Question

If $R$ is an equivalence relation on a set $A$, then $R \circ R$ is

• A. reflexive only
• B. symmetric but not transitive
• C. equivalence
• D. none of the above

Hint:

For an equivalence relation $R$, composition with itself gives the same relation: $R \circ R = R$.

Answer 1

The Correct Option is C

Step 1: Recall the properties of an equivalence relation. An equivalence relation is reflexive, symmetric, and transitive.
Step 2: Since $R$ is transitive, we have: \[ R \circ R \subseteq R \]
Step 3: Because $R$ is also reflexive, every $(a,a) \in R$, which implies: \[ R \subseteq R \circ R \]
Step 4: Hence, \[ R \circ R = R \]
Step 5: Since $R$ itself is an equivalence relation, $R \circ R$ is also an equivalence relation.