Math 220 Exam 2
November 9, 2012
S. Witherspoon
Name
There are 6 questions, for a total of 100 points. Point values are written beside each
question.
2n,
if n is even
1. Let f : Z → Z be defined by f (n) =
n + 1, if n is odd
(a) [5 points] Find f (O), where O is the set of odd integers. (You need not justify
your answer.)
(b) [5] Find f −1 (E), where E is the set of even integers. (You need not justify your
answer.)
(c) [5] Is f injective? Justify your answer.
1
2
2. (a) [5] Let A = {1, 2, 3} and B = {x, y, z}. Give an example of a surjective
function f : A → B.
(b) [10] Let f : A → B be a function for which A and B are finite sets and |A| = |B|.
Prove that f is injective if, and only if, f is surjective. (Hint: Use the Pigeonhole
Principle.)
3
3. Let ∗ be the binary operation on Z defined by a ∗ b = 2a + 3b.
(a) [5] Is ∗ commutative? Prove or disprove.
(b) [5] Is ∗ associative? Prove or disprove.
(c) [5] Is the set E of even integers closed in Z under ∗? Prove or disprove.
(d) [5] Is the set O of odd integers closed under ∗? Prove or disprove.
4
4. [15] Let R be the relation on the set R of real numbers defined by aRb if a − b ∈ Z.
Prove that R is an equivalence relation on R.
5
5. [20] Prove by induction that for each positive integer n,
1 + 5 + 9 + 13 + · · · + (4n − 3) = n(2n − 1).
6
6. (a) [5] Give an example of integers n, a, b for which n divides ab, n does not divide
a, and n does not divide b.
(b) [10] Let n be a positive integer greater than 1 with the property that for all
a, b ∈ Z, if n divides ab, then n divides a or n divides b. Prove that n is a prime
number. (Hint: Try a proof by contradiction.)