Define a relation R on the set Z of all integers by a R b if

Define a relation R on the set Z of all integers by a R b if and only if 3a + b is a multiple of 4. Prove that R is an equivalence relation. Determine the equivalence classes associated with R.

Solution

1. 3a+a=4a

Hence, aRa for all integers a

Hence, R is reflexive

2. Let, aRb

So

3a+b= multiple of 4

3(3a+b)=3b+9a is multiple of 4

3b+a+8a is multiple of 4

Hence, 3b+a is multiple of 4

Hence,bRa

So, R is symmetric

3. Let, aRb and bRc

3a+b is multiple of 4, 3b+c is multipe of 4

So,

3a+b+3b+c =3a+c+4b is multiple of 4

HEnce, 3a+c is multiple of 4

So, aRc

Hence, R is equivalence relation

3a+b=4a-a+b

Hence, 4a+(b-a) is multiple of 4

ie b-a is multipel of 4

Hence all elemenets of an equivalence class are one which give same remainder modulo 4

so there are 4 equivalence classes for each remainder modulo 4 ie [0],[1],[2],[3]