# 4.4E: Exercises for Section 4.4 - Mathematics

1)Why do you need continuity to apply the Mean Value Theorem? Construct a counterexample.

2) Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.

One example is (f(x)=|x|+3,−2≤x≤2)

3) When are Rolle’s theorem and the Mean Value Theorem equivalent?

4) If you have a function with a discontinuity, is it still possible to have (f′(c)(b−a)=f(b)−f(a)?) Draw such an example or prove why not.

Yes, but the Mean Value Theorem still does not apply

In exercises 5 - 9, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.

5) (y=sin(πx))

6) (y=dfrac{1}{x^3})

((−∞,0),(0,∞))

7) (y=sqrt{4−x^2})

8) (y=sqrt{x^2−4})

((−∞,−2),(2,∞))

9) (y=ln(3x−5))

In exercises 10 - 13, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points (c) such that (f′(c)(b−a)=f(b)−f(a).)

10) [T] (y=3x^3+2x+1) over ([−1,1])

2 points

11) [T] (y= anleft(frac{π}{4}x ight)) over (left[−frac{3}{2},frac{3}{2} ight])

12) [T] (y=x^2cos(πx)) over ([−2,2])

5 points

13) [T] (y=x^6−frac{3}{4}x^5−frac{9}{8}x^4+frac{15}{16}x^3+frac{3}{32}x^2+frac{3}{16}x+frac{1}{32}) over ([−1,1])

In exercises 14 - 19, use the Mean Value Theorem and find all points (0such that (f(2)−f(0)=f′(c)(2−0)).

14) (f(x)=x^3)

(c=frac{2sqrt{3}}{3})

15) (f(x)=sin(πx))

16) (f(x)=cos(2πx))

(c=frac{1}{2},1,frac{3}{2})

17) (f(x)=1+x+x^2)

18) (f(x)=(x−1)^{10})

(c=1)

19) (f(x)=(x−1)^9)

In exercises 20 - 23, show there is no (c) such that (f(1)−f(−1)=f′(c)(2)). Explain why the Mean Value Theorem does not apply over the interval ([−1,1].)

20) (f(x)=left|x−frac{1}{2} ight|)

Not differentiable

21) (f(x)=dfrac{1}{x^2})

22) (f(x)=sqrt{|x|})

Not differentiable

23) (f(x)=lfloor x floor) (Hint: This is called the floor function and it is defined so that (f(x)) is the largest integer less than or equal to (x).)

In exercises 24 - 34, determine whether the Mean Value Theorem applies for the functions over the given interval ([a,b]). Justify your answer.

24) (y=e^x) over ([0,1])

Yes

25) (y=ln(2x+3)) over ([−frac{3}{2},0])

26) (f(x)= an(2πx)) over ([0,2])

The Mean Value Theorem does not apply since the function is discontinuous at (x=frac{1}{4},frac{3}{4},frac{5}{4},frac{7}{4}.)

27) (y=sqrt{9−x^2}) over ([−3,3])

28) (y=dfrac{1}{|x+1|}) over ([0,3])

Yes

29) (y=x^3+2x+1) over ([0,6])

30) (y=dfrac{x^2+3x+2}{x}) over ([−1,1])

The Mean Value Theorem does not apply; discontinuous at (x=0.)

31) (y=dfrac{x}{sin(πx)+1}) over ([0,1])

32) (y=ln(x+1)) over ([0,e−1])

Yes

33) (y=xsin(πx)) over ([0,2])

34) (y=5+|x|) over ([−1,1])

The Mean Value Theorem does not apply; not differentiable at (x=0).

For exercises 35 - 37, consider the roots of each equation.

35) Show that the equation (y=x^3+3x^2+16) has exactly one real root. What is it?

36) Find the conditions for exactly one root (double root) for the equation (y=x^2+bx+c)

(b=±2sqrt{c})

37) Find the conditions for (y=e^x−b) to have one root. Is it possible to have more than one root?

