on the method of transportation that I decide to take (airplane vs. car vs. train vs. boat vs. …). It will depend on a plentiful amount of other factors such as the choice of the road to be traveled (if going by car), the atmospheric conditions (if flying), and so on. A typical “travel distance” by car is, for example, about 4,500 km, which is about 12% more than the “air distance.” Indeed, we could even design a very inefficient road trip that avoids all highways and will result in a “travel distance” of 8,000 km or even more (200% of the “air distance”). The “travel distance” is a clear example of a path function because it depends on the specific path that I decide to travel to go from NYC to LA. See Figure \(\PageIndex{1}\).


8.6: Post-test
-
These types of nutrients provide the major source of energy for our body. Examples include breads, crackers, oatmeal, rice, potatoes, peas, and pasta.
- Protein
- Carbohydrates
- Fats
-
These are the building blocks which the body needs to repair muscle, tissues, organs, and skin. Examples include beef, chicken, turkey, fish, peanut butter, and eggs.
- Protein
- Carbohydrates
- Fats
-
The body needs these to help make cell membranes, prevent heat loss, and cushion organs. Examples include butter, margarine, peanut butter, and olive oil.
- Protein
- Carbohydrates
- Fats
- True or False: The HHA/PCA should never suggest a patient take a vitamin. All patient questions should be referred to the supervisor and diet guidelines to be followed are those stated within the Care Plan.
- True or False: Unless a doctor states otherwise, patients should drink about 8 glasses of water per day for optimum health.
- True or False: Sodium, which can be found in canned soups, meats, vegetables, and packaged or frozen foods causes hypertension and fluid retention and intake should be limited.
-
Diets high in fruits and vegetables have been shown to reduce the risk of which of the following?
Select
all
that
apply
.
- Heart disease
- Certain types of cancer
- Pneumonia
- Type 2 diabetes
- Glaucoma
- Obesity
-
At what temperature should refrigerated foods be kept at?
- 55 degrees or above
- 0 degrees
- 40 degrees or below
- 45 degrees
-
At what temperature should frozen foods be kept at?
- 55 degrees or above
- 0 degrees
- 40 degrees or below
- 45 degrees
-
At which temperature should poultry be cooked to?
- 165 degrees
- 145 degrees
- 160 degrees
- 155 degrees
-
At which temperature should ground beef be cooked to?
- 165 degrees
- 145 degrees
- 160 degrees
- 155 degrees
-
At which temperature should leftovers be reheated to?
- 165 degrees
- 145 degrees
- 160 degrees
- 155 degrees
-
At which temperature should steaks, chops, and roasts be cooked to?
- 165 degrees
- 145 degrees
- 160 degrees
- 155 degrees
-
Which of these items would
not
be allowed on a clear liquid diet?
- Beef broth
- Jello
- Frozen yogurt or ice cream
- Apple juice
-
ChooseMyPlate guidelines include which of the following?
Select
all
that
apply
.
- Well-balanced meals with selections from all food groups.
- Calcium rich foods with nonfat or low fat dairy products chosen over whole milk products.
- Lean proteins that are baked or roasted rather than fried.
- Plenty of fruits and vegetables.
- Half of all grains selected as whole grains.
-
Which of the following precautions should a HHA/PCA take for safe food handling?
Which
two
are
correct
?
- Use 1 cutting board for both chopping vegetables and raw meats.
- Wash hands before food preparation and after handling raw meats and eggs.
- Leave spills from raw meats on the counter until food preparation is complete.
- Leave food on the counter after meals for several hours to cool before refrigerating.
- Never use food after the expiration date.
- True or False: It is okay for the HHA/PCA to do their own shopping while grocery shopping for their patient or to borrow money from the patient to buy something for themselves.
-
In this type of diet, food is cut up, cooked, and blended to a baby food consistency.
- Chopped
- Diced
- Pureed
- Minced
-
To help prevent choking or aspiration of food, the HHA/PCA can encourage a patient to do which of the following?
Select
all
that
apply
.
- Eat while sitting in an upright position.
- Take small bites and chew food well.
- Stay in an upright position during meals and afterwards for 30-60 minutes.
- Tuck their chin down and lean slightly forward to help food pass through the esophagus.
- True or False: The HHA/PCA should always follow dietary guidelines for the patient as set forth in the Care Plan and prepare foods as the Care Plan directs.
- Answer
-
1. B
2. A
3. C
4. True
5. True
6. True
7. A, B, D, and F
8. C
9. B
10. A
11. C
12. A
13. B
14. C
15. All of the above
16. B and E
17. False
18. C
19. All are ways to help prevent choking or aspiration of food.
20. True


9.3: Models of Chemical Bonding
Make sure you thoroughly understand the following essential ideas which have been presented.
