Sample Second Midterm

# Sample Second Midterm - then all three are empty(12 5 a Let...

This preview shows page 1. Sign up to view the full content.

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,
This is the end of the preview. Sign up to access the rest of the document.

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 b. Prove directly (do not cite any theorems) that the interval (-infinity, a) is equivalent to the interval (b, infinity).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)...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online