Definition:Graph of Mapping
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:
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 validity of the material on this page is questionable. In particular: Note that real function as defined here on $\mathsf{Pr} \infty \mathsf{fWiki}$ is a mapping specifically on the reals (or subsets thereof). The most general object of this type is Definition:Graph of Mapping which subsumes this. If this definition (that is, on a mapping from a real Euclidean space to another real Euclidean space) is actually needed (yes, it probably is), then we need to link here to that specific type of mapping. It's possible there may already be such a definition, I haven't looked. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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
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 graph 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 graph of a mapping $f$ can also be seen denoted:
- $\map G f$
- $\Gamma_f$
- $\map \Gamma f$
and so on.
Examples
Also see
- Results about graphs of mappings can be found here.
Sources
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- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): graph (of a function or mapping)