- Comment on the distinction between a theory and a model in the context of chemical bonding.
- What is meant by a classical model of chemical bonding?
Why do atoms bind together— sometimes? The answer to this question would ideally be a simple, easily understood theory that would not only explain why atoms bind together to form molecules, but would also predict the three-dimensional structures of the resulting compounds as well as the energies and other properties of the bonds themselves. Unfortunately, no one theory exists that accomplishes these goals in a satisfactory way for all of the many categories of compounds that are known. Moreover, it seems likely that if such a theory does ever come into being, it will be far from simple.
About Models in Science
When we are faced the need to find a scientific explanation for a complex phenomenon such as bonding, experience has shown that it is often best to begin by developing a model . A scientific model is something like a theory in that it should be able to explain observations and to make useful predictions. But whereas a theory can be discredited by a single contradictory case, a model can be useful even if it does not encompass all instances of the effects it attempts to explain. We do not even require that a model be a credible representation of reality; all we ask is that it be able to explain the behavior of those cases to which it is applicable in terms that are consistent with the model itself.
An example of a model that you may already know about is the kinetic molecular theory of gases. Despite its name, this is really a model (at least at the level that beginning students use it) because it does not even try to explain the observed behavior of real gases. Nevertheless, it serves as a tool for developing our understanding of gases, and as an essential starting point for more elaborate treatments.
One thing is clear: chemical bonding is basically electrical in nature, the result of attraction between bodies of opposite charge; bonding occurs when outer-shell electrons are simultaneously attracted to the positively-charged nuclei of two or more nearby atoms. The need for models arises when we try to understand why
- Not all pairs of atoms can form stable bonds
- Different elements can form different numbers of bonds (this is expressed as "combining power" or "valence".)
- The geometric arrangement of the bonds ("bonding geometry") around a given kind of atom is a property of the element.
Given the extraordinary variety of ways in which atoms combine into aggregates, it should come as no surprise that a number of useful bonding models have been developed. Most of them apply only to certain classes of compounds or attempt to explain only a restricted range of phenomena. In this section we will provide brief descriptions of some of the bonding models; the more important of these will be treated in much more detail in later lessons in this unit.
Some early views of chemical bonding
Intense speculation about “chemical affinity” began in the 18th century. Some likened the tendency of one atom to “close” with another as an expression of a human-like kind of affection. Others attributed bonding to magnetic-like forces (left) or to varying numbers of “hooks” on different kinds of atoms (right). The latter constituted a primitive (and extremely limited) way of explaining the different combining powers ( valances ) of the different elements.
"There are no such things..."
Napoleon's definition of history as a set of lies agreed on by historians seems to have a parallel with chemical bonding and chemists. At least in Chemistry, we can call the various explanations "models" and get away with it even if they are demonstrably wrong, as long as we find them useful. In a provocative article ( J Chem Educ 1990 67(4) 280-298), J. F. Ogilvie tells us that there are no such things as orbitals, or, for that matter, non-bonding electrons, bonds, or even uniquely identifiable atoms within molecules. This idea disturbed a lot of people (teachers and textbook authors preferred to ignore it) and prompted a spirited rejoinder ( J Chem Ed 1992 69(6) 519-521) from Linus Pauling, father of the modern quantum-mechanical view of the chemical bond.
But the idea has never quite gone away. Richard Bader of McMaster University has developed a quantitative "atoms in molecules" model that depicts molecules as a collection of point-like nuclei embedded in a diffuse cloud of electrons. There are no "bonds" in this model, but only "bond paths" that correspond to higher values of electron density along certain directions that are governed by the manner in which the positive nuclei generate localized distortions of the electron cloud.
Classical models of the chemical bond
By classical , we mean models that do not take into account the quantum behavior of small particles, notably the electron. These models generally assume that electrons and ions behave as point charges which attract and repel according to the laws of electrostatics. Although this completely ignores what has been learned about the nature of the electron since the development of quantum theory in the 1920’s, these classical models have not only proven extremely useful, but the major ones also serve as the basis for the chemist’s general classification of compounds into “covalent” and “ionic” categories.
The Ionic Model
Ever since the discovery early in the 19th century that solutions of salts and other electrolytes conduct electric current, there has been general agreement that the forces that hold atoms together must be electrical in nature. Electrolytic solutions contain ions having opposite electrical charges; opposite charges attract, so perhaps the substances from which these ions come consist of positive and negatively charged atoms held together by electrostatic attraction.
It turns out that this is not true generally, but a model built on this assumption does a fairly good job of explaining a rather small but important class of compounds that are called ionic solids . The most well known example of such a compound is sodium chloride, which consists of two interpenetrating lattices of Na + and Cl – ions arranged in such as way that every ion of one type is surrounded (in three dimensional space) by six ions of opposite charge.
