Definition:Identity Mapping

Definition

The identity mapping, also known as identity operator or identity transformation, of a set $S$ is the mapping $I_S: S \to S$ defined as:

$I_S = \left\{{\left({x, y}\right) \in S \times S: x = y}\right\}$

or alternatively:

$I_S = \left\{{\left({x, x}\right): x \in S}\right\}$

That is:

$I_S: S \to S: \forall x \in S: I_S \left({x}\right) = x$

Informally, it is a transformation in which every element is a fixed element.

The symbol $1_S$ is also seen, as are $i_S$, $id_S$, $\operatorname {id}_S$ and $\iota_S$.

The subscript is frequently removed if there is no danger of confusion as to which set is under discussion.

Beware of the possibility of confusing with the inclusion mapping.

Also see

• Identity Mapping is a Bijection
• Inverse of Identity Mapping
• Identity Mapping is Left Identity
• Identity Mapping is Right Identity

Note that the identity mapping on $S$ is the same as the diagonal relation $\Delta_S$ on $S$.

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
• W.E. Deskins: Abstract Algebra (1964): Exercise $1.3: 10$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 3.5$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 5$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.4$: Example $10$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
• Ian D. Macdonald: The Theory of Groups (1968): Appendix
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 10$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$: Example $4.7$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 24$
• 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 $2.11$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.5$
• For a video presentation of the contents of this page, visit the Khan Academy.