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- Question : 1 - If a, b E R and a < b, give a description in set theory notation for each of the intervals (a, b), a, b, a, b), and (a. b] (see Example 1.1.1).
- Question : 2 - If A, B, and C are sets, prove that A n (Bu C) = (A n B) U (A n C).
- Question : 3 - If A and B are two sets, then prove that A is the union of a disjoint pair of sets, one of which is contained in B and one of which is contained t from B.
- Question : 4 - What is the intersection of all the open intervals containing the closed interval 0, 1]? Justify your answer.
- Question : 5 - What is the intersection of all the closed intervals containing the open interval (0, 1 )? Justify your answer.
- Question : 6 - What is the union of all of the closed intervals contained in the open interval (0, 1)? Justify your answer.
- Question : 7 - If It is a collection of subsets of a set X, formulate and prove a theorem like Theorem 1.1.5 for the intersection and union of A.
- Question : 8 - Which of the following functions f : IR R are one-to-one an(l which ones are onto. Justify your answer. (a) f(.x) = x2; (b) f(.x) = x3; (c) f(x)=ex.
- Question : 9 - Prove part (b) of Theorem 1.1.6.
- Question : 10 - Prove part (c) of Theorem 1.1.6.
- Question : 11 - Prove part (a) of Theorem 1. 1.7.
- Question : 12 - Prove part (b) of Theorem 1.1.7.
- Question : 13 - Give an example of a function f : A
- Question : 14 - Prove that equality holds in parts (b) and (c) of Theorem 1.1.7 if the function f is one-to-one.
- Question : 15 - Prove that if f : A
- Question : 16 - Prove that a subset C of A x B is the graph of a function from A to B if and only if the following condition is satisfied: for each a e A there is exactly one b E B such that (a, b) E C.
- Question : 17 - Prove that the commutative law for addition, n + m = m + n, holds in N. Use induction and Examples 1.2.6 and 1.2.5.
- Question : 18 - Prove that if n, in E N, then m + n ?n. Hint: Use induction on n.
- Question : 19 - Use the preceding exercise to prove that if n, in E J, n ?in, and in ?n, then ii = in. This is the reflexive property of an order relation.
- Question : 20 - Prove that the order relation on N has the transitive property: if k ?ii and ii ?in, then k ?in.
- Question : 21 - Use the preceding exercise and Peano
- Question : 22 - Show how to define the product mu of two natural numbers. Hint: Use induction On m.
- Question : 23 - Use the definition of product that you gave in the preceding exercise to prove that if ii, in E N, then ii urn.
- Question : 24 - Using induction, prove that 7^n
- Question : 25 - Using induction, prove that ?_(k=1)^n??k=? (n(n+1))/2 for every ii e N.
- Question : 26 - Using induction, prove that ?_(k=1)^n??(2k-1)=n^2 ? for every n ? N.
- Question : 27 - Finish the prove of Theorem 1.2.3 by showing that there is only one sequence { x,-L} which satisfies the conditions of the theorem.
- Question : 28 - If x1 is chosen so that I < x < 2 and x is defined inductively y X[1 = /x11 + 2, then prove by induction that I < x x7 < 2 for all n E N.
- Question : 29 - Let a sequence {x} of numbers be defined recursively by x=I and x_n+1=(x_n+1)/2 Prove by induction that x for all ii E N. Vou1cl this conclusion change if we set x = 2?
- Question : 30 - 14. Let a sequence {x} of numbers be defined recursively by x1= 1 and x_n+1=1/(1+x_n) Prove by induction that 5n+2 is between x, and for each n E N.
- Question : 31 - Mathematical induction also works for a sequence Pk, Pk+1,... of propositions, indexed by the integers n? k for some k? N. The statement is: if Pk is true and P+i is true whenever P is true and ii k, then P,1 is true for all
- Question : 32 - Use induction in the form stated in the preceding exercise to prove that ,2 < 2 for all ii
- Question : 33 - Prove the identity (?(n@k-1))+(?(n@k))=?(n+1@k) which was used in the proof of Theorem 1.2.12.
- Question : 34 - Write out the binornial formula iii the case n = 4
- Question : 35 - . Prove the well ordering principal for the natural numbers: each non-empty subset S of N contains a smallest element. Hint: Apply the induction axiom to the set T=(n?N:n
- Question : 36 - Use the result of Exercise 1.2.19 to prove the division algorithm: if ii and in are natural numbers with in <
- Question : 37 - Given that N has an operation of addition which is commutative and associative, how would you define such an addition operation in Z?
- Question : 38 - Referring to the previous exercise, answer the same question for the operation of multiplication.
- Question : 39 - Prove that if Z satisfies the axioms for a commutative ring, then Q satisfies Al and Ml.
- Question : 40 - Prove that if Z satisfies the axioms for a commutative ring, then Q satisfies A2 and M2. In the next three exercises you are to prove the given statement assuming x, y, z are elements of a field. You may use the results of examples an(l theorems from this section.