In exercises 38 - 42, use a calculator to graph the function over the interval ([a,b]) and graph the secant line from (a) to (b). Use the calculator to estimate all values of (c) as guaranteed by the Mean Value Theorem. Then, find the exact value of (c), if possible, or write the final equation and use a calculator to estimate to four digits.

38) [T] (y= an(πx)) over (left[−frac{1}{4},frac{1}{4} ight])

(c approx ±0.1533)
(c=±frac{1}{π}cos^{−1}(frac{sqrt{π}}{2}))

39) [T] (y=dfrac{1}{sqrt{x+1}}) over ([0,3])

40) [T] (y=|x^2+2x−4|) over ([−4,0])

The Mean Value Theorem does not apply.

41) [T] (y=x+dfrac{1}{x}) over (left[frac{1}{2},4 ight])

42) [T] (y=sqrt{x+1}+dfrac{1}{x^2}) over ([3,8])

(dfrac{1}{2sqrt{c+1}}−dfrac{2}{c^3}=dfrac{521}{2880})
(c approx 3.133, 5.867)

43) At 10:17 a.m., you pass a police car at 55 mph that is stopped on the freeway. You pass a second police car at 55 mph at 10:53 a.m., which is located 39 mi from the first police car. If the speed limit is 60 mph, can the police cite you for speeding?

44) Two cars drive from one spotlight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.

Yes

45) Show that (y=sec^2x) and (y= an^2x) have the same derivative. What can you say about (y=sec^2x− an^2x)?

46) Show that (y=csc^2x) and (y=cot^2x) have the same derivative. What can you say about (y=csc^2x−cot^2x)?

It is constant.

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## 4.4 Reaction Yields

The relative amounts of reactants and products represented in a balanced chemical equation are often referred to as stoichiometric amounts. All the exercises of the preceding module involved stoichiometric amounts of reactants. For example, when calculating the amount of product generated from a given amount of reactant, it was assumed that any other reactants required were available in stoichiometric amounts (or greater). In this module, more realistic situations are considered, in which reactants are not present in stoichiometric amounts.

### Limiting Reactant

Consider another food analogy, making grilled cheese sandwiches (Figure 4.13):

Stoichiometric amounts of sandwich ingredients for this recipe are bread and cheese slices in a 2:1 ratio. Provided with 28 slices of bread and 11 slices of cheese, one may prepare 11 sandwiches per the provided recipe, using all the provided cheese and having six slices of bread left over. In this scenario, the number of sandwiches prepared has been limited by the number of cheese slices, and the bread slices have been provided in excess.

Consider this concept now with regard to a chemical process, the reaction of hydrogen with chlorine to yield hydrogen chloride:

The balanced equation shows the hydrogen and chlorine react in a 1:1 stoichiometric ratio. If these reactants are provided in any other amounts, one of the reactants will nearly always be entirely consumed, thus limiting the amount of product that may be generated. This substance is the limiting reactant , and the other substance is the excess reactant . Identifying the limiting and excess reactants for a given situation requires computing the molar amounts of each reactant provided and comparing them to the stoichiometric amounts represented in the balanced chemical equation. For example, imagine combining 3 moles of H2 and 2 moles of Cl2. This represents a 3:2 (or 1.5:1) ratio of hydrogen to chlorine present for reaction, which is greater than the stoichiometric ratio of 1:1. Hydrogen, therefore, is present in excess, and chlorine is the limiting reactant. Reaction of all the provided chlorine (2 mol) will consume 2 mol of the 3 mol of hydrogen provided, leaving 1 mol of hydrogen unreacted.

An alternative approach to identifying the limiting reactant involves comparing the amount of product expected for the complete reaction of each reactant. Each reactant amount is used to separately calculate the amount of product that would be formed per the reaction’s stoichiometry. The reactant yielding the lesser amount of product is the limiting reactant. For the example in the previous paragraph, complete reaction of the hydrogen would yield

Complete reaction of the provided chlorine would produce

The chlorine will be completely consumed once 4 moles of HCl have been produced. Since enough hydrogen was provided to yield 6 moles of HCl, there will be unreacted hydrogen remaining once this reaction is complete. Chlorine, therefore, is the limiting reactant and hydrogen is the excess reactant (Figure 4.14).

