Tue March 27, 2012 By:

Relation R in the set A of human beings in a town at a particular time given by R = {(x, y) : x is wife of y}. Determine if reflexive, symmetric and transitive.

Expert Reply
Wed March 28, 2012
R = {(x, y): x is the wife of y}
(x, x) ? R
Since x cannot be the wife of herself.
?R is not reflexive.

Now, let (x, y) ? R
? x is the wife of y.
Clearly y is not the wife of x.
?(y, x) ? R
If x is the wife of y, then y is the husband of x.
? R is not transitive.
Let (x, y), (y, z) ? R
? x is the wife of y and y is the wife of z.
This case is not possible. Also, this does not imply that x is the wife of z.
?(x, z) ? R
?R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.
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