Question

Which of the following pairs is NOT logically equivalent?

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

Answer 1

The correct option is B $p\vee (p\wedge q)$ and $q$

$\sim (\sim p)\equiv p$

By De Morgan's law,
$\sim (p\wedge q)\equiv (\sim p)\vee (\sim q)$

By distributive law,
$p\vee (p\wedge q)$
$\equiv (p\vee p)\wedge (p\vee q)$
$\equiv p\wedge (p\vee q)$

If truth value of $p$ and $q$ are $F$ and $T$ respectively, then truth value of $p\vee (p\wedge q)$ is $F.$
So, $p\vee (p\wedge q)$ and $q$ are not logically equivalent.