Why is the relation $R = \{(a, b), (a, c)\}$ over $X = \{a, b, c\}$ transitive?

Question

I know it is not complete and I know it is asymmetric. But how is it transitive? The textbook says it is transitive...

Instead, I used a different example of transitive, asymmetric, and not complete, which is $X = \{a, b, c, d\}$ with $R = \{(a, b), (b, c), (a, c)\}$.

Answer 1

As noted in the comments, this is indeed a transitive relation. This is because the definition of transitivity requires "$xRy \land yRz$" to hold; however, for the given relation, this never holds, and thus via a vacuous argument the relation is transitive.

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Mostly just posting this to get this out of the unanswered queue. Posting as Community Wiki in particular since I have nothing further to add.

Answer 2

Another view on this. Let $R\subseteq X\times X$. Define $R^2=\{(a,c)\,|\, \exists b\in X.(a,b),(b,c)\in R\}$.

By definition $"R \text{ is transitive}"\Leftrightarrow R^2\subseteq R$.

In your case (in the title) $R^2=\varnothing$.

For your second relation we have $R^2=\{(a,c)\}$.