Question

Show that$\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{r}\mathbf{)}\mathbf{\vee }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{r}\mathbf{)}$ and $\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{\to }\mathbf{r}$are logically equivalent

Short answer

It is shown that $\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{r}\mathbf{)}\mathbf{\vee }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{r}\mathbf{)}$and$\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{\to }\mathbf{r}$ are logically equivalent.

Step-by-step solution

Definition of Logical equivalence

Logical equivalence is a relationship between two statements in propositional logic.

The two statements are logically equivalent

We solve the given expression,

$$\begin{gathered} \mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{r}\mathbf{)}\mathbf{\vee }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{r}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{r}\mathbf{)}\mathbf{\vee }\mathbf{(}\mathbf{\neg }\mathbf{q}\mathbf{\vee }\mathbf{r}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\vee }\mathbf{(}\mathbf{r}\mathbf{\vee }\mathbf{r}\mathbf{)}............[\mathrm{Associative}\mathrm{law}] \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\neg }\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{\vee }\mathbf{r}.......................[\mathrm{DeMorgan}\text{'}\mathrm{s}\mathrm{law}] \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{\to }\mathbf{r} \end{gathered}$$

Hence it is shown that $\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{r}\mathbf{)}\mathbf{\vee }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{r}\mathbf{)}$ and$\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{\to }\mathbf{r}$ are logically equivalent.