Example of a relation

Consider the set \(F = \{\text{Peter}, \text{John}, \text{Mark}\hspace{0.15em}\}\) and the set \(S = \{\text{Luke}, \text{Mary}, \text{Esther}\hspace{0.15em}\}\).
In this scenario, it turns out that:

Luke and Mary are children of Mark. Esther is the daughter of Peter.

Let \(R\) be the relation between \(F\) and \(S\) that depicts the parent-child relation.

In general, \((p, c) \in R\) if and only if \(p\) is a parent of \(c\). We may sometimes write \(pRc\) in short for \((p, c) \in R\).
We may also draw a diagram to show this relation \(R\).

From the given information, we can express:

\[R = \{(\text{Mark}, \text{Luke}), (\text{Mark}, \text{Mary}), (\text{Peter}, \text{Esther})\}\]

We can also draw a relation diagram to represent this relation \(R\):

Relation diagram between sets F and S

Note that for a relation, it is not obliged that every element on the left is linked to some element on the right. It is also not required that a relation satisfies the condition that for every element on the left there is only \(1\) arrow emerging from it.