Definition:Injection

Definition

A mapping $f$ is an injection, or injective, or one-one, or one-to-one iff:

$\forall x_1, x_2 \in \operatorname{Dom} \left({f}\right): f \left({x_1}\right) = f \left({x_2}\right) \implies x_1 = x_2$.

That is, it is a mapping such that the output uniquely determines its input.

Alternatively, this can be put:

$\forall x_1, x_2 \in \operatorname{Dom} \left({f}\right): x_1 \ne x_2 \implies f \left({x_1}\right) \ne f \left({x_2}\right)$.

An injective mapping is sometimes written $f: S \rightarrowtail T$ or $f : S \hookrightarrow T$.

It can be seen that this definition is consistent with that of a one-to-one relation.

Thus an injection is a relation which is both one-to-one and left-total.

Also see

• Surjection
• Bijection

• In Injection iff Left Cancellable it is shown that a mapping $f$ is an injection iff it is left cancellable.

• In Injection iff Inverse of Image Mapping it is shown that a mapping $f$ is an injection iff the inverse mapping from its image is itself a mapping.

• In Injection iff Left Inverse it is shown that a mapping $f$ is an injection iff it has a left inverse.

• In Preimages All Unique iff Injection, it is shown that a mapping $f$ is an injection iff the preimage of every element of the codomain is guaranteed to have no more than one element.

• In Preimage of Image of Injection, it is shown that a mapping $f$ is an injection iff the preimage of the image of every subset of its domain equals that subset.

• Results about injections can be found here.

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 2$
• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.3$
• 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): $\text{I}$
• 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): $\S 12$
• 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): $\S 22, \ \S 22.1$
• 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 $2.2$
• 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.