Question 28 of 103MediumMCQ
Which one of the following statements is not a
tautology?
Solutions · 1
Official solution
Here you have to check all the options.
We know, p \[ \to \] q = \[ \sim \] p \[ \vee \] q
(A) (p \[ \wedge \] q) \[ \to \] (~ p) \[ \vee \] q
= \[ \sim \] (p \[ \wedge \] q) \[ \vee \] ((~ p) \[ \vee \] q)
= (\[ \sim \] p \[ \vee \] \[ \sim \] q) \[ \vee \] ((~ p) \[ \vee \] q)
= \[ \sim \] p \[ \vee \] (q \[ \vee \] \[ \sim \] q)
= \[ \sim \] p \[ \vee \] t
[(q \[ \vee \] \[ \sim \] q) = t (tautology)]
= t
(B) (p \[ \wedge \] q) \[ \to \] p
= \[ \sim \] (p \[ \wedge \] q) \[ \vee \] p
= \[ \sim \] p \[ \vee \] \[ \sim \] q \[ \vee \] p
= t \[ \vee \] \[ \sim \] q
= t
(C) ( p \[ \vee \] q) \[ \to \] ( p \[ \vee \] (~q))
= \[ \sim \] ( p \[ \vee \] q) \[ \vee \] ( p \[ \vee \] (~q))
= (\[ \sim \] p \[ \wedge \] \[ \sim \] q) \[ \vee \] ( p \[ \vee \] (~q))
= \[ \sim \] q \[ \wedge \] (p \[ \vee \] \[ \sim \]p)
= \[ \sim \] q \[ \wedge \] t
= \[ \sim \] q (not tautology)
(D) p \[ \to \] ( p \[ \vee \] q)
= \[ \sim \] p \[ \vee \] ( p \[ \vee \] q)
= t \[ \vee \] q
= t
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