150 lines
4 KiB
Markdown
150 lines
4 KiB
Markdown
**Test Yourself**
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Page 194
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1. An integer is even if, and only if, ______.
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it equals twice some integer.
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2. An integer is odd if, and only if, ______.
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it equals twice some integer plus 1.
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3. An integer $n$ is prime if, and only if, ______.
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$n$ is greater than $1$ and if $n$ equals the product of any two positive
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integers, then one of the integers equals $1$ and the other equals $n$.
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4. The most common way to disprove a universal statement is to find ______.
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a counterexample.
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5. According to the method of generalizing from the generic particular, to show
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that every element of a set satisfies a certain property, suppose $x$ is a
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______, and show that ______.
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particular but arbitrarily chosen element of the set; $x$ satisfies the given
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property.
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6. To use the method of direct proof to prove a statement of the form, "For
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every $x$ in a set $D$, if $P(x)$ then $Q(x)$," one supposes that ______ and
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one shows that ______.
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$x$ is a particular but arbitrarily chosen element of the set $D$ that makes the
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hypothesis $P(x)$ true; $x$ makes the conclusion $Q(x)$ true.
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---
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**Test Yourself**
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Page 204
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1. The meaning of every variable used in a proof should be explained with
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______.
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The body of the proof.
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2. Proofs should be written in sentences that are ______ and ______.
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complete; grammatically correct
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3. Every assertion in a proof should be supported by a ______.
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reason
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4. The following are some useful "little words and phrases" that clarify the
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reasoning in a proof:
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______, ______, ______, ______, and ______.
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because; since; then; thus; so; hence; therefore; consequently; it follows that;
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by substitution
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5. A new thought or fact that does not follow as an immediate consequence of the
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preceding statement can be introduced by writing ______, ______, ______,
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______, or ______.
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observe that; note that; recall that; but; now
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6. To introduce a new variable that is defined in terms of previous variables,
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use the word ______.
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let
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7. Displaying equations and inequalities increases the ______ of a proof.
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readability
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8. Some proof-writing mistakes are ______, ______, ______, ______, ______,
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______, and ______.
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arguing from examples; using the same letter to mean two different things;
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jumping to a conclusion; assuming what is to be proved; confusion between what
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is known and what is still to be shown; use of _any_ when the correct word is
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_some_; misuse of the word _if_
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---
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**Test Yourself**
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Page 210
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1. To show that a real number is rational, we must show that we can write it as
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______.
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The ratio of integers, where the denominator is not 0.
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2. An irrational number is a ______ that is ______.
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real number; not rational
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3. Zero is a rational number because ______.
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zero is an integer that is a ratio of integers where the denominator is not
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zero, $0 = \dfrac{0}{1}$.
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---
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**Test Yourself**
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Page 220
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1. To show that a nonzero integer $d$ divides an integer $n$, we must show that
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______.
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$n$ equals $d$ divided by some integer and $d \neq 0$.
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2. To say that $d$ divides $n$ means the same as saying that ______ is divisible
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by ______.
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$n$; $d$
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3. If $a$ and $b$ are positive integers and $a \mid b$, then ______ is less than
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or equal to ______.
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$a$; $b$
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4. For all integers $n$ and $d$, $d \nmid n$ if, and only if, ______.
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$\dfrac{n}{d}$ is not an integer.
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5. If $a$ and $b$ are integers, the notation $a \mid b$ denotes ______ and the
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notation $a/b$ denotes ______.
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the sentence "$a$ divides $b$"; the number obtained when $a$ is divided by $b$
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6. The transitivity of divisibility theorem says that for all integers $a$, $b$,
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and $c$, if ______ then ______.
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$a \mid b$ and $b \mid c$; $a \mid c$
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7. The divisibility by a prime theorem says that every integer greater than $1$
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is ______.
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divisible by some prime number.
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8. The unique factorization of integers theorem says that any integer greater
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than $1$ is either ______ or can be written as ______ in a way that is unique
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except possibly for the ______ in which the numbers are written.
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prime; a product of prime numbers; order
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