Let $f:X \rightarrow Y$ be a bijection with inverse function $f^{-1}: Y \rightarrow X.$ Show that $f^{-1}(f)$ and $f(f^{-1})$ are the identity functions on X and Y respectively.
My approach for the first part of the problem was to try to show that for $x \in X$, we need to show that $f^{-1}(f(x)) = x$ I feel like I can use the fact that f is one to one here somehow, but i'm not sure how to get started.
Best Answer
To show that $f^{-1}\circ f$ is the identity on $X$, let $x \in X$. Since $f$ is a bijection, $f$ is an injection, so $f(x)$ is the unique element in $Y$ such that $f^{-1}(f(x)) = x$. Hence $f^{-1}\circ f = \text{id}_X$. The other direction is similar. Can you see how to show $f\circ f^{-1} = \text{id}_Y$?