One can envision the formation of a solid NaCl unit by a sequence of events in which one mole of gaseous Na atoms lose electrons to one mole of Cl atoms, followed by condensation of the resulting ions into a crystal lattice:
| Na(g) → Na + (g) + e– | +494 kJ | ionization energy |
|---|---|---|
| Cl(g) + e– → Cl – (g) | –368 kJ | electron affinity |
| Na + (g) + Cl – (g) | –498 kJ | lattice energy |
| Na(g) + Cl(g) → NaCl(s) | –372 kJ | Sum: Na-Cl bond energy |
Note: positive energy values denote endothermic processes, while negative ones are exothermic.
Since the first two energies are known experimentally, as is the energy of the sum of the three processes, the lattice energy can be found by difference. It can also be calculated by averaging the electrostatic forces exerted on each ion over the various directions in the solid, and this calculation is generally in good agreement with observation, thus lending credence to the model. The sum of the three energy terms is clearly negative, and corresponds to the liberation of heat in the net reaction (bottom row of the table), which defines the Na–Cl “bond” energy.
The ionic solid is more stable than the equivalent number of gaseous atoms simply because the three-dimensional NaCl structure allows more electrons to be closer to more nuclei. This is the criterion for the stability of any kind of molecule; all that is special about the “ionic” bond is that we can employ a conceptually simple electrostatic model to predict the bond strength.
The main limitation of this model is that it applies really well only to the small class of solids composed of Group 1 and 2 elements with highly electronegative elements such as the halogens. Although compounds such as CuCl 2 dissociate into ions when they dissolve in water, the fundamental units making up the solid are more like polymeric chains of covalently-bound CuCl 2 molecules that have little ionic character.
Shared-electron (covalent) model
This model originated with the theory developed by G.N. Lewis in 1916, and it remains the most widely-used model of chemical bonding. It is founded on the idea that a pair of electrons shared between two atoms can create a mutual attraction, and thus a chemical bond.
Usually each atom contributes one electron (one of its valence electrons ) to the pair, but in some cases both electrons come from one of the atoms. For example, the bond between hydrogen and chlorine in the hydrogen chloride molecule is made up of the single 1 s electron of hydrogen paired up with one of chlorine's seven valence (3 p ) electrons. The stability afforded by this sharing is thought to derive from the noble gas configurations (helium for hydrogen, argon for chlorine) that surround the bound atoms.
The origin of the electrostatic binding forces in this model can best be understood by examining the simplest possible molecule. This is the hydrogen molecule ion H 2 + , which consists of two nuclei and one electron.
First, however, think what would happen if we tried to make the even simpler molecule H 2 2+ . Since this would consist only of two protons whose electrostatic charges would repel each other at all distances, it is clear that such a molecule cannot exist; something more than two nuclei are required for bonding to occur.
H 2 + we have a third particle, the electron. The effect of this electron will depend on its location with respect to the two nuclei. If the electron is in the space between the two nuclei (the binding region in the diagram), it will attract both protons toward itself, and thus toward each other. If the total attraction energy exceeds the internuclear repulsion, there will be a net bonding effect and the molecule will be stable. If, on the other hand, the electron is off to one side (in an antibinding region ), it will attract both nuclei, but it will attract the closer one much more strongly, owing to the inverse-square nature of Coulomb’s law. As a consequence, the electron will now actively work ag against bonding by helping to push the two nuclei apart.Polar covalent model
A purely covalent bond can only be guaranteed when the electronegativities (electron-attracting powers) of the two atoms are identical. When atoms having different electronegativities are joined, the electrons shared between them will be displaced toward the more electronegative atom, conferring a polarity on the bond which can be described in terms of percent ionic character.
Coulombic model
This is an extension of the ionic model to compounds that are ordinarily considered to be non-ionic. Combined hydrogen is always considered to exist as the hydride ion H – , so that methane can be treated as if it were C 4+ H – 4 .
This is not as bizarre as it might seem at first if you recall that the proton has almost no significant size, so that it is essentially embedded in an electron pair when it is joined to another atom in a covalent bond. This model, which is not as well known as it deserves to be, has surprisingly good predictive power, both as to bond energies and structures.
VSEPR model
The “valence shell electron repulsion” model is not so much a model of chemical bonding as a scheme for explaining the shapes of molecules. It is based on the quantum mechanical view that bonds represent electron clouds— physical regions of negative electric charge that repel each other and thus try to stay as far apart as possible. We will explore this concept in much greater detail in a later unit.
3 Quantum-mechanical models
These models of bonding take into account the fact that a particle as light as the electron cannot really be said to be in any single location. The best we can do is define a region of space in which the probability of finding the electron has some arbitrary value which will always be less than unity. The shape of this volume of space is called an orbital and is defined by a mathematical function that relates the probability to the ( x,y,z) coordinates of the molecule.
