Question

Explain, without using a truth table, why $\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{\vee }\mathbf{r}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{\vee }\mathbf{\neg }\mathbf{r}\mathbf{)}$is true when at least one of p,q, and r is true and at least one is false, but is false when all three variables have the same truth value.

Short answer

$(\mathrm{p}\vee \mathrm{q}\vee \mathrm{r})\wedge (\neg \mathrm{p}\vee \neg \mathrm{q}\vee \neg \mathrm{r})$ is true whenat least one p,q & r ofis true and at least one is false but it is false when all three variables have same truth value.

Step-by-step solution

Definition of truth table

A truth table is a mathematical table which is used in logic.

Prove the given statement

We write given statement as,

$\mathrm{p}\vee \mathrm{q}\vee \mathrm{r}$ is true if anyone ofp,q & r is true.

width="103" height="20" role="math" style="max-width: none;" localid="1668149102110" $\neg \mathrm{p}\vee \neg \mathrm{q}\vee \neg \mathrm{r}$is true if anyone of p,q & r is false for the given statement to be true at least one of the p,q,r should be true and at least one should be false.

Hence, we proved that $(\mathrm{p}\vee \mathrm{q}\vee \mathrm{r})\wedge (\neg \mathrm{p}\vee \neg \mathrm{q}\vee \neg \mathrm{r})$ is true whenat least one ofp,q & r is true and at least one is false but it is false when all three variables have same truth value.