Showing a relationship is transitive

Question

I can't figure out why the following relationship is transitive:

Consider the relation $R=\left\{(a,a),(b,b),(c, c),(d,d),(a,b),(b,a)\right\}$ on set $A =\left\{a,b, c,d\right\}$. Is $R$ reflexive? Symmetric? Transitive? If a property does not hold, say why.

Answer 1

A relation is transitive if $(a,b),(b,c) \in R \rightarrow (a,c) \in R$. In your case it suffices to try out all the possible combinations. Note that only elements with $a,b$ may give rise to any problems, because all others are pairs of two equal elements.

Answer 2

The relation $R$ is reflexive because $\forall x$, $xRx$, $(x,x)\in R$
The relation $R$ is symmetric because in every case that $xRy$, $yRx$
The relation $R$ is transitive because indeed in every case where $xRy \land yRz$ is true, indeed $xRz$ is true.