Question

Show that$\mathbf{p}\mathbf{\downarrow }\mathbf{q}$ is logically equivalent torole="math" localid="1668402236860" $\mathbf{\neg }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}$.

Short answer

For showing, $\mathbf{p}\mathbf{\downarrow }\mathbf{q}$and$\mathbf{\neg }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}$ equivalent, prepare the truth table for every statement and also take different values of pand qsuch as every possible combination of Tand F to check the result for their verification.

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.

To show p↓q and ¬(p∨q) are logically equivalent

Take two variables as inputs such aspandq.

p NOR q is denoted by $\mathbf{p}\mathbf{\downarrow }\mathbf{q}$.

Prepare the truth table for $\mathbf{p}\mathbf{\downarrow }\mathbf{q}$and $\mathbf{\neg }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}$

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

Truth Table

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

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

Therefore, it has been proved that $\mathbf{p}\mathbf{\downarrow }\mathbf{q}$is logically equivalent to$\mathbf{\neg }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}$.