Question

How many of the disjunctions $\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{\vee }\mathbf{s}\mathbf{,}\mathbf{}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{r}\mathbf{\vee }\mathbf{s}\mathbf{,}\mathbf{}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{\vee }\mathbf{\neg }\mathbf{s}\mathbf{,}\mathbf{}\mathbf{q}\mathbf{\vee }\mathbf{r}\mathbf{\vee }\mathbf{\neg }\mathbf{s}\mathbf{,}\mathbf{}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{\vee }\mathbf{\neg }\mathbf{s}\mathbf{,}\mathbf{p}\mathbf{\vee }\mathbf{r}\mathbf{\vee }\mathbf{s}\mathbf{,}$and $\mathbf{p}\mathbf{\vee }\mathbf{r}\mathbf{\vee }\mathbf{\neg }\mathbf{s}\mathbf{,}$ can be made simultaneously true by an assignment of truth values to p,q,r and s?

Short answer

For finding the number of disjunctions that can be made simultaneously true by an assignment of truth vales top,q,rands, prepare the truth table for every statement and also take different values ofp,q,randssuch as every possible combination ofTand F.

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 find the number of disjunctions to become true.

Prepare the truth table for $\mathbf{pv}\mathbf{\neg }\mathbf{q}\mathbf{\vee }\mathbf{s}\mathbf{,}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{r}\mathbf{\vee }\mathbf{s}\mathbf{,}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{\vee }\mathbf{\neg }\mathbf{s}\mathbf{,}\mathbf{q}\mathbf{\vee }\mathbf{r}\mathbf{\vee }\mathbf{\neg }\mathbf{s}\mathbf{,}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{\vee }\mathbf{\neg }\mathbf{s}\mathbf{,}\mathbf{p}\mathbf{\vee }\mathbf{\vee }\mathbf{s}\mathbf{,}$and $\mathbf{p}\mathbf{\vee }\mathbf{r}\mathbf{\vee }\mathbf{\neg }\mathbf{s}\mathbf{,}\mathbf{.}$

Take four input variablesp,q,rands.

p	q	r	s	$\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{\vee }\mathbf{s}$	$\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{r}\mathbf{\vee }\mathbf{s}$	$\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{\vee }\mathbf{\neg }\mathbf{s}$	$\mathbf{q}\mathbf{\vee }\mathbf{r}\mathbf{\vee }\mathbf{\neg }\mathbf{s}$	$\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{\vee }$	$\mathbf{p}\mathbf{\vee }\mathbf{r}\mathbf{\vee }\mathbf{s}$	$\mathbf{p}\mathbf{\vee }\mathbf{r}\mathbf{\vee }\mathbf{\neg }\mathbf{s}$
T	T	T	T	T	T	T	T	F	T	T
T	T	T	F	T	F	T	T	T	T	T
T	T	F	T	T	T	T	T	F	T	T
T	T	F	F	T	T	T	T	T	T	T
T	F	T	T	T	T	F	T	T	T	T
T	F	T	F	T	F	T	T	T	T	T
T	F	F	T	T	T	F	F	T	T	T
T	F	F	F	T	T	T	T	T	T	T
F	T	T	T	T	T	T	T	T	T	T
F	T	T	F	F	T	T	T	T	T	T
F	T	F	T	T	T	T	T	T	T	F
F	T	F	F	F	T	T	T	T	F	T
F	F	T	T	T	T	T	T	T	T	T
F	F	T	F	T	T	T	T	T	T	T
F	F	F	T	T	T	T	F	T	T	F
F	F	F	F	T	T	T	T	T	F	T

Truth Table

From the truth table, it is observed that for all nine of the disjunctions to become true.

Therefore, there are nine disjunctions that can be made simultaneously true by an assignment of truth vales top,qand r.