Question

Which of the following Boolean expressions is not a tautology:

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

Answer 1

The correct option is A $(\sim p\Rightarrow q)\vee (\sim q\Rightarrow p)$

$1.(\sim p\to q)\vee (\sim q\to p)\equiv (p\vee q)\vee (q\vee p)\equiv p\vee q\equiv ̸T$

$2.(\sim p\vee q)\vee (q\vee p)\equiv \sim (p\wedge \sim q)\vee (p\vee q)\equiv T$

$3.(\sim q\vee p)\vee (q\vee p)\equiv \sim (\sim p\wedge q)\vee (p\vee q)\equiv T$

$4.(\sim p\vee \sim q)\vee (q\vee p)\equiv \sim (p\wedge q)\vee (p\vee q)\equiv T$