Definition:Inverse of Bijection

Definition

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

Then from Bijection iff Left and Right Inverse, there exists a mapping $g$ such that:

• $g \circ f = I_T$
• $f \circ g = I_S$

$g$ is sometimes known as the two-sided inverse of $f$.

Note that from Bijection iff Inverse is Bijection, this two-sided inverse is the inverse mapping $f^{-1}$ defined as:

$\forall y \in T: f^{-1} \left({y}\right) = \left\{{x \in S: \left({x, y}\right) \in f}\right\}$

Usually we dispense with calling it the two-sided inverse, and just refer to it as the inverse.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 3.6$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
• Ian D. Macdonald: The Theory of Groups (1968): Appendix
• Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.4$: Problem $25$
• 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 $\text{A}.7$