What a relation is

A relation between two sets puts pairs of elements into correspondence: it is the first step towards the concept of a function.

Definition — Relation

A relation is a set of “arrows” that connect elements of one set (the domain set) to elements of another set (the codomain set).

Example — The relation "had a puppy with"

Let $A=\{\text{George, Rufus, Albert, Jack}\}$ and $B=\{\text{Susie, Prilly, Jay, Jill, Nerina}\}$. The relation $f\colon A\to B$ defined by ”$a$ had a puppy with $b$” is represented by arrows:

Each arrow connects an element of $A$ to an element of $B$.

The same relation can be written in equivalent ways:

• Ordered pairs: $\{(\text{George, Susie}),\,(\text{George, Prilly}),\,(\text{Rufus, Jay}),\,(\text{Jack, Jay}),\,(\text{Jack, Jill})\}$.
• Cartesian representation: each arrow corresponds to a point of the plane, with $A$ on the horizontal axis and $B$ on the vertical axis.

Links

Topics: Set theory
Concepts: Ordered pair · Cartesian plane · Relation