Invertible <=> F is 1-1 and onto

[jeff1evesque]

invertible <==> F is "1-1" and "onto"

Never mind, haha- found out

Does anyone know why the following is true:

F is invertible <==> F is "1-1" and "onto"?

 

[ThirstyDog]

Yes it is true.

Say F maps from A to B then:

1) As F is onto every element in B corresponds to at least one element in A i.e. for all $b \in B$ there is at least one $a \in A$ such that $f(a) = b$.

2) As F is 1-1 every element in B can correspond to at most one element in A ie. $f(a) = f(a') \Rightarrow a=a'$.

This is enough to be able to construct the inverse
$$f^{-1}(b) = a \mbox{ if and only if }f(a) = b$$