View this interactive simulation illustrating the concepts of limiting and excess reactants.

### Example 4.12

#### Identifying the Limiting Reactant

Which is the limiting reactant when 2.00 g of Si and 1.50 g of N2 react?

#### Solution

The provided Si:N2 molar ratio is:

The stoichiometric Si:N2 ratio is:

Comparing these ratios shows that Si is provided in a less-than-stoichiometric amount, and so is the limiting reactant.

Alternatively, compute the amount of product expected for complete reaction of each of the provided reactants. The 0.0712 moles of silicon would yield

while the 0.0535 moles of nitrogen would produce

Since silicon yields the lesser amount of product, it is the limiting reactant.

### Percent Yield

The amount of product that may be produced by a reaction under specified conditions, as calculated per the stoichiometry of an appropriate balanced chemical equation, is called the theoretical yield of the reaction. In practice, the amount of product obtained is called the actual yield , and it is often less than the theoretical yield for a number of reasons. Some reactions are inherently inefficient, being accompanied by side reactions that generate other products. Others are, by nature, incomplete (consider the partial reactions of weak acids and bases discussed earlier in this chapter). Some products are difficult to collect without some loss, and so less than perfect recovery will reduce the actual yield. The extent to which a reaction’s theoretical yield is achieved is commonly expressed as its percent yield :

Actual and theoretical yields may be expressed as masses or molar amounts (or any other appropriate property e.g., volume, if the product is a gas). As long as both yields are expressed using the same units, these units will cancel when percent yield is calculated.

### Example 4.13

#### Calculation of Percent Yield

What is the percent yield?

#### Solution

Using this theoretical yield and the provided value for actual yield, the percent yield is calculated to be

### How Sciences Interconnect

#### Green Chemistry and Atom Economy

The purposeful design of chemical products and processes that minimize the use of environmentally hazardous substances and the generation of waste is known as green chemistry. Green chemistry is a philosophical approach that is being applied to many areas of science and technology, and its practice is summarized by guidelines known as the “Twelve Principles of Green Chemistry” (see details at this website). One of the 12 principles is aimed specifically at maximizing the efficiency of processes for synthesizing chemical products. The atom economy of a process is a measure of this efficiency, defined as the percentage by mass of the final product of a synthesis relative to the masses of all the reactants used:

Though the definition of atom economy at first glance appears very similar to that for percent yield, be aware that this property represents a difference in the theoretical efficiencies of different chemical processes. The percent yield of a given chemical process, on the other hand, evaluates the efficiency of a process by comparing the yield of product actually obtained to the maximum yield predicted by stoichiometry.

The synthesis of the common nonprescription pain medication, ibuprofen, nicely illustrates the success of a green chemistry approach (Figure 4.15). First marketed in the early 1960s, ibuprofen was produced using a six-step synthesis that required 514 g of reactants to generate each mole (206 g) of ibuprofen, an atom economy of 40%. In the 1990s, an alternative process was developed by the BHC Company (now BASF Corporation) that requires only three steps and has an atom economy of

80%, nearly twice that of the original process. The BHC process generates significantly less chemical waste uses less-hazardous and recyclable materials and provides significant cost-savings to the manufacturer (and, subsequently, the consumer). In recognition of the positive environmental impact of the BHC process, the company received the Environmental Protection Agency’s Greener Synthetic Pathways Award in 1997.

## 4.4 Uniform Circular Motion

Uniform circular motion is a specific type of motion in which an object travels in a circle with a constant speed. For example, any point on a propeller spinning at a constant rate is executing uniform circular motion. Other examples are the second, minute, and hour hands of a watch. It is remarkable that points on these rotating objects are actually accelerating, although the rotation rate is a constant. To see this, we must analyze the motion in terms of vectors.

