Question

Use truth tables to verify the absorption laws.

(a)$\mathbf{p}\mathbf{\vee }\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{=}\mathbf{p}$ (b)$\mathbf{p}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}\mathbf{=}\mathbf{p}$

Short answer

For the verification of given absorption laws, prove that the outcome of both the parts of the expression have identivcal outcome in the truth table.

Step-by-step solution

Definition of Truth Table  

A logic gate truth table depicts each feasible input sequence to the gate or circuit, as well as the resulting output based on the combination of these inputs.

Verification of absorption laws using a truth table

(a) $\mathbf{p}\mathbf{\vee }\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{=}\mathbf{p}$

Prepare a truth table $\mathbf{p}\mathbf{\vee }\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}$ for and p

p	q	$\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}$	$\mathbf{p}\mathbf{\vee }\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}$
T	T	T	T
T	F	F	T
F	T	F	F
F	F	F	F

Truth Table

Check the outcomes of $\mathbf{p}\mathbf{\vee }\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}$and p..

From the truth tables, it is observed that $\mathbf{p}\mathbf{\vee }\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{=}\mathbf{p}$as their outcomes are the same.

Therefore, the absorption law has been verified using a truth table.

(b) $\mathbf{p}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}\mathbf{=}\mathbf{p}$

Prepare a truth table for $\mathbf{p}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}$ and p.

p	q	$\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}$	$\mathbf{p}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}$
T	T	T	T
T	F	T	T
F	T	T	F
F	F	F	F

Truth Table

Check the outcomes of $\mathbf{p}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}$and p..

From the truth tables, it is observed that $\mathbf{p}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}\mathbf{=}\mathbf{p}$as their outcomes are the same.

Therefore, the absorption law has been verified using a truth table.