# The Boolean expression

Question:

The Boolean expression $(p \wedge q) \Rightarrow((r \wedge q) \wedge p)$ is equivalent to :

1. $(\mathrm{p} \wedge \mathrm{q}) \Rightarrow(\mathrm{r} \wedge \mathrm{q})$

2. $(\mathrm{q} \wedge \mathrm{r}) \Rightarrow(\mathrm{p} \wedge \mathrm{q})$

3. $(\mathrm{p} \wedge \mathrm{q}) \Rightarrow(\mathrm{r} \vee \mathrm{q})$

4. $(\mathrm{p} \wedge \mathrm{r}) \Rightarrow(\mathrm{p} \wedge \mathrm{q})$

Correct Option: 1

Solution:

$(p \wedge q) \Rightarrow((r \wedge q) \wedge p)$

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

$\sim(\mathrm{p} \wedge \mathrm{q}) \vee((\mathrm{r} \wedge \mathrm{p}) \wedge(\mathrm{p} \wedge \mathrm{q})$

$\Rightarrow[\sim(p \wedge q) \vee(p \wedge q)] \wedge(\sim(p \wedge q) \vee(r \wedge p))$

$\Rightarrow t \wedge[\sim(p \wedge q) \vee(r \wedge p)]$

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

$\Rightarrow(p \wedge q) \Rightarrow(r \wedge p)$

Aliter :

given statement says

" if $\mathrm{p}$ and $\mathrm{q}$ both happen then $\mathrm{p}$ and $\mathrm{q}$ and $\mathrm{r}$ will happen"

it Simply implies

"If $\mathrm{p}$ and $\mathrm{q}$ both happen then 'r' too will happen "

i.e.

" if $\mathrm{p}$ and $\mathrm{q}$ both happen then $\mathrm{r}$ and $\mathrm{p}$ too will happen

i.e.

$(p \wedge q) \Rightarrow(r \wedge p)$