 
Summary: Sample Problems for Midterm 1
The purpose of this document is to help you study for our first midterm. It is in no way
a complete list of possible exam problems. In addition to completing these exercises, you
should review your homework, notes and chapters 1,2,4,18,19,20, and 22.
1 Prove that if A and B are sets, then
A B = A B A
2 Suppose that A is a set. Prove A
3 Suppose that f : A B and g : B C and that g f : A C is injective. Prove that
f must be injective.
4 Express the following proposition using quantifiers:
"Every real number can be approximated as closely as you like by a rational number."
5 Express the following proposition in plain English: x R, n N s.t. n > x.
(This is called the Archimedean Principle)
6 Express the following proposition using quantifiers: "Every positive integer is the sum
of two squares of integers".
7 Let a R \ Q and b Q \ {0}. Prove that ab R \ Q
8 Express the above proposition and its negation using quantifiers.
9 Disprove the following proposition (i.e. prove its negation): "Every odd integer is prime."
10 Disprove the following proposition: "The sum of any two irrational numbers is irra
tional."
