Question

# The false statement in the following is

A
B
C
(p)p is a tautology
D
p(p) is a tautology

Solution

## The correct option is B $$\left(p \rightarrow q \right) \leftrightarrow \left( q \rightarrow p \right)$$ is a contradiction(A)If $$p$$ is true then $$\sim p$$ is false $$\implies p\wedge \left( \sim p \right)$$ is falseIf $$p$$ is false then $$\sim p$$ is true$$\implies p\wedge \left( \sim p \right)$$ is falseHence, $$p\wedge \left( \sim p \right)$$ is a contradiction(B)If $$p$$ is true and $$q$$ is false then $$p\rightarrow q$$ is false and $$q\rightarrow p$$ is trueSo, $$\left(p \rightarrow q \right) \leftrightarrow \left( q \rightarrow p \right)$$ is falseIf $$p$$ is false and $$q$$ is true then $$p\rightarrow q$$ is true and $$q\rightarrow p$$ is falseSo, $$\left(p \rightarrow q \right) \leftrightarrow \left( q \rightarrow p \right)$$ is falseIf both $$p$$ and $$q$$ are same(i.e. true of false) then $$p\rightarrow q$$ is true.So, $$\left( p \rightarrow q \right) \leftrightarrow \left( q \rightarrow p \right)$$ is trueThus, the truth value of $$\left( p \rightarrow q \right) \leftrightarrow \left( q \rightarrow p \right)$$ is neither tautology nor contradiction.(C)$$\sim \left( \sim p \right) \leftrightarrow p$$ is same as $$p \leftrightarrow p$$ which is always trueThus, $$\sim \left( \sim p \right) \leftrightarrow p$$ is a tautology(D)$$p\vee \left( \sim p \right)$$ is always trueHence, $$p\vee \left( \sim p \right)$$ is a tautologyHence, only B is false.Maths

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