Question

If ( A = {1, 2, 3} ) and a relation ( R ) is defined on ( A ) as ( R = {(1, 1), (2, 2), (3, 3), (1,3), (2, 3), (3, 1), (3, 2), (1, 2), (2, 1)} ), check whether ( R ) is reflexive, symmetric, or transitive relation.

Answer 1

The relation R on set A = {1, 2, 3} is checked for being reflexive, symmetric, and transitive. R is reflexive as each element relates to itself; symmetric because for every (a, b), (b, a) also exists; and transitive as (a, b) and (b, c) lead to (a, c). Thus, R possesses all three properties.

Step-by-step explanation:

To determine if the relation R on set A = {1, 2, 3} is reflexive, symmetric, or transitive, we need to check the properties that define these types of relations:

• Reflexive: Every element in A is related to itself. Since (1, 1), (2, 2), and (3, 3) are in R, the relation is reflexive.
• Symmetric: For every (a, b) in R, there must also be a (b, a) in R. Since (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), and (3, 2) are all in R, the relation is symmetric.
• Transitive: For every (a, b) and (b, c) in R, there must also be a (a, c) in R. Since the pairs (1, 2), (2, 3), and (1, 3) are in R, along with (2, 1), (1, 3), and (2, 3), it is also transitive.

Therefore, the relation R on the set A is reflexive, symmetric, and transitive.