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Using Schröder-Bernstein Theorem. Assume there exists a 1-1 function $f:X\to Y$ and another 1-1 function $g:Y\to X$. If we define $f^{−1}(y)=x$, then $f^{−1}$ is a 1-1 function from $f(X)$ onto $X$, and similarly $g^{-1}$ is a function from $g(Y)$ onto $Y$. Follow the steps to show that there exists a 1-1, onto function $h:X\to Y$.

Let $x \in X$ be arbitrary. Let the chain $C_x$ be the set consisting of all elements of the form

$$\dots,f^{−1}(g^{−1}(x)),g^{−1}(x),x,f(x),g(f(x)),f(g(f(x))),\dots$$

Show that any two chains are either identical or completely disjoint.

I've been stuck on this problem for a while, can anyone give me a hint that will push me in the right direction?

Thanks.

2 Answers 2