### Centripetal Acceleration

We can find the magnitude of the acceleration from

The direction of the acceleration vector is toward the center of the circle (Figure 4.19). This is a radial acceleration and is called the centripetal acceleration , which is why we give it the subscript c. The word centripetal comes from the Latin words centrum (meaning “center”) and petere (meaning “to seek”), and thus takes the meaning “center seeking.”

Let’s investigate some examples that illustrate the relative magnitudes of the velocity, radius, and centripetal acceleration.

### Example 4.10

#### Solution

Solving for the radius, we find

#### Significance

A flywheel has a radius of 20.0 cm. What is the speed of a point on the edge of the flywheel if it experiences a centripetal acceleration of 900.0 cm / s 2 ? 900.0 cm / s 2 ?

Centripetal acceleration can have a wide range of values, depending on the speed and radius of curvature of the circular path. Typical centripetal accelerations are given in the following table.

### Equations of Motion for Uniform Circular Motion

A particle executing circular motion can be described by its position vector r → ( t ) . r → ( t ) . Figure 4.20 shows a particle executing circular motion in a counterclockwise direction. As the particle moves on the circle, its position vector sweeps out the angle θ θ with the x-axis. Vector r → ( t ) r → ( t ) making an angle θ θ with the x-axis is shown with its components along the x- and y-axes. The magnitude of the position vector is A = | r → ( t ) | A = | r → ( t ) | and is also the radius of the circle, so that in terms of its components,

If T is the period of motion, or the time to complete one revolution ( 2 π 2 π rad), then

Velocity and acceleration can be obtained from the position function by differentiation:

It can be shown from Figure 4.20 that the velocity vector is tangential to the circle at the location of the particle, with magnitude A ω . A ω . Similarly, the acceleration vector is found by differentiating the velocity:

### Example 4.11

#### Solution

The position of the particle at t = 2.0 × 10 −7 s t = 2.0 × 10 −7 s with A = 0.175 m is

From this result we see that the proton is located slightly below the x-axis. This is shown in Figure 4.21.

### Nonuniform Circular Motion

Circular motion does not have to be at a constant speed. A particle can travel in a circle and speed up or slow down, showing an acceleration in the direction of the motion.

In uniform circular motion, the particle executing circular motion has a constant speed and the circle is at a fixed radius. If the speed of the particle is changing as well, then we introduce an additional acceleration in the direction tangential to the circle. Such accelerations occur at a point on a top that is changing its spin rate, or any accelerating rotor. In Displacement and Velocity Vectors we showed that centripetal acceleration is the time rate of change of the direction of the velocity vector. If the speed of the particle is changing, then it has a tangential acceleration that is the time rate of change of the magnitude of the velocity:

The direction of tangential acceleration is tangent to the circle whereas the direction of centripetal acceleration is radially inward toward the center of the circle. Thus, a particle in circular motion with a tangential acceleration has a total acceleration that is the vector sum of the centripetal and tangential accelerations:

## Rotation Worksheets

Our printable rotation worksheets have numerous practice pages to rotate a point, rotate triangles, quadrilaterals and shapes both clockwise and counterclockwise (anticlockwise). In addition, pdf exercises to write the coordinates of the graphed images (rotated shapes) are given here. These handouts are ideal for students of grade 5 through high school. Swing your practice wheels with our free worksheets! Task 5th grade children to follow the instructions and rotate the shape in the clockwise or counterclockwise direction with a single quarter or half turn and draw it in the space provided. In these printable worksheets, 6th grade and 7th grade students need to draw the shapes, following the specifications carefully turning them half or quarter multiple times in both clockwise and counterclockwise directions. In these rotation worksheet pdfs for grade 6 and grade 7, graph the image of each point according to the given rule. Each worksheet has eight problems for practice. Rotate the given point about the origin (clockwise or anticlockwise) and choose the correct response from the given multiple choices. Rotate each shape clockwise or counterclockwise about the origin to draw the image of the shape on the given grid. In these printable grade 8 worksheets, rotate each triangle according to the given instruction. Graph the new position of the rotated triangle. Shapes with four sides and points (quadrilaterals) are shown. Sketch the image of the given shape after rotation. Write a rule to describe each rotation. Mention the degree of rotation (90° or 180°) and the direction of rotation (clockwise or counterclockwise). Rotate each shape. Graph the image obtained and label it. Also write the coordinates of the image. Each pdf worksheet has six problems for 8th grade and high school students. The coordinates of a figure are given in these printable handouts. Make the desired rotation and write down the new coordinates of the rotated image. Rotate the given shapes clockwise or anticlockwise and choose the correct image from the multiple choices.

