Definition:Graph of a Mapping

Definition

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

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

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:

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.

Note

Not to be confused with a graph theoretic graph.

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 2$
• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 8$: Functions
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 7.1$
• John F. Humphreys: A Course in Group Theory (1996): $\S 2$: Example $2.22$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.9$