Let A = {1, 2, 3}. The number of relations on A, containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is ______ -
Step-by-Step Solution
Key Concept: Apply the core result for equivalence relations and set operations and simplify using the given constraints.
R is reflexive \Rightarrow R have (1, 1), (2, 2), (3, 3) (3) R is transitive ∵ (1, 2), (23) \in R \therefore (1, 3) \in R \therefore R1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} Clearly R is reflexive and transitive but not symmetric. 1 Similarly, R2 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3), (3, 2)} R3 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3), (2, 1)} Therefore, 3 relations are possible
Correct Answer: 3