Question

The equivalent statement of $(p\leftrightarrow q)$ is

• A. $(p\wedge q)\vee (p\vee q)$
• B. $(p\to q)\vee (q\to p)$
• C. $(\sim p\wedge \sim q)\vee (p\vee q)$
• D. $(\sim p\vee q)\wedge (q\vee \sim p)$

Answer 1

Answer: (D)

$(\sim p\vee q)\wedge (q\vee \sim p)$

$(p\leftrightarrow q)$ is equivallent to $(\sim p\vee q)\wedge (q\vee \sim p)$we can show it by truth table but before that let use define boolean expression which are used here

1.$p\wedge q$=min(p,q)

2.$p\vee q$=max(p,q)

3.$(p\leftrightarrow q)$=1-|p-q|

by using truth table

p	q	$(p\leftrightarrow q)$
1	1	0
1	0	0

this is equivalent to $(\sim p\vee q)\wedge (q\vee \sim p)$

we show to by truth table.

p	q	∼p	∼pVq	qV∼p	$(\sim p\vee q)\wedge (q\vee \sim p)$
1	1	0	1	1	1
1	0	0	0	0	0

both truth tables have same values hence

$(p\leftrightarrow q)$ is equivallent to $(\sim p\vee q)\wedge (q\vee \sim p)$