Like other models of bonding, the quantum models attempt to show how more electrons can be simultaneously close to more nuclei. Instead of doing so through purely geometrical arguments, they attempt this by predicting the nature of the orbitals which the valence electrons occupy in joined atoms.
The hybrid orbital model
This was developed by Linus Pauling in 1931 and was the first quantum-based model of bonding. It is based on the premise that if the atomic s , p , and d orbitals occupied by the valence electrons of adjacent atoms are combined in a suitable way, the hybrid orbitals that result will have the character and directional properties that are consistent with the bonding pattern in the molecule. The rules for bringing about these combinations turn out to be remarkably simple, so once they were worked out it became possible to use this model to predict the bonding behavior in a wide variety of molecules. The hybrid orbital model is most usefully applied to the p -block elements in the first few rows of the periodic table, and is especially important in organic chemistry.
The molecular orbital model
This model takes a more fundamental approach by regarding a molecule as a collection of valence electrons and positive cores. Just as the nature of atomic orbitals derives from the spherical symmetry of the atom, so will the properties of these new molecular orbitals be controlled by the interaction of the valence electrons with the multiple positive centers of these atomic cores.
These new orbitals, unlike those of the hybrid model, are delocalized; that is, they do not “belong” to any one atom but extend over the entire region of space that encompasses the bonded atoms. The available (valence) electrons then fill these orbitals from the lowest to the highest, very much as in the Aufbau principle that you learned for working out atomic electron configurations. For small molecules (which are the only ones we will consider here), there are simple rules that govern the way that atomic orbitals transform themselves into molecular orbitals as the separate atoms are brought together. The real power of molecular orbital theory, however, comes from its mathematical formulation which lends itself to detailed predictions of bond energies and other properties.
The electron-tunneling model
A common theme uniting all of the models we have discussed is that bonding depends on the fall in potential energy that occurs when opposite charges are brought together. In the case of covalent bonds, the shared electron pair acts as a kind of “electron glue” between the joined nuclei. In 1962, however, it was shown that this assumption is not strictly correct, and that instead of being concentrated in the space between the nuclei, the electron orbitals become even more concentrated around the bonded nuclei. At the same time however, they are free to “move” between the two nuclei by a process known as tunneling .
This refers to a well-known quantum mechanical effect that allows electrons (or other particles small enough to exhibit wavelike properties) to pass (“tunnel”) through a barrier separating two closely adjacent regions of low potential energy. One result of this is that the effective volume of space available to the electron is increased, and according to the uncertainty principle this will reduce the kinetic energy of the electron.
According to this model, the bonding electrons act as a kind of fluid that concentrates in the region of each nucleus (lowering the potential energy) and at the same time is able to freely flow between them (reducing the kinetic energy). Despite its conceptual simplicity and full acknowledgment of the laws of quantum mechanics, this model is less known than it deserves to be and is unfortunately absent from most textbooks.


15.9: Why It Matters- Capital Budgeting Decisions
Capital budgeting decisions are big decisions for companies. Many times, they will have limited funds and several project options to consider. By looking at each option using several different methods, managers can get the information they need to spend limited dollars in the way that will bring the greatest rate of return.
We will discuss multiple methods that can be used to analyze various potential projects in a side-by-side way. Beginning with the screening process, which helps to determine if a project should move onto the next step,by meeting predetermined criteria of each individual business, through the post-audit, which evaluates whether the project we ultimately chose to fund, lived up to the expectations.
So these big decisions cannot be taken lightly. Whether you are buying a new piece of equipment, building a new production facility or doing a complete redesign of a website, it is important to analyze not only the cost of the project, but the future benefits of the project as well.
Are you ready?


9.7: Organizing Slides
Learning Objectives
- Add, delete, and move slides.
Besides text edits on individual slides, you may also need to make more global changes, including adding, deleting, or moving slides within a presentation. Each of these tasks is easiest to accomplish using the slide thumbnail view. The steps for these are outlined below.
Add a Slide
- To add a slide, simply click between two existing slides where you wish to insert a new blank slide.
- Add a slide using the New Slide dropdown display in the Slides group of the ribbon.
The New Slide dropdown display will show you various slide layouts you may want to use.
The layout name gives you a basic idea of when to use what format. For example, you can use the Comparison slide if you wish to present two images side-by-side for direct comparison. There is space above each comparison box to provide a descriptive title, such as “Before editing” and “After editing.”
Don’t worry; it is very easy to change the slide layout if you decide you want a different one later. Note that when you add a slide, the new slide becomes the active slide in the main window.
You can also add slides using the the New slide dropdown menu in the Insert tab.