- Question : 41 - (-x)(y)=xy
- Question : 42 - xz=yz Implies x = y, provided z?0.
- Question : 43 - xy=0 Implies x = 0 or y = 0. In the next three exercises you are to prove the given statement assuming x, y, z are elements of an ordered field. Again, you may use the results of examples and theorems from this section.
- Question : 44 - x>O and y>O imply xy >0.
- Question : 45 - x > 0 implies x
- Question : 46 - 0 < x < y implies y^(-1)
- Question : 47 - Prove that the equation x2 = 5 has no rational solution.
- Question : 48 - Generalize Theorem 1.3.9 by proving that every rational solution of a polynomial equation x^n+a_(n-1)+?+a_1 x+a_n=0, with integer coefficients
- Question : 49 - x>O and y>O imply xy >0.
- Question : 50 - x > 0 implies x
- Question : 51 - 0 < x < y implies y^(-1)
- Question : 52 - Prove that the equation x2 = 5 has no rational solution.
- Question : 53 - Generalize Theorem 1.3.9 by proving that every rational solution of a polynomial equation x^n+a_(n-1)+?+a_1 x+a_n=0, with integer coefficients
- Question : 54 - Prove that if in and ii are positive integers with no common factors other than 1 (i.e. in and ii are relatively prime), then there are integers a and b such that 1 =am+bn. hint: Let S be the set of all positive integers of the form am+bn, where a and b are integers. This set has a smallest element by Exercise 1 .2.19. Use the division algorithm (Exercise 1.2.20) to show that this smallest element divides both in and ii.
- Question : 55 - Use the result of the preceding exercise to prove that if a prime v divides the product of two positive integers, then it divides ii or it divides in.
- Question : 56 - For each of the following sets, describe the set of all upper hounds for the set: (a) of odd integers; (b) {1-1/n:n?N}; (c) {r?Q:r^3
- Question : 57 - For each of the sets in (a), (b), (c) of the preceding exercise, find the least upper bound of the set, if it exists.
- Question : 58 - Prove that if a subset A of R is bounded above, then the set of all upper bounds for A is a set of the form [x,?). What is x?
- Question : 59 - Show that the set A = {x :x^2 < 1
- Question : 60 - If F is an ordered field, prove that there is a sequence of elements ?{n?_k}k?n all different, such that n1 = 1 (the identity element of F) and nk+1 = k + 1 for each k E N. Argue that the terms of this sequence form a subset of F which is a copy of the natural numbers, by showing that the correspondence k?n_k is a one-to-one function from N onto this subset. By definition it takes the successor k + 1 of an element k E N to the successor nk + 1 of its image k.
- Question : 61 - Let F be an ordered field. We consider N to be a subset of F as described in the preceding exercise. Prove that F is Archimedean if and only if, for each pair x, y e F with x > 0, there exists a natural number n such that n x > y.
- Question : 62 - Prove that if x < y are two real numbers, then there is a rational number r with x < r < y. Hint: Use the result of Example 1.4.9.
- Question : 63 - Prove that if x is irrational and r is a non-zero rational number, then x + r and rx are also irrational.
- Question : 64 - we know that is irrational. Use this fact and the previous exercise to prove that if r < s are rational numbers, then there is an irrational number x with r
- Question : 65 - Show that if L1 and L are Dedekind cuts defining real numbers x and y, then L_x+L_y={r+s:r?L_x and s?L_y} is also a Dedekind cut (this is the Dedekind cut determining the sum x + g).
- Question : 66 - If L and L are Dedekind cuts determining positive real numbers x and y and if we set K={rs:0?r?L_y }?{t? Q:t
- Question : 67 - If L is the Dedekind cut of Example 1.4.2 and L determines the real number r (so that L = L1), prove that L2 = L2. Thus, the real number corresponding to L has square 2.
- Question : 68 - For each of the following sets, find the set of all extended real numbers x that are greater than
- Question : 69 - Find the Sup and inf of the following sets. Tell whether each set has a maximum or a minimum. (a) (
- Question : 70 - Prove that if sup A
- Question : 71 - Prove that if sup A = , then for each n E N there is an element
- Question : 72 - Formulate and prove the analog of Theorem 1.5.4 for inf.
- Question : 73 - Prove part (d) of Theorem 1.5.7.
- Question : 74 - Prove part (e) of Theorem 1.5.7.
- Question : 75 - If A and B are two non-empty sets of real numbers, then prove that sup(A?B)=max?{supA,supB} and inf?(A?B)=min?{infA.infB}.
- Question : 76 - Find sup1 f and infj f for the following functions f and sets I. Which of these is actually the maximum or the minimum of the function f on I? (a) f(x) = x2, I = [
- Question : 77 - Prove (a) of Theorem 1.5.10
- Question : 78 - Prove (b) of Theorem 1.5.10.
- Question : 79 - Prove (c) of Theorem 1.5.10.
- Question : 80 - Prove (d) of Theorem 1.5.10.

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