Proof
(related to Proposition: Injective, Surjective and Bijective Compositions)
Ad (1)
- Let $x,y\in A$ with $x\neq y$.
- By hypothesis, the composition $g\circ f$ is injective.
- Thus, $g(f(x))\neq g(f(y))$.
- Since $g$ is a function, $g$ is right-unique.
- Therefore, $f(x)\neq f(y)$, otherwise we would have $g(f(x)) = g(f(y))$, which is not the case.
Ad (2)
- By hypothesis, the composition $g\circ f$ is surjective.
- This means that there is an element $x\in A$ with $g(f(x))=z$ for every $z\in C.$
- Set $y:=f(x)$. Since $y\in B$, there is an element $y\in B$ with $g(y)=z$ for every $z\in C.$
- Therefore, $g$ is surjective.
Ad (3)
- By hypothesis, $f$ is surjective and $g\circ f$ is injective.
- Let $x,y\in B$ with $x\neq y$.
- Since $f$ is surjective, there are elements $u,w\in A$ with $f(u)=x$ and $f(w)=y.$
- Therefore, since $x\neq y$, we have $f(u)\neq f(w).$
- Since $g\circ f$ is injective, we have $g(f(u))\neq g(f(w)).$
- Altogether, it follows $g(x)\neq g(y).$
- Thus, $g$ is injective.
Ad (4)
- By hypothesis, $g\circ f$ is surjective and $g$ is injective.
- Since $g\circ f$ is surjective, there is an element $x\in A$ with $g(f(x))=y$ for every $y\in B.$
- Since $g$ is injective, we have $f(x)=y.$
- In other words, there is an element $x\in A$ with $f(x)=y$ for every $y\in B.$
- Thus, $f$ is surjective.
∎
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References
Bibliography
- Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition
- Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994
