Let A=Z∪{2}. Define a relation R in A by a R b if and only if a +  b∈z. The relation R is

We have, R={(a,b):a+b∈Z, a,b∈Z∪{2}.       2∴Z and 2+ 2 =22 ∉Z∴    (2,2) ∈ R∴ R is not reflexiveLet           (a, b) ∈ R∴  a+b∈Z⇒ b+ a ∈Z⇒(b,a)∈R⇒R is symmetric.Let      (a, b), (b, c)∈R ∴ a+b, b+c∈Z∴ None of a,b,c is equal to 2.∴      a,b,c∈Z ∴      a+c∈Z ⇒              ( a,c)∈R⇒ R is transitive ∴  . R is not an equivalence relation.