The number of relations on the set A=1,2,3, containing at most 6 elements including (1,2),…

Question

The number of relations on the set $A=\{1,2,3\}$, containing at most 6 elements including $(1,2)$, which are reflexive and transitive but not symmetric, is \_\_\_\_\_\_\_\_\_\_.

Answer: 6

Accepted Answer

Since relation needs to be reflexive the ordered pairs $(1,1),(2,2),(3,3)$ need to be there and $(1,2)$ is also to be included.

Let's call ${R}_{0}={(1,1),(2,2),(3,3),(1,2)}$ the base relation.

$\because A	imes A$ contain $3	imes 3=9$ ordered pairs, remaining 5 ordered are

$2,1),(1,3),(3,1),(2,3),(3,2)$

We have to add at most two ordered pairs to ${R}_{0}$ such that resulting relation is reflexive, transitive but not symmetric.

Following are the only possibilities.

$R={R}_{0}U{(1,3)}$

OR ${R}_{0}U{(3,2)}$

OR ${R}_{0}U{(1,3),(3,1)}$

OR ${R}_{0}U{(1,3),(3,2)}$

OR ${R}_{0}U{(3,1),(3,2)}$

Solutions · 1