Let A={1,2,3}. Then find the number of relations containing (1,2) and (1,3), which are reflexive and symmetric but not transitive is:

• (A). 1
• (B). 2
• (C). 3
• (D). 4

To solve the problem, we need to analyze the properties of the relations defined on the set A = {1, 2, 3}. We are looking for relations that are reflexive, symmetric, and not transitive, while also containing the pairs (1, 2) and (1, 3).

Understanding Reflexivity

A relation is reflexive if every element in the set relates to itself. For our set A, this means we must include the pairs (1, 1), (2, 2), and (3, 3) in our relation.

Exploring Symmetry

A relation is symmetric if for every pair (a, b) in the relation, the pair (b, a) must also be included. Since we have (1, 2) and (1, 3), we must also include (2, 1) and (3, 1).

Identifying Non-Transitivity

A relation is not transitive if there exists at least one instance where (a, b) and (b, c) are in the relation, but (a, c) is not. We need to ensure that our relations violate this property.

Constructing the Relations

Starting with the required pairs, we have:

• (1, 1), (2, 2), (3, 3) for reflexivity
• (1, 2), (2, 1) for symmetry
• (1, 3), (3, 1) for symmetry

Now, we can choose to include or exclude the pairs (2, 3) and (3, 2). Including both would make the relation transitive, so we must avoid that.

Possible Combinations

We can have the following combinations:

• Include neither (2, 3) nor (3, 2)
• Include (2, 3) but not (3, 2)
• Include (3, 2) but not (2, 3)

Counting the Valid Relations

Thus, we have three valid relations that satisfy all conditions:

• Relation 1: {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}
• Relation 2: {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3)}
• Relation 3: {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1), (3, 2)}

Therefore, the number of relations that meet the criteria is 3. The correct answer is (C). 3.