Definition:Graph of Mapping

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This page is about Graph in the context of mapping theory. For other uses, see Definition:Graph.

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 $\mathcal R = \set {\tuple {x, \map f x}: x \in S}$


Alternatively, this can be expressed:

$G_f = \set {\tuple {s, t} \in S \times T: \map f s = t}$

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


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


GraphOfFunction.png


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 point.


Also denoted as

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


Also see

  • Results about Graphs of Mappings can be found here.


Sources