Question

Determine whether the following statement pattern is a tautology or a contradiction or contingency:
$(p\wedge q)\vee (\sim p\wedge q)\vee (p\vee \sim q)\vee (\sim p\wedge \sim q)$

Answer 1

$p$	$q$	$\sim p$	$\sim q$	$p\wedge q$	$\sim p\wedge q$	$p\vee \sim q$	$\sim p\wedge \sim q$	$\left(I\right)\vee \left(II\right)\vee \left(III\right)\vee \left(IV\right)$
				$\left(I\right)$	$\left(II\right)$	$\left(III\right)$	$\left(IV\right)$
T	T	F	F	T	F	T	F	T
T	F	F	T	F	F	T	F	T
F	T	T	F	F	T	F	F	T
F	F	T	T	F	F	T	T	T

All the entries in the last column of the above truth table are T.
$\therefore (p\wedge q)\vee (\sim p\wedge q)\vee (p\vee \sim q)\vee (\sim p\wedge \sim q)$ is a tautology.