The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is

( A ) Symmetric only

( B ) Reflexive only

( C ) An equivalence relation

( D ) Transitive relation

To determine the properties of the relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3}, we need to analyze it based on the definitions of reflexivity, symmetry, and transitivity. Let's break down each property one by one.

Reflexivity

A relation is reflexive if every element in the set is related to itself. In our case, the set is {1, 2, 3}. The relation R includes (1, 1), (2, 2), and (3, 3), which means:

• 1 is related to 1
• 2 is related to 2
• 3 is related to 3

Since all elements in the set are related to themselves, R is reflexive.

Symmetry

A relation is symmetric if for every (a, b) in R, (b, a) is also in R. In our relation R, we only have pairs where both elements are the same. For example:

• (1, 1) implies (1, 1)
• (2, 2) implies (2, 2)
• (3, 3) implies (3, 3)

Since there are no pairs where a ≠ b, the condition for symmetry is trivially satisfied. Thus, R is symmetric.

Transitivity

A relation is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R. In our case, since all pairs are of the form (x, x), we can check:

• If (1, 1) and (1, 1) are in R, then (1, 1) is also in R.
• If (2, 2) and (2, 2) are in R, then (2, 2) is also in R.
• If (3, 3) and (3, 3) are in R, then (3, 3) is also in R.

Since this holds true for all elements, R is transitive.

Equivalence Relation

A relation is an equivalence relation if it is reflexive, symmetric, and transitive. Since we have established that R satisfies all three properties, we can conclude that R is indeed an equivalence relation.

Final Assessment

Based on our analysis, the correct answer to the question is:

• R is reflexive.
• R is symmetric.
• R is transitive.
• R is an equivalence relation.

Therefore, the most accurate choice is (C) An equivalence relation.