Definition:Inverse Mapping

Definition

If the inverse $f^{-1}$ of a mapping $f$ is itself a mapping, then it is called the inverse mapping of $f$.

Thus, from the definition of a mapping, for $f^{-1}$ to be the inverse mapping of $f$:

$\forall y \in T: \left({x_1, y}\right) \in f \land \left({x_2, y}\right) \in f \implies x_1 = x_2$

and

$\forall y \in T: \exists x \in S: \left({x, y}\right) \in f$

When $f^{-1}$ is a mapping, we say that $f$ has an inverse mapping.

Invertible Mapping

If $f$ has an inverse mapping, then $f$ is an invertible mapping.

Also see

• Bijection iff Inverse is Bijection, where is shown that $f^{-1}$ is a mapping iff $f$ is a bijection, and that $f^{-1}$ it itself a bijection.

• Inverse of Bijection

• Left and Right Inverses of Mapping are Inverse Mapping, which some sources use as the definition of an inverse mapping.

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 2$
• W.E. Deskins: Abstract Algebra (1964): $\S 1.3$: Definition $1.9 \ \text {(a)}$
• 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): $\S 3.3$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.3$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 13$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 7.11$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 22$