## Suffixes

A suffix is a word part added to the end of a word to create a new meaning. Study the suffix rules in the following boxes.

When adding the suffixes –ness and –ly to a word, the spelling of the word does not change.

When the word ends in y, change the y to i before adding –ness and –ly.

When the suffix begins with a vowel, drop the silent e in the root word.

When the word ends in ce or ge, keep the silent e if the suffix begins with a or o.

When the suffix begins with a consonant, keep the silent e in the original word.

When the word ends in a consonant plus y, change the y to i before any suffix not beginning with i.

When the suffix begins with a vowel, double the final consonant only if (1) the word has only one syllable or is accented on the last syllable and (2) the word ends in a single vowel followed by a single consonant.

• tan + ing = tanning (one syllable word)
• regret + ing = regretting (The accent is on the last syllable the word ends in a single vowel followed by a single consonant.)
• cancel + ed = canceled (The accent is not on the last syllable.)
• prefer + ed = preferred

### Exercise 3

On your own sheet of paper, write correctly the forms of the words with their suffixes.

1. refer + ed
2. refer + ence
3. mope + ing
4. approve + al
5. green + ness
6. benefit + ed
7. resubmit + ing
8. use + age
9. greedy + ly
10. excite + ment

### Key Takeaways

• A prefix is a word part added to the beginning of a word that changes the word’s meaning.
• A suffix is a word part added to the end of a word that changes the word’s meaning.
• Learning the meanings of prefixes and suffixes will help expand your vocabulary, which will help improve your writing.

#### Writing Application

Write a paragraph describing one of your life goals. Include five words with prefixes and five words with suffixes. Exchange papers with a classmate and circle the prefixes and suffixes in your classmate’s paper. Correct each prefix or suffix that is spelled incorrectly.

1. Given two strings, write a program that efficiently finds the longest common subsequence.
2. Given an array with numbers, write a program that efficiently answers queries of the form: “Which is the nearest larger value for the number at position i ?”, where distance is the difference in array indices. For example in the array [1,4,3,2,5,7] , the nearest larger value for 4 is 5. After linear time preprocessing you should be able to answer queries in constant time.
3. Given two strings, write a program that outputs the shortest sequence of character insertions and deletions that turn one string into the other.
4. Write a function that multiplies two matrices together. Make it as efficient as you can and compare the performance to a polished linear algebra library for your language. You might want to read about Strassen’s algorithm and the effects CPU caches have. Try out different matrix layouts and see what happens.
5. Implement a van Emde Boas tree. Compare it with your previous search tree implementations.
6. Given a set of d-dimensional rectangular boxes, write a program that computes the volume of their union. Start with 2D and work your way up.
• Write a program that displays a bouncing ball.
• Write a Memory game.
• Write a Tetris clone

## David Marker

The second theme is illustrated by Morley's Categoricity Theorem, which says that if T is a theory in a countable language and there is an uncountable cardinal $kappa$ such that, up to isomorphism, T has a unique model of cardinality $kappa$, then T has a unique model of cardinality $lambda$ for every uncountable $kappa$. This line has been extended by Shelah, who has developed deep general classification results.

For some time, these two themes seemed like opposing directions in the subject, but over the last decade or so we have come to realize that there are fascinating connections between these two lines. Classical mathematical structures, such as groups and fields, arise in surprising ways when we study general classification problems, and ideas developed in abstract settings have surprising applications to concrete mathematical structures. The most striking example of this synthesis is Hrushovski's application of very general model-theoretic methods to prove the Mordell--Lang Conjecture for function fields.

