Kenneth Rosen Edition 7 Exercise 1.3 Question 18 (Page No. 35)

Question

show that $p \rightarrow q$ and $\sim q \rightarrow \sim p$ are logically equivalent.

Answer 1

${\color{Green}{p\rightarrow q\equiv\sim p\vee q} }$  ----------$>(1)$

 $\sim q\rightarrow \sim p\equiv\sim(\sim q)\vee \sim p\equiv q\vee \sim p\equiv\sim p\vee q$     ----------$>(2)$

From equation $(1)$ and $(2)$ 

we can write    ${\color{Magenta}{p\rightarrow q\equiv\sim q\rightarrow \sim p} }$​​​

Actually, implication$(p\rightarrow q)$ and contrapositive$(\sim q\rightarrow \sim p)$ are equivalent.

We can show that the above relation using the truth table also.

Answer 2

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