Answer :

(x–x) = 0 is divisible by 3 for all x z. So, (x, x) R

R is reflexive

(x – y) is divisible by 3 implies (y – x) is divisible by 3

So (x, y) R implies (y, x) R, x, y z

R is symmetric

(x – y) is divisible by 3 and (y – z) is divisible by 3

So (x – z) = (x – y) + (y – z) is divisible by 3

Hence (x, z) R R is transitive

R is an equivalence relation


Operation table a*b in A: –

Now for all a in A

a*0 = a + 0 = a a in A0 is the identity element for*.

Now, for a’ 0 let, b = 6–a.

Since, a + b = a + 6–a = 6≥6

a*b = a + 6–a–6 = b*a = 0.

Hence, b = 6–a is the inverse of a.

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