Question

The only statement among the following which is a tautology, is

• A. $B\to \left[A\wedge \left(A\to B\right)\right]$
• B. $A\wedge \left(A\vee B\right)$
• C. $A\vee \left(A\wedge B\right)$
• D. $\left[A\wedge \left(A\to B\right)\to B\right]$

Answer 1

The correct option is D

$\left[A\wedge \left(A\to B\right)\to B\right]$

Explanation for the correct option

Option (D):$\left[A\wedge \left(A\to B\right)\to B\right]$

$A$	$B$	$A\to B$	$A\wedge \left(A\to B\right)$	$\left[A\wedge \left(A\to B\right)\to B\right]$
$T$	$T$	$T$	$T$	$T$
$T$	$F$	$F$	$F$	$T$
$F$	$T$	$T$	$F$	$T$
$F$	$F$	$T$	$F$	$T$

Explanation for the incorrect option

Option (A): $B\to \left[A\wedge \left(A\to B\right)\right]$

$B$	$A$	$A\to B$	$A\wedge \left(A\to B\right)$	$B\to \left[A\wedge \left(A\to B\right)\right]$
$T$	$T$	$T$	$T$	$T$
$T$	$F$	$T$	$F$	$F$
$F$	$T$	$F$	$F$	$T$
$F$	$F$	$T$	$F$	$T$

Option (B): $A\wedge \left(A\vee B\right)$

Clearly it is a set operation and so it is not a tautology.

Option (C): $A\vee \left(A\wedge B\right)$

Clearly it is a set operation and so it is not a tautology.

Hence, the correct option is $\left(\mathrm{D}\right)$