Question

Let S = Z and R be the relation defined by R = {Z times Z - (n, n)|n Element Z}. (a) Define the relation R, that is aRb if and only if ..... (b) Prove that R^2 = Z times Z

Answer 1

Here relation R is defined over Z.

in Z x Z, for all oder pairs (p, q) where p,q belongs to Z, if p = q then (p, q) is not in relation R.

a) aRb is in relation R if and only if a != b .

b) R² is defined by a R² b if and only if there is some x in Z such that a R x and x R b both are ture.

now according to relation R , if a R x is true then x != a .

if x R b is true then x != b.

Now, we can say that there has to be some order pair (x, a) in R such that x R a is true , as x != a.

so in other words : R can have aRx and xRa both.

In that case R² will be defined on aRa too.

Hence  R² will have all order pairs of elements a and b from Z even when a = b.

so we can say that  R² = Z x Z.

If you have any doubts, you can ask in comment section.