I understand that a relation is transitive if $(a,b), (b,c)$ and $(a,c)$ are elements of the relation $X$.
However, I don't understand why the following relation is transitive?
$X = \{(1,2),(1,3),(1,4)\}$
I know that this relation is anti-symmetric.
Any help is much appreciated, thanks.
A relation $R$ is transitive if for every couple of pairs $\{ (x,y),(y,z)\} \subseteq R$ (i.e. second element of first pair is first element of second pair), you also have that $(x,z) \in R$.
In your case, no such couple of pairs exists, so no pair of the form $(x,z)$ must be in $R$ for it to be transitive!
To make this more formal, the definition of transitivity is $$\forall x,y,z \in X: \left((x,y) \in R \wedge (y,z) \in R\right) \implies (x,z) \in R$$ But, in your example, $\left((x,y) \in R \wedge (y,z) \in R\right)$ is never satisfied. Hence the implication is always true!