A relation R is defined on Z by aRb if 5a b is even a Prove

A relation R is defined on Z by aRb if 5a - b is even. (a) Prove that R is an equivalence relation. (b) Describe the distinct equivalence classes resulting from R.

Solution

a> aRa holds as 5a-a = 4a is even. So R is reflexive

aRb holds .

Then 5a-b is even

=> 5a-b = 2m for some integer m   ---(1)

Now 5b-a

= 4b-5a+4a+b

= 4b+4a -(5a-b)

=4(b+a)-2m (by 1)

= 2( 2b+2a-m)

So 5b-a is even

Thus bRa holds.

R is symmetric.

Let aRb & bRc hold.

Then 5a-b and 5b-c both are even.

5a-c = 5a-b+5b-c -4b

= 2k+2l -4b since 5a-b and 5b-c both are even and 5a-b=2k and 5b-c=2l for some integer k,l

= 2(k+l-2b)

So 5a-c is even.

Thus aRc holds.

So R is transitive.

As R is reflexive , symmetric and transitive, R is an equivalence relation