Surjection if Composite is Surjection

Theorem

Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be mappings such that $g \circ f$ is a surjection.

Then $g$ is a surjection.

Let $g \circ f$ be surjective.

Fix $z \in S_3$.

Now find an $x \in S_1: \map {g \circ f} x = z$.

The surjectivity of $g \circ f$ guarantees this can be done.

Now find an $y \in S_2: f \paren x = y$.

$f$ is a mapping and therefore a left-total relation; which guarantees this too can be done.

It follows that:

$\ds \map g y$

$=$

$\ds \map g {\map f x}$

$\ds $

$=$

$\ds \map {g \circ f} x$

Definition of Composition of Mappings

$\ds $

$=$

$\ds z$

Choice of $x$

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.6$. Products of bijective mappings. Permutations
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 5$: Composites and Inverses of Functions: Theorem $5.3: \ 2^\circ$
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{H}$
1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Exercise $14 \ \text{(b)}$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Mappings: Exercise $13$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings: Exercise $2$
2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions: Exercise $2.4 \ \text{(e)}$