A,
if x R y and y R z, then x R z. 8. Recall that a relation R on a set A is transitive
if, and only if, for all x, y and zA,
if x R y and y R z, then x R z.
Consider the statement If R is transitive then R-1 is transitive.
The contrapositive of this statement is If R-1 is not transitive then R is not transitive.
Suppose that the relation R-1 on a set A is
not transitive.
So there exist elements
x, y and zA,
such that (x,y) R-1,
(y,z) R-1
and
x is not
related to z by R-1.
Then in R, we must have
(y,x) R,
(z,y) R
and z is not related to x by R.
Thus R is not transitive.
Therefore, by a proof by contraposition we have shown that If R is transitive then R-1 is transitive.