discrete_mathematics_with_applications/chapter_4/test_yourself.md

Test Yourself

Page 194

1. An integer is even if, and only if, ______.

it equals twice some integer.

2. An integer is odd if, and only if, ______.

it equals twice some integer plus 1.

3. An integer n is prime if, and only if, ______.

n is greater than 1 and if n equals the product of any two positive integers, then one of the integers equals 1 and the other equals n.

4. The most common way to disprove a universal statement is to find ______.

a counterexample.

5. According to the method of generalizing from the generic particular, to show that every element of a set satisfies a certain property, suppose x is a ______, and show that ______.

particular but arbitrarily chosen element of the set; x satisfies the given property.

6. To use the method of direct proof to prove a statement of the form, "For every x in a set D, if P(x) then Q(x)," one supposes that ______ and one shows that ______.

x is a particular but arbitrarily chosen element of the set D that makes the hypothesis P(x) true; x makes the conclusion Q(x) true.