Question

Use a truth table to verify the first De Morgan law.

$\mathbf{\neg }\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{\equiv }\mathbf{\neg }\mathbf{p}\mathbf{}\mathbf{v}\mathbf{\neg }\mathbf{q}$

Short answer

The given De Morgan law can be verified using truth tables. Prepare the truth table for every statement and also take different values of$\mathbf{p}$and $\mathbf{q}$such as every possible combination of $\mathbf{T}$and $\mathbf{F}$to check the result for their verification.

Step-by-step solution

Definition of Truth Table  

A logic gatetruth 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 Distributive laws using Truth Table

Prepare a truth table for $\mathbf{\neg }\left(p\wedge q\right)$and $\mathbf{\neg }\mathbf{pv}\mathbf{\neg }\mathbf{q}$

$\mathbf{p}$	$\mathbf{q}$	$\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}$	$\mathbf{\neg }\left(p\wedge q\right)$	$\mathbf{\neg }\mathbf{p}$	$\mathbf{\neg }\mathbf{q}$	$\mathbf{\neg }\mathbf{pv}\mathbf{\neg }\mathbf{q}$
T	T	T	F	F	F	F
T	F	F	T	F	T	T
F	T	F	T	T	F	T
F	F	F	T	T	T	T

Truth Table

Check the outcomes of $\mathbf{\neg }\left(p\wedge q\right)$ and $\mathbf{\neg }\mathbf{pv}\mathbf{\neg }\mathbf{q}$

From the truth tables, it is observed that $\mathbf{\neg }\left(p\wedge q\equiv \right)\mathbf{\neg }\mathbf{pv}\mathbf{\neg }\mathbf{q}$as their outcomes are the same.

Therefore, the first De Morgan law has been verified using a truth table.