Practice Question
Delete a Slide
- To delete a slide, click on the thumbnail image of the slide you wish to delete in the thumbnail view.
- Then use either the Backspace or Delete button on your keyboard to remove that slide. Compare the two screenshots below. In the first there are four slides visible in the thumbnail. Slide 2 is selected and deleted, so you do not see it in the second screenshot.
Note that when you delete a slide, the next slide in the presentation becomes the active slide in the main window.
Practice Question
Move a Slide
- To move a slide, click on the slide you wish to move in the slide thumbnail view.
- Drag that slide to a new position in the slide thumbnail view.
Compare the screenshots below. Slides #2 and #3 have been switched.
Note that the slide you move remains the active slide in the main window.
Contributors and Attributions
- Organizing Slides. Authored by : Shelli Carter. Provided by : Lumen Learning. License : CC BY: Attribution


1.3: Content
Types of Content
Content in art takes the form of portraits, landscapes, still-lifes, genre art, and narrative art.
Describe different categories of figurative or abstract art.
Key Points
- Content in a work of art refers to what is being depicted and might be helpful in deriving a basic meaning. It appears in the visual arts in several forms , all of which may be figurative (realistic) or abstract (distorted). Among them are portraits, landscapes, still-lifes, genre art, and narrative art.
- Portraits represents the likeness of a person and can include a study of the sitter’s mood or personality.
- Landscapes depict natural scenery such as mountains, valleys, trees, rivers, and forests, especially where the main subject is a wide view.
- A still-life is a work of art depicting mostly inanimate subject matter, typically commonplace objects that may be either natural or man-made.
- Genre art involves the pictorial representation in any of various media of scenes or events from everyday life, whereas narrative art tells a story that may be real or imagined.
Content in a work of art refers to what is being depicted and might be helpful in deriving a basic meaning. Sometimes content is straightforward; in other cases, however, it is less obvious and requires additional information. Content appears in the visual arts in several forms, all of which may be figurative (realistic) or abstract (distorted). Among them are portraits, landscapes, still-lifes, genre art, and narrative art.
Portraits
A portrait is a painting, photograph, sculpture, or other artistic representation of a person, in which the face and its expression are predominant. The intent is to display the likeness, personality, and even the mood of the person. For this reason, in photography a portrait is generally not a snapshot, but a composed image of a person in a still position. A portrait often shows a person looking directly at the painter or photographer in order to most successfully engage the subject with the viewer .
Landscapes
Landscape painting, also known as landscape art, is the depiction in art of landscapes—natural scenery such as mountains, valleys, trees, rivers, and forests, especially where the main subject is a wide view—with its elements arranged into a coherent composition . In other works, landscape backgrounds for figures can still form an important part of the work. Sky is almost always included in the view, and weather is often an element of the composition. Detailed landscapes as a distinct subject are not found in all artistic traditions and develop when there is already a sophisticated tradition of representing other subjects.
Still Lifes
A still life (plural still lifes) is a work of art depicting mostly inanimate subject matter, typically commonplace objects that may be either natural (food, flowers, dead animals, plants, rocks, or shells) or man-made (drinking glasses, books, vases, jewelry, coins, pipes, and so on). Early still-life paintings, particularly before 1700, often contained religious and allegorical symbolism relating to the objects depicted. Some modern still lifes break the two-dimensional barrier and employ three-dimensional mixed media, and use found objects, photography, computer graphics, as well as video and sound.
Genre Art
Genre art is the pictorial representation in any of various media of scenes or events from everyday life, such as markets, domestic settings, interiors, parties, inn scenes, and street scenes. Such representations (also called genre works, genre scenes , or genre views) may be realistic, imagined, or romanticized by the artist.
Narrative Art
Narrative art is art that tells a story, either as a moment in an ongoing story or as a sequence of events unfolding over time. Some of the earliest evidence of human art suggests that people told stories with pictures. However, without some knowledge of the story being told, it is very hard to read ancient pictures because they are not organized in a systematic way like words on a page, but rather can unfold in many different directions at once.
Figurative and Abstract Art
Art exists along a continuum from realistic representational work to fully non-representational work.
Distinguish between figurative and abstract art
Key Points
- Representational art, or figurative art, references objects or events in the real world.
- Romanticism , Impressionism , and Expressionism contributed to the emergence of abstract art in the nineteenth century.
- Even representational work is abstracted to some degree; entirely realistic art is elusive.
Key Terms
- verisimilitude :The property of seeming true, of resembling reality; resemblance to reality, realism.
Painting and sculpture can be divided into the categories of figurative (or representational) and abstract (or non-representational). Figurative art describes artwork – particularly paintings and sculptures – which are clearly derived from real object sources, and therefore are, by definition, representational. Since the arrival of abstract art in the early twentieth century, the term “figurative” has been used to refer to any form of modern art that retains strong references to the real world.
