Question

Let R be the real line. Consider the following subsets of the plane R x R.

S = {(x, y) : y = x + 1 and 0 < x < 2}, T = {(x, y) : x - y is an integer}. Which one of the following is true?

(A)  Neither S nor T is an equivalence relation on R

(B)  Both S and T are equivalence relations on R 

(C)  S is an equivalence relation on R but T is not 

(D)  T is an equivalence relation on R but S is not

Answer 1

Correct option  (d) T is an equivalence relation on R but S is not

Explanation :

We have

T = {(x,y): x - y ∈/}

As 0 ∈/ T is a reflexive relation. If

x - y ∈ / ⇒ y -  x ∈/

Then T is symmetrical as well.

If x - y = I₁ and y - z = I₂, then x - z − = (x - y) + (x - z) = I₁ + I₂ ∈I;

therefore, T is transitive as well.

Hence, T is an equivalence relation. Clearly,

x ≠ x + 1 ⇒ (x,x) ∉ S.

Therefore, S is not reflexive.