367 lines
11 KiB
Markdown
367 lines
11 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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---
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**Test Yourself**
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Page 232
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1. The quotient-remainder theorem says that for all integers $n$ and $d$ with
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$d \geq 0$, there exists ______ $q$ and $r$ such that ______ and ______.
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integers; $n = dq + r$; $0 \leq r < d$
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2. If $n$ and $d$ are integers with $d > 0$, $n\ div\ d$ is ______ and
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$n \mod d$ is ______.
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the quotient obtained when $n$ is divided by $d$; the nonnegative remainder
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obtained when $n$ is divided by $d$
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3. The parity of an integer indicates whether the integer is ______.
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even or odd
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4. According to the quotient-remainder theorem, if an integer $n$ is divided by
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a positive integer $d$, the possible remainders are ______. This implies that
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$n$ can be written in one of the forms ______ for some integer $q$.
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$0, 1, 2, \dots d - 1$; $dq + 1, dq + 2, \dots dq + (d - 1)$
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5. To prove a statement of the form "If $A_1$ or $A_2$ or $A_3$, then $C$,"
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prove ______ and ______ and ______.
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If $A_1$ then $C$; If $A_2$ then $C$; If $A_3$ then $C$
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6. The triangle inequality says that for all real numbers $x$ and $y$, ______.
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$|x + 6| \leq |x| + |y|$
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---
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**Test Yourself**
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Page 239
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1. Given any real number $x$, the floor of $x$ is the unique integer $n$ such
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that ______.
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$n \leq x < n + 1$
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2. Given any real number $x$, the ceiling of $x$ is the unique integer $n$ such
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that ______.
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$n - 1 < x \leq n$
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---
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**Test Yourself**
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Page 248
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1. To prove a statement by contradiction, you suppose that ______ and you show
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that ______.
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the statement is false; this supposition leads to a contradiction
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2. A proof by contraposition of a statement of the form
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"$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$" is a direct proof of
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______.
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$\forall x \in D, \text{ if } \neg Q(x) \text{ then } \neg P(x)$
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3. To prove a statement of the form
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"$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$" by contraposition,
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you suppose that ______ and you show that ______.
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$Q(x)$ is false; $P(x)$ is false.
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---
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**Test Yourself**
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Page 256
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1. The ancient Greeks discovered that in a right triangle where both legs have
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length $1$, the ratio of the length of the hypotenuse to the length of one of
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the legs is not equal to a ratio of ______.
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two integers.
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2. One way to prove that $\sqrt{2}$ is an irrational number is to assume that
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$\sqrt{2} = \dfrac{m}{n}$ for some integers $m$ and $n$ that have no common
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factor greater than $1$, use the lemma that says that if the square of an
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integer is even then ______, and eventually show that $m$ and $n$ ______.
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that integer is even; have a common factor greater than 1.
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3. One way to prove that there are infinitely many prime numbers is to assume
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that there is a largest prime number $p$, construct the number ______, and
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then show that this number has to be divisible by a prime number that is
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greater than ______.
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$N = (2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot \dots \cdot p) + 1$; $p$
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---
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**Test Yourself**
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Page 265
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1. The total degree of a graph is defined as ______.
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The sum of the degrees of all the vertices of the graph
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2. The handshake theorem says that the total degree of a graph is ______.
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equal to the number of edges of the graph
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3. In any graph the number of vertices of odd degree is ______.
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an even number
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4. A simple graph is ______.
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a graph with no loops or parallel edges
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5. A complete graph on $n$ vertices is a ______.
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a simple graph with $n$ vertices whose set of edges contains exactly one edge
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for each pair of vertices
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6. A complete bipartite graph on $(m, n)$ vertices is a simple graph whose
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vertices can be divided into two distinct, non-overlapping sets, say $V$ with
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$m$ vertices and $W$ with $m$ vertices, in such a way that (1) there is
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______ from each vertex of $V$ to each vertex of $W$, (2) there is ______
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from any one vertex of $V$ to any other of $V$, and (3) there is ______ from
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any one vertex of $W$ to any other vertex of $W$.
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one edge; no edge; no edge
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---
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**Test Yourself**
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Page 277
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1. When an algorithm statement of the form $x := e$ is executed, ______.
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The expression $e$ is evaluated (using the current values of all the variables
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in the expression), and this value is placed in the memory location
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corresponding to $x$ (replacing any previous contents of the location)
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2. Consider an algorithm statement of the following form.
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$\text{\textbf{if }(condition)}\\ \text{\textbf{then }} s_1\\ \text{\textbf{else }} s_2$
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then the algorithm at $s_1$ is executed; then the algorithm at $s_2$ is executed
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When such a statement is executed, the truth or falsity of the _condition_ is
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evaluated. If _condition_ is true, ______. If _condition_ is false, ______.
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3. Consider an algorithm statement of the following form.
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$\text{\textbf{while }(condition)}$
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_[statements that make up the body of the loop]_
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$\text{\textbf{end while}}$
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When such a statement is executed, the truth or falsity of the _condition_ is
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evaluated. If _condition_ is true, ______. If _condition_ is false, ______.
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the statements that make up the body of the loop are executed in order and then
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execution moves back to the beginning of the loop and the process repeats; the
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loop ends and execution passes to the next algorithm statement following the
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loop
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4. Consider an algorithm statement of the following form.
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the statements that make up the body of the loop are executed in order, the
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variable's value is assigned to the next (iterated), and then execution moves
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back to the beginning of the loop and the process repeats; the loop ends and
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execution passes to the next algorithm statement following the loop
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$\text{\textbf{for } variable } := \text{initial expression \textbf{to} final expression.}$
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_[statements that make up the body of the loop]_
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$\text{\textbf{next } (same) variable}$
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When such a statement is executed, _variable_ is set equal to the value of the
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_initial expression_, and a check is made to determine whether the value of
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_variable_ is less than or equal to the value of _final expression_. If so,
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______. If not, ______.
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5. Given a nonnegative integer $a$ and a positive integer $d$ the division
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algorithm computes ______.
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Integers $q$ and $r$ with the property that $n = dq + r$ and $0 \leq r < d$.
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6. Given integers $a$ and $b$, not both zero, $\text{gcd}(a, b)$ is the integer
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$d$ that satisfies the following two conditions: ______ and ______.
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$d \mid a$ and $d \mid b$; if $c$ is a common divisor of both $a$ and $b$, then
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$c \leq d$.
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7. If $r$ is a positive integer, then $gcd(r, 0) =$ ______.
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$r$
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8. If $a$ and $b$ are integers not both zero and if $q$ and $r$ are nonnegative
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integers such that $a = bq + r$ then $\text{gcd}(a, b) =$ ______.
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$gcd(b, r)$
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9. Given positive integers $A$ and $B$ with $A > B$, the Euclidean algorithm
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computes ______.
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the greatest common divisor of $A$ and $B$, $\text{gcd}(A, B)$.
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