Question

•Determine whether (p ⇔ q) and (¬p ∨ q) ∧ (¬q ∨ p) are logically equivalent.

•Determine whether p ∧ (p ⇔ q) ∧ ¬q is a tautology, contradiction, neither.

Answer 1

1) $(p\Leftrightarrow q)$ and $(\neg p\lor q)\land (\neg q\lor p)$ are logically equivalent.

$p\Leftrightarrow q$

$(p\rightarrow q)\land (q\rightarrow p)$

$(\neg p\lor q)\land (\neg q\lor p)$

2) $p\land (p\Leftrightarrow q)\land \neg q$ is a contradiction.

$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} p & q & p\Leftrightarrow q&\neg q&p\land (p\Leftrightarrow q)\land \neg q \\ \hline T & T & T&F&F \\ \hdashline T & F & F&T&F \\ \hdashline F&T&F&F&F \\ \hdashline F&F&T&T&F \end{array}$