Definition:Surjection
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Definition
A mapping $f: S \to T$ is described as onto, or a surjection, or surjective, iff:
- $\forall y \in T: \exists x \in \operatorname{Dom} \left({f}\right): f \left({x}\right) = y$
That is, if it is right-total, i.e. every element in the codomain of $f$ is mapped to by at least one element in the domain.
That is, a surjection is a relation which is:
The notation $f: S \twoheadrightarrow T$ is sometimes used to emphasize surjectivity.
If $f$ is not a surjection, then $f$ is described as into.
Also see
- In Surjection iff Image equals Codomain, it is shown that a mapping $f$ is a surjection iff its image equals its codomain.
- In Surjection iff Right Cancellable it is shown that a mapping $f$ is a surjection iff it is right cancellable.
- In Surjection iff Right Inverse it is shown that a mapping $f$ is a surjection iff it has a right inverse.
- In Preimages All Exist iff Surjection, it is shown that a mapping $f$ is a surjection iff the preimage of every element is guaranteed not to be empty.
- In Image of Preimage of Surjection, it is shown that a mapping $f$ is a surjection iff the image of the preimage of every subset of its codomain equals that subset.
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
- Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.3$
- W.E. Deskins: Abstract Algebra (1964): $\S 1.3$: Definition $1.9 \ \text{(b)}$
- Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 3.3$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 5$
- Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.4$
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$: Theorem $8$
- A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis (1968): $\S 1.3$
- Ian D. Macdonald: The Theory of Groups (1968): Appendix
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 11$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 5$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
- Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.4$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 21.2$
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.6$
- John F. Humphreys: A Course in Group Theory (1996): $\S 2$: Definition $2.3$
- H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.5$
- René L. Schilling: Measures, Integrals and Martingales (2005)... (previous)... (next) $\S 2$
- For a video presentation of the contents of this page, visit the Khan Academy.
