Definition:Graph of Mapping

Definition
Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

The graph of $f$ is the relation $\mathcal R \subseteq S \times T$ defined as $\mathcal R = \left\{{\left({x, f \left({x}\right)}\right): x \in S}\right\}$

Alternatively, this can be expressed:
 * $G_f = \left\{{\left({s, t}\right) \in S \times T: f \left({s}\right) = t}\right\}$

where $G_f$ is the graph of $f$.

The word is usually used in the context of a diagram:


 * GraphOfFunction.png

The defining nature of a mapping means that each vertical line through a point in $A$ intersects the graph at one and only one place, corresponding to a single point in $B$.

Graph of a Relation
The concept can still be applied when $f$ is a relation, but in this case a vertical line through a point in the graph is not guaranteed to intersect the graph at one and only one place.

Also denoted as
The symbol $\Gamma_f$ is sometimes seen to denote the graph of $f$.