My goal was to write an introductory text in model theory that, in addition to developing the basic material, illustrates the abstract and applied directions of the subject and the interaction of these two programs.

Chapter 1 begins with the basic definitions and examples of languages, structures, and theories. Most of this chapter is routine, but, because studying definability and interpretability is one of the main themes of the subject, I have included some nontrivial examples. Section 1.3 ends with a quick introduction to $MM^< m eq>$. This is a rather technical idea that will not be needed until Chapter 6 and can be omitted on first reading.

The first results of the subject, the Compactness Theorem and the Lowenheim--Skolem Theorem, are introduced in Chapter 2. In Section 2.2 we show that even these basic results have interesting mathematical consequences by proving the decidability of the theory of the complex field. Section 2.4 discusses the back-and-forth method beginning with Cantor's analysis of countable dense linear orders and moving on to Ehrenfeucht--Fra"ss'e Games and Scott's result that countable structures are determined up to isomorphism by a single infinitary sentence.

Chapter 3 shows how the ideas from Chapter 2 can be used to develop a model-theoretic test for quantifier elimination. We then prove quantifier elimination for the fields of real and complex numbers and use these results to study definable sets.

Chapters 4 and 5 are devoted to the main model-building tools of classical model theory. We begin by introducing types and then study structures built by either realizing or omitting types. In particular, we study prime, saturated, and homogeneous models. In Section 4.3, we show that even these abstract constructions have algebraic applications by giving a new quantifier elimination criterion and applying it to differentially closed fields. The methods of Sections 4.2 and 4.3 are used to study countable models in Section 4.4, where we examine $aleph_0$-categorical theories and prove Morley's result on the number of countable models. The first two sections of Chapter 5 are devoted to basic results on indiscernibles. We then illustrate the usefulness of indiscernibles with two important applications---a special case of Shelah's Many-Models Theorem in Section 5.3 and the Paris--Harrington independence result in Section 5.4. Indiscernibles also later play an important role in Section 6.5.

Chapter 6 begins with a proof of Morley's Categoricity Theorem in the spirit of Baldwin and Lachlan. The Categoricity Theorem can be thought of as the beginning of modern model theory and the rest of the book is devoted to giving the flavor of the subject. I have made a conscious pedagogical choice to focus on $omega$-stable theories and avoid the generality of stability, superstability, or simplicity. In this context, forking has a concrete explanation in terms of Morley rank. One can quickly develop some general tools and then move on to see their applications. Sections 6.2 and 6.3 are rather technical developments of the machinery of Morley rank and the basic results on forking and independence. These ideas are applied in Sections 6.4 and 6.5 to study prime model extensions and saturated models of $omega$-stable theories.

Chapters 7 and 8 are intended to give a quick but, I hope, seductive glimpse at some current directions in the subject. It is often interesting to study algebraic objects with additional model-theoretic hypotheses. In Chapter 7 we study $omega$-stable groups and show that they share many properties with algebraic groups over algebraically closed fields. We also include Hrushovski's theorem about recovering a group from a generically associative operation which is a generalization of Weil's theorem on group chunks. Chapter 8 begins with a seemingly abstract discussion of the combinatorial geometry of algebraic closure on strongly minimal sets, but we see in Section 8.3 that this geometry has a great deal of influence on what algebraic objects are interpretable in a structure. We conclude with an outline of Hrushovski's proof of the Mordell--Lang Conjecture in one special case.

Because I was trying to write an introductory text rather than an encyclopedic treatment, I have had to make a number of ruthless decisions about what to include and what to omit. Some interesting topics, such as ultraproducts, recursive saturation, and models of arithmetic, are relegated to the exercises. Others, such as modules, the $p$-adic field, or finite model theory, are omitted entirely. I have also frequently chosen to present theorems in special cases when, in fact, we know much more general results. Not everyone would agree with these choices.

