Non-transitive relation

Question

I have the set $A = \left \{1, 6, 0 \right \}$ and the relation $R = \left \{(1,6), (6,1), (1,0), (0,6), (6,1) \right \}$ Is that relation transitive? If I am right it is not because I don't have $(1,1)$. Is that a correct reason for saying that it is not transitive? And why it is antisymmetric?

Answer 1

Yes, you are correct.

If $(a,b) $ and $ (b,c)$ are in the relation, and if $(a,c)$ is not, then the relation not transitive.

In your case, where $(a,b)$ and $(b,a)$ are in the relation, both $(a,a)$ and $(b,b)$ need to be present for the relation to be transitive.

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Anti-symmetric means that when $(a,b)$ is in the relation, where $a\ne b$, then $(b,a)$ must not be in the relation. As you can see, that is not the case in the given relation, so the given relation is not anti-symmetric.