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

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] Reflexivity gives the diagonal and transitivity forces $(1,3)$. Only $(2,1),(3,2),(3,1)$ can still be added. Checking transitivity, the valid additions are $\varnothing$, $\{(2,1)\}$, $\{(3,2)\}$ and all three. The last one makes an equivalence relation (symmetric), so the non-symmetric ones number $3$. Answer: 3.

Q. 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 _______ -

Explanation

Reviewed by Dr. Abubacker

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