Question

The only statement among the following that is tautology is:-
A.$A \wedge \left( {A \vee B} \right)$
B.$A \wedge \left( {A \wedge B} \right)$
C.$\left[ {A \wedge \left( {A \to B} \right)} \right] \to B$
D.$B \to \left[ {A \wedge \left( {A \to B} \right)} \right]$

Answer 1

Hint: First of all, we will draw the truth tables of each of the given Boolean expressions mentioned in the options. The Boolean expression for which the output values of all the possible cases is true is known as tautology.

Complete step-by-step answer:
First of all, let us write the meaning of each symbol used in the given Boolean expression.
$ \vee $ represents OR logical operation
\[ \wedge \] represents AND logical operation
\[ \to \] represents if-then (implies) logical operation
Next, we will draw the truth tables of each of the given Boolean expressions mentioned in the options.
Let us draw the table for $A \wedge \left( {A \vee B} \right)$

$A$	$B$	$\left( {A \vee B} \right)$	$A \wedge \left( {A \vee B} \right)$
True	True	True	True
True	False	True	True
False	True	True	False
False	False	False	False

 Now, we will the draw the table for $A \wedge \left( {A \wedge B} \right)$

$A$	$B$	$\left( {A \wedge B} \right)$	$A \wedge \left( {A \wedge B} \right)$
True	True	True	True
True	False	False	False
False	True	False	False
False	False	False	False

Now, we will the draw the table for $\left[ {A \wedge \left( {A \to B} \right)} \right] \to B$

$A$	$B$	$\left( {A \to B} \right)$	$A \wedge \left( {A \to B} \right)$	$\left[ {A \wedge \left( {A \to B} \right)} \right] \to B$
True	True	True	True	True
True	False	False	False	True
False	True	True	False	True
False	False	True	False	True

Now, we will the draw the table for $B \to \left[ {A \wedge \left( {A \to B} \right)} \right]$

$A$	$B$	$\left( {A \to B} \right)$	$A \wedge \left( {A \to B} \right)$	$B \to \left[ {A \wedge \left( {A \to B} \right)} \right]$
True	True	True	True	True
True	False	False	False	True
False	True	True	False	False
False	False	True	False	True

As, the truth-table for the $\left[ {A \wedge \left( {A \to B} \right)} \right] \to B$, gives all outputs as true, so we can say that the Boolean expression $\left[ {A \wedge \left( {A \to B} \right)} \right] \to B$ is a tautology.
Hence, option C is the correct one.
Note: While calculating the truth-table, many students interpret the wrong meaning of AND operator and OR operator. Also, the if-then operator (\[ \to \]) is non-commutative and one has to be careful about the order of input values. But the AND operator and OR operator are commutative in nature.