Fundamentals of Mathematical Analysis

Rod Haggarty

Chapter 1 Preliminaries - all with Video Answers

Educators

Section 1

Logic

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Problem 1

Use truth tables to determine which of the following pairs of composite statements are logically equivalent.
(a) $(\operatorname{not}(P$ and $(\operatorname{not} P))) ;(P$ or $(\operatorname{not} P))$
(b) $(P \Rightarrow Q) ;(\operatorname{not}(P$ and $(\operatorname{not} Q)))$
(c) $((P \Rightarrow Q)$ and $R) ;(P \Rightarrow(Q$ and $R))$

James Kiss

Numerade Educator

01:46

Problem 2

What conclusion, if any, can be drawn from
(a) the truth of $((\operatorname{not} P) \Rightarrow P$ )
(b) the truth of $P$ and the truth of $(P \Rightarrow Q)$
(c) the truth of $Q$ and the truth of $(P \Rightarrow Q)$
(d) the truth of $(\operatorname{not} Q)$ and the truth of $(P \Rightarrow Q)$

Hossam Mohamed

Numerade Educator

05:33

Problem 3

A tautology is a statement that is true no matter what the truth values of its constituent statements are. Decide which of the following are tautologies.
(a) $(P$ or $(\operatorname{not} P))$
(b) $(P$ and $(\operatorname{not} P))$
(c) $(P \Rightarrow(\operatorname{not} P))$
(d) $(((P \Rightarrow Q)$ or $(Q \Rightarrow P))$ and $(\operatorname{not} Q))$

Rosina Dapaah

Numerade Educator

04:21

Problem 4

Let $n$ be a positive whole number. Which of the following conditions imply that the $n$ is divisible by $6 ?$
(a) $n$ is divisible by 3
(b) $n$ is divisible by 9
(c) $n$ is divisible by 12
(d) $n^{2}$ is civisible by 12
(e) $n=24$
(f) $n$ is even and divisible by 3
(g) $n=m^{3}-m$ for some positive whole number $m$
Which of $(a)-(g)$ are logically equivalent to the statement ' $n$ is divisible by $6^{\prime} ?$

Julian Wong

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03:25

Problem 5

Let $n$ be a positive whole number. Find three different proofs (as illustrated in Example 3 ) of the fact that $n^{2}$ even $\Rightarrow n$ even.

Julian Wong

Numerade Educator

01:23

Problem 6

Let $m$ and $n$ be positive whole numbers. Prove that $m n^{2} \operatorname{cvcn} \Rightarrow$ at least one of $m$ and $n$ is cven.

Carson Merrill

Numerade Educator