Question

Which of the following is an equivalence relation on R, for a, b ∈ Z?

(a) (a-b) ∈ Z

(b) (a^2+c) ∈ Z

(c) (ab+cd)/2 ∈ Z

(d) (2c^3)/3 ∈ Z

The question was posed to me in an online quiz.

The above asked question is from Relations topic in section Relations of Discrete Mathematics

Answer 1

Correct option is (b) (a^2+c) ∈ Z

To explain I would say: Let a ∈ R, then a−a = 0 and 0 ∈ Z, so it is reflexive. To see that a-b ∈ Z is symmetric, then a−b ∈ Z -&gt  say, a−b = m, where m ∈ Z ⇒  b−a = −(a−b)=−m and −m ∈ Z. Thus, a-b is symmetric. To see that a-b is transitive, let a, b, c ∈ R. Thus, a−b ∈ Z; b−c ∈ Z. Let a−b = i and b−c = j, for integers i,j ∈ Z. Then a−c ='(a−b)+(b−c)=i + j. So, a−c ∈ Z. Therefore a – c is transitive. Hence, (a-b) is an equivalence relation on the set R. Rest of the options are not equivalence relations.