Give an example of two sets A, B and functions $f : A \rightarrow B$ and $g : B \rightarrow A$ that satisfy these conditions:

Question

Give an example of two sets A, B and functions $f: A \rightarrow B$ and $g: B \rightarrow A$ that satisfy the following three conditions

1. Both $f$ and $g$ are onto;
2. $f(g(x))=x$ for all $x$ in B; and
3. There exists $y$ in A such that $g(f(y)) \ne y$.

If you think that such sets and functions do not exist explain why.

Answer 1

Hint: Given (1), if $y\in A$, and $g$ is onto, $y=g(x)$ for some $x\in B$.

Given (2), what can you say about $g(f(y))=g(f(g(x)))$?