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 _______.

Question

Let \(\mathrm{A}=\{1,2,3\}.\) The number of relations on A , containing ( 1,2 ) and (2, 3), which are reflexive and transitive but not symmetric, is _______.

Answer 1

Answer is: 3  

Transitivity

\((1,2) \in \mathrm{R},(2,3) \in \mathrm{R} \Rightarrow(1,3) \in \mathrm{R}\)

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

Now (2, 1), (3, 2), (3, 1)

(3, 1) cannot be taken

(1) (2, 1) taken and (3, 2) not taken

(2) (3, 2) taken and (2, 1) not taken

(3) Both not taken

therefore 3 relations are possible.