Definition:Bijection
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Definition
A mapping $f: S \to T$ is a bijection iff $f$ is both a surjection and an injection.
It is clear that a bijection is a relation which is:
Also known as
The terms
- biunique correspondence
- bijective correspondence
are sometimes seen for bijection.
If a bijection exists between two sets $S$ and $T$, then $S$ and $T$ are said to be in one-to-one correspondence.
The symbol $f: S \leftrightarrow T$ is sometimes seen to denote that $f$ is a bijection from $S$ to $T$.
Also seen sometimes is the notation $f: S \cong T$ or $S \stackrel f \cong T$ but this is cumbersome and the symbol has already got several uses.
Also see
- Results about bijections can be found here.
Basic Properties of a Bijection
- In Bijection iff Left and Right Inverse, it is shown that a mapping $f$ is a bijection iff it has both a left inverse and a right inverse, and that these are the same, called the two-sided inverse.
- In Bijection iff Inverse is Bijection, it is shown that the inverse mapping $f^{-1}$ of a bijection $f$ is also a bijection, and that it is the same mapping as the two-sided inverse.
- In Bijection Composite with Inverse, it is established that the inverse mapping $f^{-1}$ and the two-sided inverse are the same thing.
- In Bijection iff Left and Right Cancellable, it is shown that a mapping $f$ is a bijection iff it is both left cancellable and a right cancellable.
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 8$: Functions
- 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): $\S 5$
- 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): $\S 13$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 5$
- 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): $\S 22$
- 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 $2.4$
- 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) $\S 2$
- Barile, Margherita. "One-to-One." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/One-to-One.html
