# 1.5 Quantifiers & Predicate Logic

## Propositions with Quantifiers

Determine which propositions are true when the variables range over:

1. the nonnegative integers.

Exercise 1

Counterexample for 2: Let $$x = 1$$.

Counterexample for 4: Let $$x = 100$$.

2. the integers.

Exercise 2

Counterexample for 2: Let $$x = 1$$.

3. the real numbers.

Exercise 3

Counterexample for 3: Let $$x = 9.5$$.