Question

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

Short answer

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

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,

$$\begin{gathered} \mathbf{\neg }\mathbf{q}\mathbf{\to }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{r}\mathbf{)}\mathbf{=}\mathbf{p}\mathbf{\vee }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{r}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{p}\mathbf{\vee }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{r}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\neg }\mathbf{q}\mathbf{\vee }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{r}\mathbf{)}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{[}\mathit{A}\mathit{s}\mathit{s}\mathit{o}\mathit{c}\mathit{i}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{i}\mathit{t}\mathit{y}\mathit{}\mathit{a}\mathit{n}\mathit{d}\mathit{}\mathit{c}\mathit{o}\mathit{m}\mathit{m}\mathit{u}\mathit{t}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{i}\mathit{t}\mathit{y}\mathbf{]} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{q}\mathbf{\to }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{r}\mathbf{)} \end{gathered}$$

Thus, $\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{r}\mathbf{)}$and $\mathbf{q}\mathbf{\to }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{r}\mathbf{)}$are logically equivalent.