Math 220 Final Exam
May 7, 2012
S. Witherspoon
Name
There are 9 questions, for a total of 100 points. Point values are written beside each
question.
1. Consider the statement: For all real numbers x and y, if xy is rational, then x is
rational.
(a) [3 points] Write the converse of this statement.
(b) [3] Write the contrapositive of this statement.
(c) [3] Write the negation of this statement.
(d) [3] Which of the above four statements (the proposition, its converse (a), its
contrapositive (b), its negation (c)) are true? (You need not justify your answer.)
1
2
2. [10] Prove that for all integers n, n is divisible by 3 if, and only if, n2 is divisible
by 3.
3
3. Let f : Z → Z be defined by f (n) =
2n − 1,
2n,
(a) [3] Find f ({1, 2, 3, 4}).
(b) [3] Find f −1 ({1, 2, 3, 4}).
(c) [3] Is f injective? Justify your answer.
(d) [3] Is f surjective? Justify your answer.
if n is even
if n is odd
4
4. Let A and B be sets and let Y be a subset of B.
(a) [5] Let f : A → B be a surjective function. Prove that Y = f (f −1 (Y )).
(b) [5] Show that the assumption that f be surjective in part (a) is necessary, by
giving an example of sets A, B, a subset Y of B, and a function f : A → B for which
Y 6= f (f −1 (Y )).
5
5. [14] Prove by induction that for each positive integer n,
1 + 3 + 5 + · · · + (2n − 1) = n2 .
6
6. (a) [5] Use the Euclidean algorithm to find (91, 35).
(b) [5] Find integers x and y such that (91, 35) = 91x + 35y.
7
7. [12] Let R be the relation on Z defined by aRb if a ≤ b + 1. Determine whether R
is reflexive, symmetric, or transitive. Justify your answer.
8
8. [10] Find the least positive integer x that satisfies the congruence
4x ≡ 32(mod 9).
9
1
defines a bijection from R+ (the set
2
1+x
of positive real numbers) to the open interval (0, 1).
9. [10] Prove that the function f (x) =