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.

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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}.

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Test Yourself

Page 220

1. To show that a nonzero integer d divides an integer n, we must show that ______.

2. To say that d divides n means the same as saying that ______ is divisible by ______.

3. If a and b are positive integers and a \mid b, then ______ is less than or equal to ______.

4. For all integers n and d, d \nmid n if, and only if, ______.

5. If a and b are integers, the notation a \mid b denotes ______ and the notation a/b denotes ______.

6. The transitivity of divisibility theorem says that for all integers a, b, and c, if ______ then ______.

7. The divisibility by a prime theorem says that every integer greater than 1 is ______.

8. The unique factorization of integers theorem says that any integer greater than 1 is either ______ or can be written as ______ in a way that is unique except possibly for the ______ in which the numbers are written.