Question

Show that$(p\to q)\to r$ and$p\to (q\to r)$are not logically equivalent

Short answer

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

Step-by-step solution

Step1:Definition of Logical equivalence

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

The two statements are not logically equivalent

We write given expression,

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

Now we write other expression,

$\begin{array}{l}p\to (q\to r)=\neg p\vee (\neg q\vee r)\\ =(\neg p\vee \neg q)\vee r\dots \dots \dots \dots \dots \dots \dots \left[Associativity\right]\end{array}$

We can see the two statements are not equal.

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