Artistic independence was advanced during the nineteenth century, resulting in the emergence of abstract art. Three movements which contributed heavily to the development of these styles were Romanticism, Impressionism, and Expressionism.
Abstraction indicates a departure from reality in its depiction of imagery . Abstraction exists along a continuum; it can formally refer to compositions that are derived (or abstracted) from figurative or other natural sources, or it can refer to non-representational art and non-objective art that has no derivation from figures or objects.
Even art that aims for verisimilitude of the highest degree can be said to be abstract, at least theoretically, since perfect representation is likely to be exceedingly elusive. Artwork which takes liberties, altering, for instance, color and form in ways that are conspicuous, can be said to be partially abstract.
Non-representational art refers to total abstraction, bearing no trace of any reference to anything recognizable. In geometric abstraction, for instance, one is unlikely to find references to naturalistic entities. Figurative art and total abstraction are nearly mutually exclusive, but figurative or representational art often contains at least one element of abstraction.
Meaning in Nonrepresentational Art
Meaning in nonrepresentational art is highly subjective and can be difficult to define.
Relate the meaning of nonrepresentational art, its goals, and its specific expressions
Key Points
- Nonrepresentational artwork refers to art that does not attempt to represent or reference reality.
- In the late 19th century, artists began to move toward increasing abstraction as a means of communicating subjective experience more personally and creatively.
- Artists such as Kandinsky and Mondrian viewed art as an expression of spirituality.
Key Terms
- expressionism :A movement in the arts in which the artist does not depict objective reality, but rather the subjective expression of inner experience.
- nonrepresentational :Not intended to represent a physical object in reality.
Nonrepresentational art refers to compositions which do not rely on representation or mimesis to any extent. Abstract art , nonfigurative art, nonobjective art, and nonrepresentational art are related terms that indicate a departure from reality in the depiction of imagery in art. Meaning in nonrepresentational art is highly subjective and can be difficult to define. We can focus on the elements of the artwork (form, shape, line , color, space , and texture) in terms of the aesthetic value of the work, but the meaning will always be personal to the viewer unless the artist has made a statement about his or her intentions.
Generally, we can look at nonrepresentational art as the personal expression of an artist’s subjective experience. Certain movements have described their intentions as an aim to evoke moods or emotions in the viewer. A good example are the expressionists of the early 20th century, who aimed to present the world solely from a subjective perspective , distorting it radically for emotional effect.
Nonrepresentational art has often been explored by artists as a means to spiritual expression. Wassily Kandinsky, a Russian painter, printmaker, and art theorist, is one of the most famous 20th century artists and is generally considered the first important painter of modern abstract art. As an early modernist in search of new modes of visual expression and spiritual expression, he theorized (as did contemporary occultists and theosophists) that pure visual abstraction had corollary vibrations with sound and music. He posited that pure abstraction could express pure spirituality.
Piet Mondrian’s art was also related to his spiritual and philosophical studies. In 1908 he became interested in the theosophical movement launched by Helena Petrovna Blavatsky, who believed that it was possible to attain a knowledge of nature more profound than that provided by empirical means, and much of Mondrian’s work for the rest of his life was inspired by his search for that spiritual knowledge.
Iconography
Iconography is the scholarly study of the content of images, including identification, description, and interpretation.
Define iconography and interpret or perform an iconographical analysis of an image
Key Points
- Academic studies of iconography in painting emerged in the 19th century in France and Germany.
- Iconographical scholarship became particularly prominent in art history after 1940.
- In the 20th century, studies of iconography have become of interest to a broad public beyond the scholarly community.
Key Terms
- iconography :The branch of art history which studies the identification, description, and interpretation of the content of images.
Iconography is the branch of art history which studies the identification, description, and interpretation of the content of images such as the subjects that are depicted, particulars of composition , and other elements that are distinct from artistic style .
Iconography as an academic art historical discipline developed in the nineteenth century in the works of scholars such as Adolphe Napoleon Didron (1806–1867), Anton Heinrich Springer (1825–1891), and Émile Mâle (1862–1954). Christian religious art was the main focus of study throughout this period, and French scholars were especially prominent. They looked back to earlier attempts to classify and organize subjects encyclopedically, as guides to understanding works of art, both religious and profane, in a more scientific manner than the popular aesthetic approach of the time. These early contributions paved the way for encyclopedias, manuals, and other publications useful in identifying the content of art.
