Sample Second Midterm

Sample Second Midterm - then all three are empty. (12) 5....

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Sample Second Midterm MATH 108 Sample Second Midterm 1. In each part below, give the definition if it is one of our concepts. Or, if there is a problem, indicate carefully what it is: a. A relation. b. The characteristic function of a set A. c. A surjection. (15) 2. Suppose we define ordered triples in terms of ordered pairs by (a, b, c) = ((a, b), c). Use this definition to prove that (a, b, c) = (x, y, z) iff a = x and b = y and c = z. (12) 3. a. Define a relation on Q as follows: xRy iff for some natural number k, x = 3^k y. Determine if R is an equivalence relation. If so find 1/2 /R. b. Define R in the same way as in part a., but let k be an integer and answer the same questions. (18) 4. a. Prove phi: phi -> phi. b. Prove that if f: A -> B and any one of f, A, or Rng(f) is empty,
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Unformatted text preview: then all three are empty. (12) 5. a. Let f:NxN->N be given by f(m,n) = 2^m(2n+1). Find f-1(B) where B = {4,6,8,10,12,14}. b. Suppose f:A->B. Then, if we consider the induced set function f from P(A) to P(B) and the induced set function f-1 from P(B) to P(A), where P denotes the power set, indicate if it is true or false that f-1(f(X)) = X, for all subsets X of A. If so, why? If not, why not? (9) 6. a. Prove that any set equivalent to a finite set must be finite. b. Prove directly (do not cite any theorems) that the interval (-infinity, a) is equivalent to the interval (b, infinity). (20) 7. Prove that if A is a denumerable set, then AxAxAx . . . xA (A appears n times) is denumerable, for any natural number n. (14)...
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