discrete_mathematics_with_applications/chapter_5/test_yourself.md

Test Yourself

Page 296

1. The notation \sum_{k = m}^{n}{a_k} is read "_____."

The summation from k equals m to n of a sub k.

2. The expanded form of \sum_{k = m}^{n}{a_k} is _____.

 a_m + a_{m + 1} + a_{m + 2} + \dots + a_n 

3. The value of a_1 + a_2 + a_3 + \dots + a_n when n = 2 is "_____."

 a_1 + a_2 

4. The notation \prod_{k = m}^{n}{a_k} is read "_____."

The product from k equals m to n of a sub k.

5. If n is a positive integer, then n! = _____.

 n \cdot (n - 1) \dots \cdot 3 \cdot 2 \cdot 1 

6. \sum_{k = m}^{n}{a_k} + c\sum_{k = m}^{n}{b_k} = _____.

 \sum_{k = m}^{n}{a_k + cb_k} 

7. \left(\prod_{k = m}^{n}{a_k}\right)\left(\prod_{k = m}^{n}{b_k}\right) = _____.

 \prod_{k = m}^{n}{a_kb_k} 

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Test Yourself

Page 309

1. Mathematical induction is a method for proving that a property defined for integers n is true for all values of n that are _____.

greater than or equal to some initial value.

2. Let P(n) be a property defined for integers n and consider constructing a proof by mathematical induction for the statement "P(n) is true for all n \geq a."

a. In the basis step one must show _____.

that P(a) is true.

b. In the inductive step one supposes that _____ for a particular but arbitrarily chosen value of an integer k \geq a. This supposition is called the _____. One then has to show that _____.

P(k) is true; inductive hypothesis; P(k + 1) is true.

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Test Yourself

Page 320

1. Mathematical induction differs from the kind of induction used in the natural sciences because it is actually a form of _____ reasoning.

deductive

2. Mathematical induction can be used to _____ conjectures that have been made using inductive reasoning.

prove

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Test Yourself

Page 333

1. In a proof by strong mathematical induction the basis step may require checking a property P(n) for more _____ value of n.

than one

2. Suppose that in the basis step for a proof by strong mathematical induction the property P(n) was checked for every integer n from a through b. Then in the inductive step one assumes that for any integer k \geq b, the property P(n) is true for all values of i from _____ through _____ and one shows that _____ is true.

a; k; P(k + 1)

3. According to the well-ordering principle for the integers, if a set S of integers contains at least _____ and if there is some integer that is less than or equal to every _____, then _____.

one integer; integer in S; S contains a least element.

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Test Yourself

Page 346

1. A pre-condition for an algorithm is _____ and a post-condition for an algorithm is _____.

2. A loop is defined as correct with respect to its pre- and post-conditions if, and only if, whenever the algorithm variables satisfy the pre-condition for the loop and the loop terminates after a finite number of steps, then _____.

3. For each iteration of a loop, if a loop invariant is true before iteration of the loop, then _____.

4. Given a while loop with guard G and a predicate I(n) if the following four properties are true, then the loop is correct with respect to its pre- and post-conditions:

(a) The pre-condition for the loop implies that _____ before the first iteration of the loop.

(b) For every integer k \geq 0, if the guard G and the predicate I(k) are both true before an iteration of the loop, then _____.

c After a finite number of iterations of the loop, _____.

(d) If N is the least number of iterations after which G is false and I(N) is true, then the values of the algorithm variables will be as specified _____.