Chapter B.2, Problem 4E

Solution

If an implication and its converse can both be false or not.

It has been determined that an implication and its converse cannot both be false at the same time.

Given:

Any implication of the form $p\to q$ .

Concept used:

The converse of an implication of the form $p\to q$ , is $q\to p$ .

Calculation:

The truth table of an implication, $p\to q$ and its converse, $q\to p$ , is given as follows:

$p$	$q$	$p\to q$	$q\to p$
T	T	T	T
T	F	F	T
F	T	T	F
F	F	T	T

It can be seen from the above truth table, that there is no combination of truth values for $p$ and $q$ , such that the implication, $p\to q$ and its converse $q\to p$ , are both false.

Thus, an implication and its converse cannot both be false at the same time.

Conclusion:

It has been determined that an implication and its converse cannot both be false at the same time.