Practice Test III, Math 314, Spring 2016

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1 Practice Test III, Math 314, Spring 2016 Dr. Holmes April 26, 2016 This is the 2014 test reorganized to be more readable. I like it as a review test. The students who took this test had to do four sections of their choice (doing both problems in each section, with the usual grading scheme). Good performance on a section repeated from earlier exams could improve the grade on the earlier exam as well. They could do more sections if they wished, with best work counting. Your test will probably be shorter, but again, with more than four sections, some being review, and you having the choice of which four to do. 1

2 1 Logic Questions A set of logic strategies and rules is attached. 1. Give a proof of ((P Q) (Q R)) (P R) in the style taught in class. 2

3 2. State and prove one of de Morgan s laws in the style taught in class. 3

4 2 Formal arithmetic questions Some axioms and theorems of formal arithmetic are attached. 1. Do one of the following relatively easy formal arithmetic proofs. If you do both your best work will count. (a) Prove S(x) + y = S(x + y) using the axioms alone. (b) Prove 1x = x using the axioms alone (well, you may also use the definition of 1 and the theorem S(x) = x + 1). 4

5 2. Prove the right distributive law of multiplication using the axioms alone. You may if you wish choose one of the other main proofs in the formal arithmetic homework assignment (whose text is attached). If you choose to do one of the others, you are allowed to use the theorems appearing before it in the list in your proof. 5

6 3 General algebra questions 1. Do one of the following three (I hope relatively easy) problems. If you do more than one your best work will count. A list of axioms from Spivak is attached. (a) Prove the lemma If x+y = 0 then y = x from Spivak s axioms (P1)-(P9) (which will be given). Then prove a( b) = (ab). Your proofs should show full justifications from P1-9. You may use the theorem x0 = 0 without proving it. [This is not the same as this part of the original Test I, but similar in difficulty and very similar in approach]. (b) Prove that for any real numbers x, y if xy = 1 then y = x 1. Your proof should show full justifications from P1-9. Remember that you can only write x 1 if you know that x 0. You can use the theorem x0 = 0 without proving it. (c) Prove using either Spivak s order axioms (P10)-(P12) and algebra skills, or his alternative set of axioms and algebra skills, that if 0 < a < b it follows that a 2 < b 2 (yes, you know the definition of a 2 as aa). You should show justifications mentioning the order axioms (from whichever of the two sets you use); you do not need to show justifications from P1-9 (you can use algebra without comment). 6

7 2. Prove using familiar properties of even and odd numbers and your algebra skills that the square root of 2 is not a rational number. There is no need to mention any Spivak axioms by name in this proof. 7

8 4 Limit proofs Limit proofs are to include scratch work. Where you put it is up to you, as long as I can tell what is scratch work and what is proof. 1. Prove that lim x 2 x 2 = 4 in complete detail (yes, this is harder. You should be up to it). This question should demonstrate your acquaintance with the exact meaning of limit notation: in this or any limit proof, start by expanding out the statement using the definition. 8

9 2. State and prove a limit theorem, your choice from the constant multiple property of limits (computing lim x a cf(x)), the addition property of limits or the subtraction property of limits. 9

10 5 Chapter 5 Limit Questions 1. Explain why lim f(x) lim g(x) = lim f(x) g(x) x a x a x a is not a good way to state the multiplication property of limits. State it in a better way. Write an example with specific functions and numbers where this equation has one side defined and one side undefined. 10

11 2. Define D(x) = 0 if x is irrational and 1 if x is rational. Prove that lim x a D(x) does not exist for any number a. 11

12 3. Prove that lim x 0 1 x does not exist. 12

13 6 Continuity questions 1. Use the Intermediate Value Theorem (along with general algebra and simple calculus knowledge) to prove that there is a cube root of 5. Hint: you want to choose a suitable closed interval and a suitable function. Your proof should make it clear that you know the full statement of some version of the IVT. 13

