Give an example of sets $X, Y$ and functions $f: X \rightarrow Y$ and $g: Y \rightarrow X$ that satisfy the following...
- $g \circ f$ is a bijection
- $g \circ f$ is different from the identity function
- $f$ is not a surjection
- $g$ is not an injection
- $X$ contains two elements
Nothing I try satisfies all conditions simultaneously. There must be a better way to solve this than guessing/checking. Any help you can give would be awesome! Thanks!