A. l The Logic of Propositions 593CONVERSE, INVERSE AND CONTRAPOSITIVE
Associated with a given implication, P ::::} Q, there a re three related impli-
cations:
Given implication: P::::} Q
Its inverse:
Its converse: Q::::} P (sometimes written P {= Q)
Its contrapositive: "'Q ::::} "'P.These are not all logically equivalent, but from Example A.l.20 we see the
following two equivalences:
Theorem A.1.21 Converse, Inverse, and Contrapositive:
(a) An implication is logically equivalent to its contrapositive.(b) The inverse and converse of an implication are logically equiva lent.Contrary to what many people expect, the negation of an implication
is not another implication. In fact, it is quite different. From Example A.l.19,
we have (P::::} Q) = "'P V Q. Thus, using de Morgan's law,
rv(P=}Q) := rv(rvP V Q)
:=rv(rvP)/\rvQ
:= p /\ rv Q.Therefore, we have proved the following theorem.Theorem A.1.22 Negation of an Implication:
rv (P ::::} Q) := P /\ rv Q.
Examples A.1.23 Prove the distributive laws:
(a) P /\ (Q v R) = (P /\ Q) V (P /\ R);
(b) P v (Q /\ R) = (P V Q) /\ (P V R).
We prove (a) and leave (b) as Exercise 20.