Question

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

Short answer

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

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,

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

Now we write other expression,

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

We can see the two statements are not equal.

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