Q. Which one of the following is a tautology ?

JEE MainJEE Main 2020Mathematical Reasoning Report Error

$(p ∧ (P \to Q)) \to Q$

$P ∧ (P ∨ Q)$

$Q \to (P ∧ (P \to Q))$

$P ∨ (P ∧ Q)$

Solution:

$\left(1\right)\,P\,\wedge \left(P\,\vee\,Q\right)=P$
$\left(2\right)\,P\,\vee \left(P\,\wedge\,Q\right)\equiv P$
$\left(3\right)\,Q \rightarrow \left(P\,\wedge\left(P \rightarrow Q\right)\right)$
$\equiv Q \rightarrow \left(P\,\wedge\left(\sim P\,\vee\,Q\right)\right)\equiv Q \rightarrow \left(P\,\wedge\,Q\right)$
$\equiv\left(\sim Q\right)\vee\left(P\,\wedge\,Q\right)\equiv\left(P\,\vee\left(\sim Q\right)\right)$
$\left(4\right)\,\left(P\,\wedge \left(P \rightarrow Q\right)\right) \rightarrow Q$
$\equiv\left(P\,\wedge \left(\sim P\,\vee\,Q\right)\right) \rightarrow Q\equiv \left(P\,\wedge\,Q \right) \rightarrow Q$
$\equiv\left(\left(\sim P\right)\vee\left(\sim Q\right)\right)\vee\,Q\equiv\left(\sim P\right)\vee\,t\equiv t$

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$R:$ There exist x$_1$,x$_2$ $\in$ X such that x$_1$ $\ne$ x$_2$ and $f$ (x$_1$) = $f$ (x$_2$).
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