Question

The boolean expression $\left(\right(p\wedge q)\vee (p\vee \sim q\left)\right)\wedge (\sim p\wedge \sim \text{q})$ is equivalent to:

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

Answer 1

The correct option is C $(\sim p)\wedge (\sim q)$

Solving it using venn diagram analogy i.e., '$\wedge$' denotes intersection and ${}^{\prime }{\vee }^{\prime }$ denotes union.
So, considering the venn diagram for two sets $p$ and $q$ as represnted below

$\therefore p\wedge q=\left\{b\right\};\sim q=\left\{a,d\right\},\sim p=\{c,d\}$
$\text{p}\vee \sim \text{q}=\{a,b\}\vee \{a,d\}=\{a,b,d\}$
$\sim \text{p}\wedge \text{q}=\{c,d\}\wedge \{b,c\}=\left\{c\right\}$
$\therefore \left(\right(\text{p}\wedge \text{q})\vee (\text{p}\vee \text{q}\left)\right)=\left\{b\right\}\vee \left\{\text{a,b,d}\right\}=\left\{\text{a,b,d}\right\}$

The Boolean expression $\left(\right(\text{p}\wedge \text{q})\wedge (\text{p}\vee \sim \text{q}\left)\right)\wedge (\sim \text{p}\sim \text{q})$ is -
$=\left\{\text{a,b,d}\right\}\wedge \left\{\text{d}\right\}$
$=\left\{\text{d}\right\}$
$=\sim \text{p}\wedge \sim \text{q}(\because \sim \text{p}\wedge \sim \text{q}=\{d\left\}\right)$