Math 109 September 1, 2016

Math 109 September 1, 2016

Question 1 Given that the proposition P Q is true. Which of the following must also be true? A. (not P ) or Q. B. (not Q) implies (not P ). C. Q implies P. D. A and B E. A, B, and C

Question 2 De Morgan s Law is the logical relationship A. not (P or Q) (not P ) and (not Q). B. not (P and Q) (not P ) or (not Q). C. A and B: both are part of De Morgan s Law(s). D. Neither A nor B: both equivalencies are false. E. There s no De Morgan s Law! Who was De Morgan anyway?

Question 3 Theorem: n 2 n for every positive integer n. Proof: Clearly, 1 2 1. Suppose k 2 k for some positive integer k. If (k + 1) 2 (k + 1), then k 2 (k + 1) 2 1 (k + 1) 1 = k. Thus, (k + 1) 2 k + 1 must be true. A. The theorem is balderdash. For example, 2 2 > 2. B. Since the proof is clear, this has become known as the Square Paradox. C. The proof is faulty because it assumes the conclusion as part of the argument. D. A and B E. A and C

Question 4 Axiom Given a statement P (n) involving a general positive integer n. positive integers n if (i) P (1) is true, and Then P (n) is true for all (ii) if k is a positive integer and P (m) is true for all all positive integers m k, then P (k + 1) is true. This axiom A. is called the strong induction principle. B. is not a valid axiom because it contradicts the induction principle. C. is logically equivalent to the induction principle. D. makes some theorems easier to prove than using the induction principle directly. E. A, C, and D

Question 5 Consider the following statement: For every real number x, there exists a real number y such that x y = 1. Which of the following is a correct negation of this statement? A. For every real number x, there is no real number y such that x y = 1. B. There exists a real number x and a real number y such that x y 1. C. For every real number x, there exists a real number y such that x y 1. D. There exists a real number x such that for every real number y, x y 1. E. None of the above.

Question 6 Given a set of real numbers A, its power set P(A) A. is the set of all subsets of A. B. is the set of all integer powers of the numbers in A. That is, P(A) = {x n x A and n Z}. C. has 2 N elements when A is a finite set with N elements. D. A and C. E. B and C.

Question 7 Given integers a and b, at least one of which is non-zero. A. a and b are coprime precisely when gcd(a, b) = 1 B. a and b are coprime precisely when they have no common prime factor. C. gcd(a, 0) = a for every integer a. D. A and B E. A, B, and C

Question 8 Given non-zero integers a and b. The general solution to the linear diophantine equation am + bn = c A. is empty when c is not a multiple of gcd(a, b). B. is unique, provided that c is a multiple of gcd(a, b). C. is of the form (p + kr, q + ks), where k is any integer, (p, q) is a particular solution, and (r, s) is any solution to the corresponding homogeneous equation. D. A and B E. A and C

Question 9 Given non-zero integers a and b. A. The existence of integers m, n such that am+bn = 1 is necessary for gcd(a, b) = 1. B. The existence of integers m, n such that am+bn = 1 is sufficient for gcd(a, b) = 1. C. The linear diophantine equation am + bn = 1 has a unique solution (m, n) when gcd(a, b) = 1. D. A and B. E. All of the above.

Question 10 Given integers a and b, and m a positive integer. A. If a divides m, then ab 1 ab 2 mod m if and only if b 1 b 2 mod ( ) m a. B. If gcd(a, m) = 1, then ab 1 ab 2 mod m if and only if b 1 b 2 mod m. C. If gcd(a, m) = 1, then the linear congruence ax b mod m has a unique solution modulo m. D. A and B E. A, B, and C

Question 11 Let X be a set. A partition Π of X A. can only be defined when X is a nonempty set. B. is a subset of P(X), the power set of X. C. defines a corresponding equivalence relation on X. D. B and C E. A, B, and C

Question 12 only if mn > 0. Define on Z {0} by m n if and A. is not an equivalence relation on Z {0} because it is not transitive. B. is not an equivalence relation on Z {0} because it is not reflexive. C. is an equivalence relation on Z {0}. D. (Z {0}) / has two elements: [ 1] and [1]. E. C and D

Question 13 Given a set X. Then, A. X = n for some positive integer n. B. X = 0 or there exists a bijection f : N n X for some positive integer n. C. X is the number of elements in X. D. X = 0, or there exists a bijection f : N n X for some positive integer n, or X is infinite. E. A, B, and C

Question 14 Let X and Y be non-empty finite sets and f : X Y a function. The pigeonhole principle principle asserts that A. if X > Y, then f is not an injection. B. if f is an injection, then X Y. C. if X = Y, then there exists a bijection f : X Y. D. A and B; B is the contrapositive of A E. None of the above: Where are the pigeons?

Question 15 Given a set X. A. A permutation of X is a bijection σ : X X. B. n! is the number of permutations of X when X = n. C. σ(n) = n + 1 is a permutation of Z. D. A, B, and C E. A and B : Permutations only make sense for finite sets.

Question 16 The binomial theorem A. asserts that (x + y) n = n k=0 ( n k) x k y n k. B. can be proved by induction on n. C. can be proved by observing that the number of times x k y n k occurs in the product (x + y) n is the number of ways x can be chosen k times out of n choices. D. implies that n ( n k) k=0 = 2 n. E. All of the above.

Question 17 A set X is infinite if and only if it is A. equipotent to a proper subset of itself. B. denumerable. C. uncountable. D. A and B E. A and C