Question

If a relation R on the set {1, 2, 3} be defined by R = {(1, 1)}, then R is

• A. Reflexive and symmetric
• B. Reflexive and transitive
• C. symmetric and transitive
• D. Only symmetric

Answer 1

The Correct Option is C

Given a relation R on the set {1, 2, 3} defined by R = {(1, 1)}, we need to determine the properties of R.

Reflexive: A relation R is reflexive if for all a in the set, (a, a) belongs to R. In this case, the set is {1, 2, 3}. For R to be reflexive, (1, 1), (2, 2), and (3, 3) must be in R. However, R = {(1, 1)} does not contain (2, 2) and (3, 3), so R is not reflexive.

Symmetric: A relation R is symmetric if for every (a, b) in R, (b, a) is also in R. In this case, R = {(1, 1)}. Since (1, 1) is in R, and (1, 1) is equal to (1, 1), R is symmetric.

Transitive: A relation R is transitive if for every (a, b) and (b, c) in R, (a, c) is also in R. In this case, R = {(1, 1)}. If (1, 1) and (1, 1) are in R, then (1, 1) must be in R, which it is. Thus, R is transitive.

Since R is symmetric and transitive, but not reflexive, the correct option is (C) Symmetric and transitive.