Question

Show that$(p\wedge q)\to r$ and$(p\to r)\wedge (q\to r)$are not logicallyequivalent

Short answer

$(p\wedge q)\to r$and$(p\to r)\wedge (q\to r)$are not equivalent logically.

Step-by-step solution

Step1:Definition of Logical equivalence

Logical equivalence is a relationship between two statementsin propositional logic.

The two statements are not logically equivalent

We write given expression,

$\begin{array}{l}(p\wedge q)\to r=\neg (p\wedge q)\vee r\\ =(\neg p\vee \neg q)\vee r\end{array}$

Now we write other expression,

$\begin{array}{l}(p\to r)\wedge (q\to r)=(\neg p\vee r)\wedge (\neg q\vee r)A\\ =(\neg p\wedge \neg q)\vee r......\left[Distributivelaw\right]\end{array}$

We can see the two statements are not equal.

$(p\wedge q)\to r$and $\mathbf{(}\mathit{p}\mathbf{\to }\mathit{r}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathit{q}\mathbf{\to }\mathit{r}\mathbf{)}$are not logically equivalent.