# Retraction function -> Onto proof

1. Mar 3, 2014

### scorpius1782

Wrong forum! sorry please move to math!

1. The problem statement, all variables and given/known data
Prove that if $r : X \rightarrow A$ is a retraction, then r is onto.

Please let me know if I did this correctly!

2. Relevant equations
$i_A(x)=x$ is the identity function of A

3. The attempt at a solution

If 'r' is a retraction function then there is a function 's' such that $A \rightarrow X$ and $r \circ s=i_A$

Then regardless of the input we put into $r(x)$ we will always get x out. Meaning that there is one, and only one, result for each element in 'r'. Then if 'r' is a function in $\mathbb{R}$ it will result in one output for each element in $\mathbb{R}$.

2. Mar 3, 2014

### PeroK

I'm not sure about your proof. To show r is onto, you need to show that for every element a of A, there exists an element x of X, such that r(x) = a.

Note that r need not be 1-1.