• graduate students considering doing research in model theory
• graduate students in logic outside of model theory
• mathematicians in areas outside of logic where model theory has had interesting applications.

The graduate student in logic outside of model theory should, in addition to learning the basics, get an idea of some of the main directions of the modern subject. I have also included a number of special topics that I think every logician should see at some point, namely the random graph, Ehrenfeucht--Fraisse Games, Scott's Isomorphism Theorem, Morley's result on the number of countable models, Shelah's Many-Models Theorem, and the Paris--Harrington Theorem.

For the mathematician interested in applications, I have tried to illustrate several of the ways that model theory can be a useful tool in analyzing classical mathematical structures. In Chapter 3, we develop the method of quantifier elimination and show how it can be used to prove results about algebraically closed fields and real closed fields. One of the areas where model-theoretic ideas have had the most fruitful impact is differential algebra. In Chapter 4, we introduce differentially closed fields. Differentially closed fields are very interesting $omega$-stable structures. Chapters 6, 7, and 8 contain a number of illustrations of the impact of stability-theoretic ideas on differential algebra. In particular, in Section 7.4 we give Poizat's proof of Kolchin's theorem on differential Galois groups of strongly normal extensions. In Chapter 7, we look at classical mathematical objects---groups--- under additional model-theoretic assumptions---$omega$-stability. We also use these ideas to give more information about algebraically closed fields. In Section 8.3, we give an idea of how ideas from geometric model theory can be used to answer questions in Diophantine geometry.

### Prerequisites

Chapter 1 begins with the basic definitions of languages and structures. Although a mathematically sophisticated reader with little background in mathematical logic should be able to read this book, I expect that most readers will have seen this material before. The ideal reader will have already taken one graduate or undergraduate course in logic and be acquainted with mathematical structures, formal proofs, G"odel's Completeness and Incompleteness Theorems, and the basics about computability. Shoenfield's cite or Ebbinghaus, Flum, and Thomas' cite are good references.

I will assume that the reader has some familiarity with very basic set theory, including Zorn's Lemma, ordinals, and cardinals. Appendix A summarizes all of this material. More sophisticated ideas from combinatorial set theory are needed in Chapter 5 but are developed completely in the text.

Many of the applications and examples that we will investigate come from algebra. The ideal reader will have had a year-long graduate algebra course and be comfortable with the basics about groups, commutative rings, and fields. Because I suspect that many readers will not have encountered the algebra of formally real fields that is essential in Section 3.3, I have included this material in Appendix B. Lang's cite is a good reference for most of the material we will need. Ideally the reader will have also seen some elementary algebraic geometry, but we introduce this material as needed.

### Using This Book as a Text

I suspect that in most courses where this book is used as a text, the students will have already seen most of the material in Sections 1.1, 1.2, and 2.1. A reasonable one-semester course would cover Sections 2.2, 2.3, the beginning of 2.4, 3.1, 3.2, 4.1--4.3, the beginning of 4.4, 5.1, 5.2, and 6.1. In a year-long course, one has the luxury of picking and choosing extra topics from the remaining text. My own choices would certainly include Sections 3.3, 6.2--6.4, 7.1, and 7.2.

### Exercises and Remarks

Each chapter ends with a section of exercises and remarks. The exercises range from quite easy to quite challenging. Some of the exercises develop important ideas that I would have included in a longer text. I have left some important results as exercises because I think students will benefit by working them out. Occasionally, I refer to a result or example from the exercises later in the text. Some exercises will require more comfort with algebra, computability, or set theory than I assume in the rest of the book. I mark those exercises with a dagger. The Remarks sections have two purposes. I make some historical remarks and attributions. With a few exceptions, I tend to give references to secondary sources with good presentations rather than the original source. I also use the Remarks section to describe further results and give suggestions for further reading.

