[Math] Proving Equality of 2 Functions

Question

I have a general question, illustrated by a specific example.

The general question is how to methodically prove that two functions are equal. Much like trying to prove an "if-and-only-if" statement (by first proving the positive, then the converse), or writing a proof by induction (by proving the base case, then writing the induction step), I think there is a general method for this.

The specific example is how to show that for a bijection $f:A \rightarrow B$, the following is true: $f^{-1} \circ f = {\text {id}}_{A}$, where $\text{id}$ is the identity function. The intuition itself seems obvious to me, I just don't know how to best write the proof!

Answer 1

Two functions $f$ and $g$ are equal if and only if they have the same domain and $f(x) = g(x)$ for all $x$ in the domain. So in your example you will have to prove the following assertions:

• The domain of $f^{-1} \circ f$ is equal to $A$ (the domain of id$_A$)
• For any $x \in A$ we have $(f^{-1} \circ f) (x) =$ id$_A(x)$