Question

The number of relations, on the set {1,2,3} containing (1,2) and (2,3), which are reflexive and transitive but not symmetric, is______

Hint:

For problems involving relations, carefully consider each property (reflexive, sym metric, transitive) separately. List out the required ordered pairs and enumerate the possible relations

Answer 1

Correct Answer: 3

Let \( A = \{1, 2, 3\} \).

Step 1: Reflexive Property

For the relation to be reflexive:

\( (1,1), (2,2), (3,3) \in R \)

Step 2: Transitive Property

For the relation to be transitive:

\( (1,2) \text{ and } (2,3) \in R \implies (1,3) \in R \)

Step 3: Symmetric Property

The relation is not symmetric because:

\( (2, 1) \in R \text{ but } (3,2) \notin R \)

Relations:
\( R_1 = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\} \)

\( R_2 = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3), (2,1)\} \)

\( R_3 = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3), (3,2)\} \)