14 2. Prove that if f is continuous at a and f(a) > 0, then f is bounded above and bounded below on some open interval (a δ, a + δ) You cannot use any Big Theorems: this is just an ɛ δ proof from the definitions of continuity and limit. You should also make it clear that you know what bounded above and bounded below mean. 14

15 3. A set D is dense iff every open interval contains an element of D. Suppose that a function f is continuous at every number a and further that f(x) = 0 for each x D, where D is a dense set. Prove that f(x) = 0 for every x. 15

16 7 Least upper bound questions 1. Determine the greatest lower bound and least upper bound of each of the following sets, if any. Explain in each case whether each of the bounds is actually an element of the set. Give brief explanations of your answers. (a) The set of fractions n 1 for n a positive integer. This set is not n unbounded in either direction! (b) The set of negative even integers (c) The set { 1 n + ( 1)n : n N}. This is an example from Spivak; I think the best approach is to list some values and see how they act; I definitely expect an explanation of each of the two answers. Zero is not a natural number. 16

17 2. Prove that for every positive real ɛ > 0 there is a natural number n such that 1 < ɛ. You can prove this by first proving Spivak s theorem n that for every real number x there is a natural number n greater than x (using P13) or you could consider the greatest lower bound of the set { 1 : n N}: first show that it exists (you may assume and use the n theorem about greatest lower bounds, but you do need to show that you know what it says) and then show that it cannot be greater than zero. 17

18 3. Suppose that A and B are nonempty sets and every element of A is less than every element of B. First, show that the least upper bound of A, sup A is less than or equal to any y B. Then show that the least upper bound of A (sup(a)) is less than or equal to the greatest lower bound of B (sup(b)). This is problem 12 from the chapter 8 homework, which we went over in class. It is really an exercise in understanding the definitions of least upper bound and greatest lower bound. 18

19 8 Sequence questions 1. Prove the addition property for limits of sequences (hint: the letter δ should not appear in your work at all!) 19

20 2. Prove the Monotone Convergence Theorem: a sequence which is nondecreasing and bounded above must converge. Your presentation should include definitions of what it means for a sequence to be nondecreasing and what it means for a sequence to be bounded above. It will use the Least Upper Bound axiom. 20

21 3. A Cauchy sequence is a sequence {a n } such that for each ɛ > 0 there is N such that for any m, n if m, n > N then a m a n < ɛ. Prove that any sequence with a limit is Cauchy. 21

22 9 Propositional Logic Style Sheet Not all of these are relevant to the assigned proofs; you need to recognize which rules and strategies are appropriate. Conjunction (and): To prove A B, prove A (part 1), then prove B (part 2). The parts are not cases, and you will lose credit if you call something a proof by cases which isn t one. From A B, deduce A. From A B, deduce B. You need to explicitly break apart assumptions or lemmas which are and statements to use their components; you will lose credit if you don t. You are permitted to break apart statements by numbering the component statements directly without copying them over. Disjunction (or): To prove A B, assume A and adopt the new goal B [or assume B and adopt the new goal A; you do not need to do both]. From A, deduce A B. From B, deduce A B (rule of weakening). From A B and A, deduce B. From A B and B, deduce A. This is disjunctive syllogism. To deduce a conclusion C from an assumption or lemma A B, use proof by cases: in the first part (case 1) assume A and prove C; in the second part (case 2) assume B and prove C. Implication (if): To prove A B, assume A and adopt the new goal B. alternative strategy: to prove A B, assume B and adopt the new goal A. Given A and A B, deduce B (modus ponens). Given B and A B, deduce A (modus tollens). If you have an assumption A B you may want to try proving A (so that you can further conclude B). Negation (not): To prove A, assume A and try to prove (contradiction). From A deduce A (double negation). To prove any statement A, assume A and try to prove. This is proof by contradiction. 22

