# Which of the following Boolean expressions

Question:

Which of the following Boolean expressions is not a tautology?

1. $(\mathrm{p} \Rightarrow \mathrm{q}) \vee(\sim \mathrm{q} \Rightarrow \mathrm{p})$

2. $(\mathrm{q} \Rightarrow \mathrm{p}) \vee(\sim \mathrm{q} \Rightarrow \mathrm{p})$

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

4. $(\sim \mathrm{p} \Rightarrow \mathrm{q}) \vee(\sim \mathrm{q} \Rightarrow \mathrm{p})$

Correct Option: , 4

Solution:

(1) $(\mathrm{p} \rightarrow \mathrm{q}) \vee(\sim \mathrm{q} \rightarrow \mathrm{p})$

$=(\sim \mathrm{p} \vee \mathrm{q}) \vee(\mathrm{q} \vee \mathrm{p})$

$=(\sim \mathrm{p} \vee \mathrm{p}) \vee \mathrm{q}$

$=\mathrm{t} \vee \mathrm{q}=\mathrm{t}$

$(2)(\mathrm{q} \rightarrow \mathrm{p}) \vee(\sim \mathrm{q} \rightarrow \mathrm{p})$

$=(\sim \mathrm{q} \vee \mathrm{p}) \vee(\mathrm{q} \vee \mathrm{p})$

$=(\sim \mathrm{q} \vee \mathrm{q}) \vee \mathrm{p}$

$=\mathrm{t} \vee \mathrm{p}=\mathrm{t}$

(3) $(\mathrm{p} \rightarrow \sim \mathrm{q}) \vee(\sim \mathrm{q} \rightarrow \mathrm{p})$

$=(\sim \mathrm{p} \vee \sim \mathrm{q}) \vee(\mathrm{q} \vee \mathrm{p})$

$=(\sim \mathrm{p} \vee \mathrm{p}) \vee(\sim \mathrm{q} \vee \mathrm{q})$

$=\mathrm{t} \vee \mathrm{t}=\mathrm{t}$

(4) $(\sim \mathrm{q} \rightarrow \mathrm{q}) \vee(\sim \mathrm{q} \rightarrow \mathrm{p})$

$=(p \vee q) \vee(q \vee p)$

$=(p \vee p) \vee(q \vee p)$

$=\mathrm{p} \vee \mathrm{q}$

Which is not a tautology.