We have just shown that a composition of injective (resp. surjective, resp. bijective) functions are again injective (resp. surjective resp. bijective). In general, we cannot presume that the converse of these statements is true. However, the following statements can be proven:

Proposition: Injective, Surjective and Bijective Compositions

Let $f:A\mapsto B$ and $g:B\mapsto C$ be functions. 1. If the composition $g\circ f$ is injective, then $f$ is injective. 1. If the composition $g\circ f$ is surjective, then $g$ is surjective. 1. If $f$ is surjective and $g\circ f$ is injective, then $g$ is injective. 1. If $f$ is surjective and $g\circ f$ is injective, then $g$ is injective.

Proofs: 1


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References

Bibliography

  1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition
  2. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994