How Do I Do #4? Best and simplest Detail. 298CHAPTEby buying a collection of 5-stamp packagesand 8-stamp packages.4. For each positive integer n, let P(n) be the sentencethat describes the following divisibility property:5"-1 is divisible by 4.a. Write P(0). Is P(0) true?b. Write P(k).c. Write P(k+1).d. In a proof by mathematical induction that this di-visibility property holds for every integer n ≥ 0,what must be shown in the inductive step?5. For each positive integer n, let P(n) be the inequality2" < (n+1)!.a. Write P(2). Is P(2) true?b. Write P(k).c. Write P(k+ 1).d. In a proof by mathematical induction that thisinequality holds for every integer n ≥ 2, whatmust be shown in the inductive step?6. For each positive integer n, let P(n) be the sentenceAny checkerboard with dimensions 2 X 3n canbe completely covered with L-shaped trominoes.a. Write P(1). Is P(1) true?b. Write P(k).c. Write P(k+ 1).d. In a proof by mathematical induction that P(n)is true for each integer n ≥ 1, what must beshown in the inductive step?7. For each positive integer n, let P(n) be the sentenceIn any round-robin tournament involving nteams, the teams can be labeled T₁, T2, T3,..., Tnso that T; beats Ti+1 for every i = 1, 2,..., n-1.a. Write P(2). Is P(2) true?b. Write P(k).c. Write P(k+1).d. In a proof by mathematical induction that P(n)is true for each integer n ≥ 2, what must beshown in the inductive step?Prove each statement in 8-23 by mathematical induction.8. 5"-1 is divisible by 4, for every integer n ≥ 0.9. 7-1 is divisible by 6, for each integer n ≥ 0.10. n³-7n+3 is divisible by 3, for each integer n ≥ 0.11. 32n12. ForH 13. ForwhoH 14. n³15. n(n16. 2"17. 1-18. 5"219. n20. 2221. Vo n22. 1ar23. a.b24. AaS25. Fb26.27.28.SExerlengfor29.

The Complete answer of 4 is given below:

and 4. For each positive integer n, let P(n) be the se that describes the following divisibility property: 5"-1 is divisible by 4. a. Write P(0). Is P(0) true? b. Write P(k). c. Write P(k+1). d. In a proof by mathematical induction that this di- visibility property holds for every integer n ≥ 0, what must be shown in the inductive step? H 1 H1

and 4. For each positive integer n, let P(n) be the se that describes the following divisibility property: 5"-1 is divisible by 4. a. Write P(0). Is P(0) true? b. Write P(k). c. Write P(k+1). d. In a proof by mathematical induction that this di- visibility property holds for every integer n ≥ 0, what must be shown in the inductive step? H 1 H1

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Concept explainers

Polynomial Time

The problems for which the deterministic polynomial-time algorithm exists belong to the complexity class (p) and it is central in the field of computational complexity theory. According to Cobham’s thesis, polynomial time is a synonym for “feasible”, “fast”, and “efficient”.

Question

How Do I Do #4? Best and simplest Detail.

298
CHAPTE
by buying a collection of 5-stamp packages
and 8-stamp packages.
4. For each positive integer n, let P(n) be the sentence
that describes the following divisibility property:
5"-1 is divisible by 4.
a. Write P(0). Is P(0) true?
b. Write P(k).
c. Write P(k+1).
d. In a proof by mathematical induction that this di-
visibility property holds for every integer n ≥ 0,
what must be shown in the inductive step?
5. For each positive integer n, let P(n) be the inequality
2" < (n+1)!.
a. Write P(2). Is P(2) true?
b. Write P(k).
c. Write P(k+ 1).
d. In a proof by mathematical induction that this
inequality holds for every integer n ≥ 2, what
must be shown in the inductive step?
6. For each positive integer n, let P(n) be the sentence
Any checkerboard with dimensions 2 X 3n can
be completely covered with L-shaped trominoes.
a. Write P(1). Is P(1) true?
b. Write P(k).
c. Write P(k+ 1).
d. In a proof by mathematical induction that P(n)
is true for each integer n ≥ 1, what must be
shown in the inductive step?
7. For each positive integer n, let P(n) be the sentence
In any round-robin tournament involving n
teams, the teams can be labeled T₁, T2, T3,..., Tn
so that T; beats Ti+1 for every i = 1, 2,..., n-1.
a. Write P(2). Is P(2) true?
b. Write P(k).
c. Write P(k+1).
d. In a proof by mathematical induction that P(n)
is true for each integer n ≥ 2, what must be
shown in the inductive step?
Prove each statement in 8-23 by mathematical induction.
8. 5"-1 is divisible by 4, for every integer n ≥ 0.
9. 7-1 is divisible by 6, for each integer n ≥ 0.
10. n³-7n+3 is divisible by 3, for each integer n ≥ 0.
11. 32n
12. For
H 13. For
who
H 14. n³
15. n(n
16. 2"
17. 1-
18. 5"
2
19. n
20. 22
21. V
o n
22. 1
ar
23. a.
b
24. A
a
S
25. F
b
26.
27.
28.
S
Exer
leng
for
29.

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