A relation R is defined on zopf by a R b if 5 2a 3b Show th

A relation R is defined on zopf by a R b if 5| (2a + 3b). Show that R is an equivalance relation. Find the equivalence classes [0], [1], [2].

Solution

Given that R is defined on Z by a R b if 5 I ( 2a+ 3b )

a) If R is an equivalence relation, then it is

Reflexive: For any a N, we have 2a + 3a = 5a. Since 5a is divisible by 5, it is congruent to 0 (mod 5). It follows that a R a.

Symmetric: For any a, b N, we have 2a + 3b = 3b + 2a. It follows that if a R b then b R a.

Transitive: For any a, b, c N, we have (2a+3b)+(3b+2c) = (2a+3c)+5b = 2a + 3c (mod 5), since 5b = 0 (mod 5). Then if 2a + 3b 0 (mod 5) and 2b + 3c = 0 (mod 5), it follows that 2a + 2c = 0 (mod 5).

Hence ,

R is an equivalence relation