# Thread: composition of functions

1. ## composition of functions

I am so rusty at this:
Theorem: Let f: A $\longrightarrow$B and g:B $\longrightarrow$C.

Prove that if f and g are onto, then g $\circ$f is onto.

Since g is onto $\forall$ c $\in$ C $\exists$ b $\in$ B such that g(b) = c.

Since f is onto $\forall$ b $\in$ B $\exists$ a $\in$ A such that f(a) = b.

Therefore $\forall$ c $\in$ C $\exists$ a $\in$ A such that g(f(a)) = c.

2. It is a bit rough for my taste, but it works.