(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [itex]f:X\rightarrow~Y[/itex] and [itex]g:Y\rightarrow~Z[/itex] be surjections. Show that [itex]g\circ~f[/itex] is surjective.

2. Relevant equations

3. The attempt at a solution

Proof:

Suppose f and g are surjections.

Then (1)[itex]\forall~y\in~Y \exists~x\in~X\textnormal{ st. }f(x)=y[/itex]

And (2) [itex]\forall~z\in~Z \exists~y\in~Y\textnormal{ st. }g(y)=z[/itex]

(1) guarantees that we can write any y as f(x) for some x, so placing this into (2) gives:

(3)[itex]\forall~z\in~Z \exists~x\in~X\textnormal{ st. }g(f(x))=g\circ~f=z[/itex]

And (3) shows that [itex]g\circ~f[/itex] is surjective.

Is my logic correct?

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# Homework Help: Surjective Proof

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