Question

${\text{R}} \subseteq {\text{A}} \times {\text{A}}$ is an equivalence relation if R is-
A. Reflexive, symmetric but not transitive
B. Reflexive, neither symmetric nor transitive
C. Reflexive, symmetric, transitive
D. None of the above

Answer 1

Hint: Any relation can be classified as reflexive, symmetric and transitive. If aRa exists in the relation, then it is said to be reflexive. If aRb and bRa both exist in the relation, then it is said to be symmetric. If aRb and bRc exist implies that aRc also exists, the relation is transitive.

Complete step-by-step answer:
It is given that the relation R is an equivalence relation. An equivalence relation is the one which is reflexive, symmetric and transitive. Some examples of an equivalence relation are-
R = {(x, y): x - y is an integer}
R= {(1, 1), (2, 2), (1, 2), (2, 1)} on a set A = {1, 2}

In the second example, (1, 1), (2, 2) exist, so R is a reflexive relation. Also, (1, 2) and (2, 1) exist, so it is symmetric as well. We can clearly see that (1, 2) and (2, 1) exist and (1, 1) also exists, hence the relation is transitive as well. Hence it is an equivalence relation.
The correct option is C. Reflexive, symmetric, transitive

Note:It is important to check carefully for each condition. It is also recommended to check and verify each condition using a suitable example. Even if one case is false, the condition is not verified. Also if it is not possible to prove that relation is symmetric, reflexive or transitive, then use a suitable example to show that it is not.