 
Summary: Math 2020 Spring 2011
Solving Discrete Problems P. Achar
Notes on Chapter 3
Universal Quantifier. A sentence with a universal quantifier can often be rewritten using "Let" or "If
. . . then." The four sentences below are all different ways of expressing the same thing. Note that in the
fourth example, the quantified variable (the subset of N) isn't given a name.
A N, if A is nonempty, then A has a smallest element.
Let A N. If is a nonempty, then A has a smallest element.
If A N is nonempty, then A has a smallest element.
Every nonempty subset of N has a smallest element.
Universal quantifiers in proofs. Remember that:
· when proving a statement involving x, you're not allowed to impose any extra conditions or restric
tions on x that aren't already in the statement. (But you can break up the proof into parts where you
consider different restrictions on x, as long as they cover all possible cases when taken together.)
· when using a statement involving x in the proof of something else, you get to pick a specific value
for x.
Existential Quantifier. Here are four more sentences that all have the same meaning as each other:
x Z such that x2
 1 = 0.
There is an integer x such that x2