### Acknowledgments

My approach to model theory has been greatly influenced by many discussions with my teachers, colleagues, collaborators, students, and friends. My thesis advisor and good friend, Angus Macintyre, has been the greatest influence, but I would also like to thank John Baldwin, Elisabeth Bouscaren, Steve Buechler, Zo'e Chatzidakis, Lou van den Dries, Bradd Hart, Leo Harrington, Kitty Holland, Udi Hrushovski, Masanori Itai, Julia Knight, Chris Laskwoski, Dugald Macpherson, Ken McAloon, Margit Messmer, Ali Nesin, Kobi Peterzil, Anand Pillay, Wai Yan Pong, Charlie Steinhorn, Alex Wilkie, Carol Wood, and Boris Zil'ber for many enlightening conversations and Alan Taylor and Bill Zwicker, who first interested me in mathematical logic.

I would also like to thank John Baldwin, Amador Martin Pizarro, Dale Radin, Kathryn Vozoris, Carol Wood, and particularly Eric Rosen for extensive comments on preliminary versions of this book.

Finally, I, like every model theorist of my generation, learned model theory from two wonderful books, C. C. Chang and H. J. Keisler's Model Theory and Gerald Sacks Saturated Model Theory. My debt to them for their elegant presentations of the subject will be clear to anyone who reads this book.

## Popularity of Programming Language Worldwide, Jul 2021 compared to a year ago:

`
Rank Change Language Share Trend
1 Python 30.32 % -1.8 %
2 Java 17.79 % +1.0 %
3 Javascript 9.03 % +1.1 %
4 C# 6.55 % -0.2%
5 C/C++ 6.02 % +0.3 %
6 PHP 5.94 % +0.0 %
7 R 3.96 % -0.0 %
8 TypeScript 2.26 % +0.3 %
9 Objective-C 2.24 % -0.3 %
10 Swift 1.78 % -0.4 %
11 Kotlin1.75 % +0.3 %
12 Matlab 1.72 % -0.2 %
13 VBA 1.38 % +0.1 %
14 Go 1.28 % -0.1 %
15 Rust 1.26 % +0.3 %
16 Ruby 1.01 % -0.2 %
17 Visual Basic 0.76 % -0.1 %
18 Ada 0.74 % +0.3 %
19 Scala 0.72 % -0.3 %
20 Dart 0.61 % +0.1 %
21 Lua 0.54 % +0.1 %
22 Abap 0.44 % -0.0 %
23 Perl 0.38 % -0.0 %
24 Julia 0.36 % -0.0 %
25 Groovy 0.34 % -0.1 %
26 Cobol 0.3 % -0.1 %
27 Delphi/Pascal 0.27 % -0.0 %
28 Haskell 0.24 % -0.0 %

TIOBE Index for June 2021

Jun 2021 Jun 2020 Change Programming Language Ratings Change
1 1 C 12.54% -4.65%
2 3 Python 11.84% +3.48%
3 2 Java 11.54% -4.56%
4 4 C++ 7.36% +1.41%
5 5 C# 4.33% -0.40%
6 6 Visual Basic 4.01% -0.68%
7 7 JavaScript 2.33% +0.06%
8 8 PHP 2.21% -0.05%
9 14 Assembly language 2.05% +1.09%
10 10 SQL 1.88% +0.15%
11 19 Classic Visual Basic 1.72% +1.07%
12 31 Groovy 1.29% +0.87%
13 13 Ruby 1.23% +0.25%
14 9 R 1.20% -0.99%
15 16 Perl 1.18% +0.36%
16 11 Swift 1.10% -0.35%
17 37 Fortan 1.07% +0.80%
18 22 Delphi/Object Pascal 1.06% +0.47%
19 15 MATLAB 1.05% +0.15%
20 12 Go 0.95% -0.06%

List of Exercises with Solutions :

Note : The solution of the exercises described here are not the only ways to do stuff. Rather, it would be great, if this helps you anyway to choose your own methods.

[ Want to contribute to C# Sharp exercises? Send your code (attached with a .zip file) to us at w3resource[at]yahoo[dot]com. Please avoid copyrighted materials.]