Suppose .
Then .
So .
Similarly,
gives us
.
So .
If f:A→B is 1-1 and im f=B prove that (f^-1 ○f)(a)=a for all a element of A and (f^-1 ○f)(b)=b for each b an element of B.
Here is what I've got out of this.
f is 1-1, so for each b, there is one and only a. So
Im f=b. Thus, f is onto. Therefore, the range is equal to the set B. Would this imply f(a)=b and f(b)=b?
I'm just not sure where to go with this problem.
One reason I did not respond before now is that I consider this question to have notational problems. Those problems are reflected in the icemanfan’s answer, he assumed the ordinary meaning of the symbols.
However is these set theory proofs, . That is the inverse image is a set.
The fact that is a bijection simply means that .
By injectivity .
Which means that each is a singleton set.
So .
Now I understand that this notation is not what the question implies.
However, I hope it may give you some guidance.