Definition:Graph of Mapping

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This page is about Graph in the context of Mapping Theory. For other uses, see Graph.

Definition

Let $S$ and $T$ be sets.

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


The graph of $f$ is the relation $\RR \subseteq S \times T$ defined as $\RR = \set {\tuple {x, \map f x}: x \in S}$


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 Real Function

Let $U \subseteq \R^n$ be an open subset of $n$-dimensional Euclidean space.

Let $f : U \to \R^k$ be a real function.


The graph $\map \Gamma f$ of the function $f$ is the subset of $\R^n \times \R^k$ such that:

$\map \Gamma f = \set {\tuple {x, y} \in \R^n \times \R^k: x \in U \subseteq \R^n : \map f x = y}$

where $\times$ denotes the Cartesian product.


Graph of Relation, also known as Truth Set

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.


Let $\RR$ be a relation on $S \times T$.


The truth set of $\RR$ is the set of all ordered pairs $\tuple {s, t}$ of $S \times T$ such that $s \mathrel \RR t$:

$\map \TT \RR = \set {\tuple {s, t}: s \mathrel \RR t}$


Also denoted as

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


Examples

Graph of Mapping/Examples

Also see

  • Results about Graphs of Mappings can be found here.


Sources

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