Exam 1 - Solutions
Unless otherwise noted, unjustied answers will receive zero credit. It should go without saying that
all responses must be fully justied in complete sentences, that liquid water is wet, and that cheating
is prohibited.
1. (a) (10pt) Giv

Exam 3
1. (a) (5pt) State the Monotone Convergence Theorem for sequences of real numbers.
(b) (5pt) Give the N denition of convergence of a sequence of real numbers.
(c) (15pt) For a function f : A B, and subsets C A and D B, dene f (C) and
f 1 (D). Prove

Final Exam
1. (a) (5pt) Let cfw_an nN be a sequence and x L R. State the denition of lim an = L.
n
(b) (5pt) Let f : A B be a function. Let C A and D B. Dene f (C) and f 1 (D).
(c) (5pt) Let R be a relation on a set A. Dene what it means for R to be reexi

Exam 2
1. (a) (5pt) Dene symmetric, reexive and transitive relations.
(b) (5pt) State the completeness axiom for the real numbers.
(c) (15pt) For a function f : A B, and subsets C A and D B, dene f (C) and f 1 (D).
Prove carefully that for subsets E and H

Exam 1
1. (a) (5pt) Let a, b Z and assume that a = 0. State the denition of a divides b.
(b) (5pt) State the triangle inequality for real numbers x, y.
(c) (5pt) Negate the following sentence: If I pass this test, then none of my classmates will
study ton

Quiz 4 Solution
1. (10pt) Suppose D R is a nonempty, closed, bounded set. Let f : D
function. Prove that f pDq is closed.
R be a continuous
Solution: Let tyn unPN be a convergent sequence in f pDq, say yn y for some y R.
We want to show that y f pDq. Now

Quiz 2
1. (15pt) Evaluate the proposed proof of the following result. If it is right, say so. If it is wrong,
say what is wrong. Can the proof be xed? If so, how? If not, nd a counterexample.
Z. Then 3|x and 3|px yq, if and only if 3|y.
Proof. We do forw

Quiz 1
1. (5pt) Let A be the set cfw_, cfw_, cfw_. List the elements of P(A) and compute |A P(A)|.
Solution: The power set P(A) consists of all subsets of A, and should have cardinality
2|A| . Since |A| = 3, we expect |P(A)| = 23 = 8, so that |A P(A)| = 3

Quiz 3
1. (10pt) Let f : A B and g : B C be functions. Prove or disprove: if g f : A C is
bijective, then f is injective.
Solution:
Proof. Let a, a1 A with f paq f pa1 q. Now since g is well-dened and f paq B, it follows
that
g pf paqq g pf pa1 qq,
or in

Exam 2
Unless otherwise noted, unjustied answers will receive zero credit. It should go without saying that
all responses must be fully justied in complete sentences, that liquid water is wet, and that cheating
is prohibited.
1. (a) (5pt) For a nonempty s