Kenneth Rosen Edition 7 Exercise 1.3 Question 24 (Page No. 35)

Question

Show that $(p \rightarrow q) \vee (p \rightarrow r)$ and $p \rightarrow (q \vee r)$ are logically equivalent.

Answer 1

$(p\rightarrow q)\vee(p\rightarrow r)\equiv(\sim p\vee q)\vee(\sim p\vee r)$

Now we can change propositional operator into boolean operator for easy calculation

$\wedge\equiv\cdot,\vee\equiv +$

$(p\rightarrow q)\vee(p\rightarrow r)\equiv(\bar {p}+ q)+(\bar{p}+ r)$

$(p\rightarrow q)\vee(p\rightarrow r)\equiv(\bar {p}+ q)+(\bar{p}+ r)$

$(p\rightarrow q)\vee(p\rightarrow r)\equiv\bar {p}+ (q+ r)$

we can write like this

$(p\rightarrow q)\vee(p\rightarrow r)\equiv\sim {p}\vee (q\vee r)$

Hence

${\color{Magenta}{(p\rightarrow q)\vee(p\rightarrow r)\equiv p\rightarrow(q\vee r)} }$

Answer 2

..........

Answer 3