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Question

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


Solution

$$p\ \ \ $$$$q\ \ \ $$$$\sim
p\ \ \ $$
$$\sim
q\ \ \ $$
$$p
\wedge
q\ \ \ $$
$$\sim 
p \wedge
q \ \ \ $$
$$p
\vee 
\sim q
\ \ \ $$
$$ \sim p
\wedge 
\sim q\ \ \ \ $$
$$(I) \vee
(II) \vee 
(III) \vee
(IV) $$




$$(I)$$$$(II)$$$$(III)$$$$(IV)$$
TTFFTFTFT
TFFTFFTFT
FTTFFTFFT
FFTTFFTTT

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.

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