Composite of Injections is Injection

Theorem

A composite of injections is an injection.


That is:

If $f$ and $g$ are injections, then so is $f \circ g$.


Proof

Let $f$ and $g$ be injections.

Then:

\(\ds \map {f \circ g} x\) \(=\) \(\ds \map {f \circ g} y\)
\(\ds \leadsto \ \ \) \(\ds \map f {\map g x}\) \(=\) \(\ds \map f {\map g y}\) Definition of Composition of Mappings
\(\ds \leadsto \ \ \) \(\ds \map g x\) \(=\) \(\ds \map g y\) as $f$ is injective
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds y\) as $g$ is injective

$\blacksquare$


Sources

  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.6$. Products of bijective mappings. Permutations: $\text{(i)}$
  • 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Factoring Functions
  • 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.14$: Composition of Functions: Exercise $6$
  • 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Theorem $5.10 \ (1)$
  • 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.4$: Functions: Theorem $\text{A}.5$
  • 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings: Exercise $2$
  • 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Exercise $2 \ \text{(a)}$
  • 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions
  • 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions: Exercise $2.4 \ \text{(b)}$
  • 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 2$: Problem $8$
  • 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Functions
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