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Let's say I have the relation $R = \{ (0,0), (1,1), (2,2), (3,3) \}$ defined on the set $X = \{ 0, 1, 2, 3 \}$.

A relation is transitive if, whenever $(a, b) \in R$ and $(b, c) \in R$, $(a, c) \in R$.

Since all of the elements of the relation are reflexive in the sense that $a = b = c$ for any sets $(a, b)$ and $(b, c)$, would we classify this relation as transitive?

I'm really just looking for confirmation as to whether it is valid to say that the relation is transitive since we have $a = b = c$ in the context of the definition of transitivity (or is it the case that $a, b, c$ must be different elements of $X$?).

The Pointer
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    it is valid to say that the relation is transitive Yes. or is it the case that a,b,c must be different elements of X No, that's nowhere specified in the definition of transitivity. – dxiv Mar 19 '18 at 05:30
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    @dxiv Ok, thank you for the clarification. I just wanted to make sure. :) – The Pointer Mar 19 '18 at 05:33

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