In early twentieth-century Germany, Aby Warburg (1866–1929) and his followers Fritz Saxl (1890–1948) and Erwin Panofsky (1892–1968) elaborated the practice of identification and classification of motifs in images to using iconography as a means of understanding meaning. Panofsky codified an influential approach to iconography in his 1939 Studies in Iconology , where he defined it as “the branch of the history of art which concerns itself with the subject matter or meaning of works of art, as opposed to form”. The distinction he and other scholars drew between particular definitions of “iconography” (put simply, the identification of visual content) and “iconology” (the analysis of the meaning of that content) has not been generally accepted, though it is still used by some writers.
While most iconographical scholarship remains highly dense and specialized, some analyses began to attract a much wider audience; for example, Panofsky’s theory (now generally out of favor with specialists) is that the writing on the rear wall in The Arnolfini Portrait by Jan van Eyck turned the painting into the record of a marriage contract. Holbein’s The Ambassadors has been the subject of books for a general market with new theories as to its iconography; as well as being a double portrait, the painting contains a still life of several meticulously rendered objects, the meaning of which is the cause of much debate. The most notable and famous of Holbein’s symbols in the work is the distorted skull which is placed in the bottom center of the composition. The skull, rendered in anamorphic perspective , another invention of the Early Renaissance , is speculated to have been a reminder of death and mortality.
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
- Vanitas-Still_Life,_Oosterwijck.jpg. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Vanitas-Still_Life,_Oosterwijck.jpg. License : Public Domain: No Known Copyright
- 364px-The_idle_servant.jpg. Provided by : Wikipedia. Located at : goo.gl/1TMtsK. License : Public Domain: No Known Copyright
- Matisse_Les_toits.jpg. Provided by : Wikipedia. Located at : en.Wikipedia.org/w/index.php?curid=21463164%20. License : Public Domain: No Known Copyright
- 499px-Laocoon_Pio-Clementino_Inv1059-1064-1067.jpg. Provided by : Wikimedia Commons. Located at : commons.wikimedia.org/w/index.php?curid=1302927%20. License : Public Domain: No Known Copyright
- 345px-Philip_Burne-Jones_holding_cat.jpg. Provided by : Wikimedia Commons. Located at : commons.wikimedia.org/w/index.php?curid=6517545. License : Public Domain: No Known Copyright
- Form and Content. Provided by : Saylor Academy. Located at : https://learn.saylor.org/mod/page/view.php?id=4034 . License : CC BY: Attribution
- Portrait. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Portrait. License : CC BY-SA: Attribution-ShareAlike
- Landscape Painting. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Landscape_painting. License : CC BY-SA: Attribution-ShareAlike
- Form and Content. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Form_and_content. License : CC BY-SA: Attribution-ShareAlike
- Narrative Art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Narrative_art. License : CC BY-SA: Attribution-ShareAlike
- Genre Art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Genre_art. License : CC BY-SA: Attribution-ShareAlike
- Still-Life. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Still_life. License : CC BY-SA: Attribution-ShareAlike
- Figurative art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Figurative_art. License : Public Domain: No Known Copyright
- Abstract art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Abstract_art. License : Public Domain: No Known Copyright
- Abstract art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Abstract_art. License : CC BY-SA: Attribution-ShareAlike
- Figurative art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Figurative_art. License : CC BY-SA: Attribution-ShareAlike
- Abstract art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Abstract_art. License : CC BY-SA: Attribution-ShareAlike
- verisimilitude. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/verisimilitude. License : CC BY-SA: Attribution-ShareAlike
- Wassily Kandinsky. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Wassily_Kandinsky. License : Public Domain: No Known Copyright
- Abstract art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Abstract_art. License : CC BY-SA: Attribution-ShareAlike
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- iconography. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/iconography. License : CC BY-SA: Attribution-ShareAlike


3.12.E: Problems on Cluster Points, Closed Sets, and Density
Complete the proof of Theorem 1\((\text { ii })\).
Prove that \(\overline{R}=E^{1}\) and \(\overline{R^{n}}=E^{n}(\text { Example }(\mathrm{a}))\).
Prove Theorem 2 for \(E^{3} .\) Prove it for \(E^{n}\left(^{*} \text { and } C^{n}\right)\) by induction on \(n .\)
Verify Note 2.
Prove Theorem 3.
Prove Corollaries 1 and 2.
Prove that \((A \cup B)^{\prime}=A^{\prime} \cup B^{\prime}\).
[Hint: Show by contradiction that \(p \notin\left(A^{\prime} \cup B^{\prime}\right)\) excludes \(p \in(A \cup B)^{\prime} .\) Hence \((A \cup B)^{\prime} \subseteq A^{\prime} \cup B^{\prime} .\) Then show that \(A^{\prime} \subseteq(A \cup B)^{\prime},\) etc. \(]\)
From Problem \(7,\) deduce that \(A \cup B\) is closed if \(A\) and \(B\) are. Then prove Corollary \(4 .\) By induction, extend both assertions to any finite number of sets.