23 From A and A, deduce. If you have a negative assumption A, the commonest way to use it is to wait until your goal is a contradiction, then try to prove A to get the contradiction. From deduce anything! 23

24 10 Axioms and Theorems of Formal Arithmetic Here are the axioms of formal arithmetic is a natural number (in symbols, 0 N). 2. If x and y are natural numbers, so are S(x), x + y, and x y. ( xy N.S(x) N x + y N x y N) is not a successor. ( x.s(x) 0). Here we understand that x y abbreviates x = y. Here and in the following axioms we write our quantifiers unrestricted: we could write ( x N.S(x) 0) instead, but in this context we are only talking about natural numbers, so we can leave the restriction on our quantifiers implicit. 4. Numbers with the same successor are the same. ( xy.s(x) = S(y) x = y). 5. Let P(x) be any sentence about a natural number variable x. We assert P (0) ( y.p (y) P (S(y))) ( x.p (x)). This is a symbolic presentation of the familiar principle of mathematical induction. From an extremely technical standpoint, this is an infinite collection of axioms, one for each sentence P (x). If we are also willing to talk about sets of natural numbers, we can state it as a single axiom: ( A P(N).0 A ( y N.y A S(y) A) A = N). We will not use the set formulation now but we might use it later. P(N) is a notation for the collection of all sets of natural numbers. 6. ( x.x + 0 = x) 7. ( xy.x + S(y) = S(x + y)) 8. ( x.x 0 = 0) 9. ( x.x S(y) = x y + x) Here we assume the usual order of operations. Here are some theorems and a definition. Not all of these are necessarily of any use. definition of 1: 1 is defined as S(0). 24

25 Theorem 1: ( x.x + 1 = S(x)) Theorem 2: ( x.x = 0 ( y.s(y) = x)) Instructions: You may if you wish write any of the following proofs to meet requirements for the second problem in section 2. The commutative laws are not recommended. You may write a second one of these if you wish, to replace any other question on the test but the last one (you cannot escape the major limit theorem). You are only allowed to use one of these theorems without proving it if you are proving one that appears later on this list. commutativity of addition: ( xy.x + y = y + x) (not recommended for you to write this proof) right distributivity of multiplication over addition: ( xyz.x(y + z) = xy + xz) associativity of addition: ( xyz.(x + y) + z = x + (y + z)) left distributivity of multiplication over addition: ( xyz.(x + y)z = xz + yz) associativity of multiplication: ( xyz.(xy)z = x(yz)) commutativity of multiplication: ( xy.xy = yx) (I do not recommend that you write this proof). 25

26 11 Axioms from Spivak algebra axioms (P1): For any numbers x, y, z, x+(y +z) = (x+y)+z. [associative property of addition] (P2): For any number x, x + 0 = 0 + x = x [identity property of addition] (P3): For any number x, x + ( x) = ( x) + x = 0 [inverse property of addition] (P4): For any numbers x, y, x+y = y+x [commutative property of addition] (P5): For any numbers x, y, z, x(yz) = (xy)z. [associative property of multiplication] (P6): For any number x, x1 = 1x = x. 1 0 [identity property of multiplication] (P7): For any number x 0, xx 1 = x 1 x = 1 [inverse property of multiplication] (P8): For any numbers x, y, xy = yx [commutative property of multiplication] (P9): For any numbers x, y, z, x(y + z) = xy + xz [distributive property of multiplication over addition] order axioms P10 For each number x, exactly one of the following is true: x P ; x = 0; x P. P11: For any numbers x,y, if x P and y P, then x + y P. P11: For any numbers x,y, if x P and y P, then xy P. alternative order axioms P10 For any numbers x, y, exactly one of the following is true: x < y;x = y;x > y 26

27 P11 For any numbers x, y, z, if x < y and y < z, then x < z. P12 For any numbers x, y, z, if x < y then x + z < y + z. P13 For any numbers x, y, z, if x < y and z > 0, then xz < yz. 27

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