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

Question

I am confused here. For the set $\{ a, b, c\}$ how is the relation $\{(a, b), (b, c), (a, c)\}$ transitive ?

Answer 1

You only have two pairs of the form $\;(a,b),\,\,(b,c)\in R\;$ , and also $\;(a,c)\in R\;$ , so it is true that whenever $\;(x,y),\,(y,z)\in R\;$ , also $\;(x,z)\in R\;$, and that's transitivity

Answer 2

Because no matter how you select elements $x,y,z \in \{a, b, c\}$ such that $xRy$ and $yRz$ (hint: there's only one way), you have $xRz$, which is what it means to be transitive.