Consider the set of ordered pairs of natural numbers Thus th

Consider the set of ordered pairs of natural numbers. Thus the set is {(a, b) | a N and b N}. Note that (a, b) 6= (b, a) in general.

Define a relation between an element (a, b) and an element (c, d). The pair ((a, b),(c, d)) P if a d = b c.

Question:  Say that we are in a world in which we can only add multiply and subtract numbers. What does an equivalent class correspond to here? What ”new” operator does it define?

Solution

According to difinition  

If A={a, b, c}, B= { b, c, d} and C={ a, c, }, verify that Ax(B C) = (A x B) (A x C)
B C = {c}
A x (B C) = { a, b, c} x{c}
= {(a, c), (b, c), (c, c)} ………..(1)
A x B = {(a, b), (a, c), (a, d), (b, c), (b, d), (c, b), (b, b), (c, c), (c, d)}
A x C = {(a, a), (a, c), (a, ), (b, a), (b, c), (b, ), (c, a), (c, c), (c, )}
( A x B) ( A x C) = {(a, c), (b, c), (c, c)}……….(2)
From equations (1) and (2) A x (B C) = ( A x B) ( A x C)

Thus the set is {(a, b) | a N and b N}