129 lines
3.6 KiB
Markdown
129 lines
3.6 KiB
Markdown
**Test Yourself**
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Page 296
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1. The notation $\sum_{k = m}^{n}{a_k}$ is read "_____."
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The summation from $k$ equals $m$ to $n$ of $a$ sub $k$.
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2. The expanded form of $\sum_{k = m}^{n}{a_k}$ is _____.
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$$ a_m + a_{m + 1} + a_{m + 2} + \dots + a_n $$
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3. The value of $a_1 + a_2 + a_3 + \dots + a_n$ when $n = 2$ is "_____."
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$$ a_1 + a_2 $$
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4. The notation $\prod_{k = m}^{n}{a_k}$ is read "_____."
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The product from $k$ equals $m$ to $n$ of $a$ sub $k$.
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5. If $n$ is a positive integer, then $n! =$ _____.
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$$ n \cdot (n - 1) \dots \cdot 3 \cdot 2 \cdot 1 $$
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6. $\sum_{k = m}^{n}{a_k} + c\sum_{k = m}^{n}{b_k} =$ _____.
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$$ \sum_{k = m}^{n}{a_k + cb_k} $$
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7. $\left(\prod_{k = m}^{n}{a_k}\right)\left(\prod_{k = m}^{n}{b_k}\right) =$
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_____.
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$$ \prod_{k = m}^{n}{a_kb_k} $$
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---
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**Test Yourself**
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Page 309
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1. Mathematical induction is a method for proving that a property defined for
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integers $n$ is true for all values of $n$ that are _____.
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greater than or equal to some initial value.
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2. Let $P(n)$ be a property defined for integers $n$ and consider constructing a
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proof by mathematical induction for the statement "P(n) is true for all
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$n \geq a$."
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a. In the basis step one must show _____.
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that $P(a)$ is true.
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b. In the inductive step one supposes that _____ for a particular but
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arbitrarily chosen value of an integer $k \geq a$. This supposition is called
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the _____. One then has to show that _____.
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$P(k)$ is true; inductive hypothesis; $P(k + 1)$ is true.
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---
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**Test Yourself**
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Page 320
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1. Mathematical induction differs from the kind of induction used in the natural
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sciences because it is actually a form of _____ reasoning.
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deductive
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2. Mathematical induction can be used to _____ conjectures that have been made
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using inductive reasoning.
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prove
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---
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**Test Yourself**
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Page 333
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1. In a proof by strong mathematical induction the basis step may require
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checking a property $P(n)$ for more _____ value of $n$.
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than one
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2. Suppose that in the basis step for a proof by strong mathematical induction
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the property $P(n)$ was checked for every integer $n$ from $a$ through $b$.
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Then in the inductive step one assumes that for any integer $k \geq b$, the
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property $P(n)$ is true for all values of $i$ from _____ through _____ and
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one shows that _____ is true.
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$a$; $k$; $P(k + 1)$
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3. According to the well-ordering principle for the integers, if a set $S$ of
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integers contains at least _____ and if there is some integer that is less
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than or equal to every _____, then _____.
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one integer; integer in $S$; $S$ contains a least element.
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---
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**Test Yourself**
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Page 346
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1. A pre-condition for an algorithm is _____ and a post-condition for an
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algorithm is _____.
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2. A loop is defined as correct with respect to its pre- and post-conditions if,
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and only if, whenever the algorithm variables satisfy the pre-condition for
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the loop and the loop terminates after a finite number of steps, then _____.
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3. For each iteration of a loop, if a loop invariant is true before iteration of
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the loop, then _____.
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4. Given a **while** loop with guard $G$ and a predicate $I(n)$ if the following
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four properties are true, then the loop is correct with respect to its pre-
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and post-conditions:
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(a) The pre-condition for the loop implies that _____ before the first iteration
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of the loop.
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(b) For every integer $k \geq 0$, if the guard $G$ and the predicate $I(k)$ are
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both true before an iteration of the loop, then _____.
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\(c\) After a finite number of iterations of the loop, _____.
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(d) If $N$ is the least number of iterations after which $G$ is false and $I(N)$
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is true, then the values of the algorithm variables will be as specified _____.
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