Question

Let A={1,2,3}. Then 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

Answer 1

The Correct Option is A

The given set is A = {1, 2, 3}. 
The smallest relation containing (1, 2) and (1, 3) which is reflexive and symmetric, 
but not transitive is given by: R = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 1), (3, 1)}
This is because relation R is reflexive as (1, 1), (2, 2), (3, 3) ∈ R.
Relation R is symmetric since (1, 2), (2, 1) ∈R and (1, 3), (3, 1) ∈R.
But relation R is not transitive as (3, 1), (1, 2) ∈ R, but (3, 2) ∉ R.
Now, if we add any two pairs (3, 2) and (2, 3) (or both) to relation R, then relation R will
become transitive.
Hence, the total number of desired relations is one.

The correct answer is A (1)