From Theorem \(4,\) prove that if the sets \(A_{i}(i \in I)\) are closed, so is \(\bigcap_{i \in I} A_{i}\).
Prove Corollary 3 from Theorem 3. Deduce that \(\overline{\overline{A}}=\overline{A}\) and prove footnote \(3 .\)
[Hint: Consider Figure 7 and Example \((1)\) in §12 when using Theorem 3 (twice). \(]\)
Prove that \(\overline{A}\) is contained in any closed superset of \(A\) and is the intersection of all such supersets.
[Hint: Use Corollaries 2 and \(3 . ]\)
(i) Prove that a bounded sequence \(\left\{\overline{x}_{m}\right\} \subseteq E^{n}\left(^{*} C^{n}\right)\) converges to \(\overline{p}\) iff \(\overline{p}\) is its only cluster point.
(ii) Disprove it for
(a) unbounded \(\left\{\overline{x}_{m}\right\}\) and
(b) other spaces.
[Hint: For \((\mathrm{i}),\) if \(\overline{x}_{m} \rightarrow \overline{p}\) fails, some \(G_{\overline{p}}\) leaves out infinitely many \(\overline{x}_{m} .\) These \(\overline{x}_{m}\) form a bounded subsequence that, by Theorem \(2,\) clusters at some \(\overline{q} \neq \overline{p} .\) (Why? \()\) Thus \(\overline{q}\) is another cluster point (contradiction!)
For (ii), consider (a) Example (f) in §14 and (b) Problem 10 in §14, with (0,2] as a subspace of \(E^{1} . ]\)
In each case of Problem 10 in §14, find \(\overline{A}\). Is \(A\) closed? (Use Theorem 4.)
Prove that if \(\left\{b_{n}\right\} \subseteq B \subseteq \overline{A}\) in \((S, \rho),\) there is a sequence \(\left\{a_{n}\right\} \subseteq A\) such that \(\rho\left(a_{n}, b_{n}\right) \rightarrow 0 .\) Hence \(a_{n} \rightarrow p\) iff \(b_{n} \rightarrow p .\)
[Hint: Choose \(a_{n} \in G_{b_{n}}(1 / n) .]\)
We have, by definition,
\[
p \in A^{0} \text { iff }(\exists \delta>0) G_{p}(\delta) \subseteq A;
\]
hence
\[
p \notin A^{0} \text { iff }(\forall \delta>0) G_{p}(\delta) \nsubseteq A, \text { i.e., } G_{p}(\delta)-A \neq \emptyset .
\]
(See Chapter \(1,§§1-3 . )\) Find such quantifier formulas for \(p \in \overline{A}, p \notin \overline{A}\), \(p \in A^{\prime},\) and \(p \notin A^{\prime}\).
[Hint: Use Corollary 6 in \(§ 14,\) and Theorem 3 in \(§16 .]\)
Use Problem 15 to prove that
(i) \(-(\overline{A})=(-A)^{0}\) and
(ii) \(-\left(A^{0}\right)=\overline{-A}\).
Show that
\[
\overline{A} \cap(\overline{-A})=\mathrm{bd} A(\text { boundary of } A);
\]
cf. \(§ 12,\) Problem \(18 .\) Hence prove again that \(A\) is closed iff \(A \supseteq\) bd \(A .\)
[Hint: Use Theorem 4 and Problem 16 above. \(]\)
A set \(A\) is said to be nowhere dense in \((S, \rho)\) iff \((\overline{A})^{0}=\emptyset .\) Show that Cantor's set \(P(§14, \text { Problem } 17)\) is nowhere dense.
\([\text { Hint: } P \text { is closed, so } \overline{P}=P .]\)
Give another proof of Theorem 2 for \(E^{1}\).
[Hint: Let \(A \subseteq[a, b] .\) Put
\[
Q=\{x \in[a, b] | x \text { exceeds infinitely many points (or terms) of } A\}.
\]
Show that \(Q\) is bounded and nonempty, so it has a glb, say, \(p=\inf A .\) Show that \(A\) clusters at \(p . ]\)
For any set \(A \subseteq(S, \rho)\) define
\[
G_{A}(\varepsilon)=\bigcup_{x \in A} G_{x}(\varepsilon).
\]
Prove that
\[
\overline{A}=\bigcap_{n=1}^{\infty} G_{A}\left(\frac{1}{n}\right).
\]
Prove that
\[
\overline{A}=\{x \in S | \rho(x, A)=0\} ; \text { see } \$ 13, \text { Note } 3.
\]
Hence deduce that a set \(A\) in \((S, \rho)\) is closed iff
\((\forall x \in S) \quad \rho(x, A)=0 \Longrightarrow x \in A\).