Let A= {1, 2, 3}, then the possible equivalence relations on A are:

(A) {(1, 1), (2, 2) , (3, 3)}

(B) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}

(C) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3)}

(D) {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)}

Choose the correct answer from the options given below:

A relation is an equivalence relation if it satisfies three properties:

Reflexive: Every element must be related to itself. Must contain $(1,1)$, $(2,2)$, $(3,3)$.

Symmetric: If $(a,b)$ exists, then $(b,a)$ must also exist.

Transitive: If $(a,b)$ and $(b,c)$ exist, then $(a,c)$ must exist.

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Checking option (A): $\{(1,1), (2,2), (3,3)\}$

Reflexive: All three elements relate to themselves.

Symmetric: No additional pairs to check.

Transitive: No chain of relations to check.

Option (A) is an equivalence relation.

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Checking option (B): $\{(1,1), (2,2), (3,3), (1,2), (2,1)\}$

Reflexive: $(1,1)$, $(2,2)$, $(3,3)$ are all present.

Symmetric: $(1,2)$ exists and $(2,1)$ also exists.

Transitive:

• $(1,2)$ and $(2,1)$ require $(1,1)$, which is present.
• $(2,1)$ and $(1,2)$ require $(2,2)$, which is present.

Option (B) is an equivalence relation.

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Checking option (C): $\{(1,1), (2,2), (3,3), (1,2), (2,1), (1,3)\}$

Reflexive: All three elements relate to themselves.

Symmetric: $(1,2)$ exists and $(2,1)$ exists. However, $(1,3)$ exists but $(3,1)$ is missing.

Option (C) is not an equivalence relation.

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Checking option (D): $\{(1,1), (2,2), (3,3), (1,3), (3,1)\}$

Reflexive: All three elements relate to themselves.

Symmetric: $(1,3)$ exists and $(3,1)$ also exists.

Transitive:

• $(1,3)$ and $(3,1)$ require $(1,1)$, which is present.
• $(3,1)$ and $(1,3)$ require $(3,3)$, which is present.

Option (D) is an equivalence relation.

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The valid equivalence relations are (A), (B), and (D).