**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.

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**Test Yourself**

Page 204

1. The meaning of every variable used in a proof should be explained with
   ______.

The body of the proof.

2. Proofs should be written in sentences that are ______ and ______.

complete; grammatically correct

3. Every assertion in a proof should be supported by a ______.

reason

4. The following are some useful "little words and phrases" that clarify the
   reasoning in a proof:

   ______, ______, ______, ______, and ______.

because; since; then; thus; so; hence; therefore; consequently; it follows that;
by substitution

5. A new thought or fact that does not follow as an immediate consequence of the
   preceding statement can be introduced by writing ______, ______, ______,
   ______, or ______.

observe that; note that; recall that; but; now

6. To introduce a new variable that is defined in terms of previous variables,
   use the word ______.

let

7. Displaying equations and inequalities increases the ______ of a proof.

readability

8. Some proof-writing mistakes are ______, ______, ______, ______, ______,
   ______, and ______.

arguing from examples; using the same letter to mean two different things;
jumping to a conclusion; assuming what is to be proved; confusion between what
is known and what is still to be shown; use of _any_ when the correct word is
_some_; misuse of the word _if_

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**Test Yourself**

Page 210

1. To show that a real number is rational, we must show that we can write it as
   ______.

The ratio of integers, where the denominator is not 0.

2. An irrational number is a ______ that is ______.

real number; not rational

3. Zero is a rational number because ______.

zero is an integer that is a ratio of integers where the denominator is not
zero, $0 = \dfrac{0}{1}$.