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I am working on a set equivalent (the textbook refers as "equinumerous" denoted by ~) as follows:

If $S$ and $T$ are sets, prove that if $(S\backslash T) \sim (T\backslash S)$, then $S \sim T$.

Here is my own proof, I am posting it here wanting to know if it is valid. (It may not be as elegant as the textbook's proof though.)

(1) We do not consider the cases of $S \subseteq T$, $T \subseteq S$ or $S \cap T = \emptyset$ since they lead to trivial solutions. We consider only the case where $S \cap T \ne \emptyset$. See the Venn diagram above.

(2) $(S \backslash T) \sim (T \backslash S)$ implies the existence of bijection $f: (S\backslash T) \to (T\backslash S)$. Here we need to prove the existence of bijection $g: S \rightarrow T$ to show $S \sim T$.

(3) Looking at the diagram, we need to prove only the existence of bijection $h: (S \cap T) \rightarrow T$ in order to prove the existence of bijection $g: S \rightarrow T$, since the bijection $f: (S \backslash T) \rightarrow (T \backslash S)$ is given.

(4) For all $x \in S \cap T,$ we have $ x \in S$ and $x \in T$. Hence the function $h: (S \cap T) \rightarrow T$ is in fact $h: T \rightarrow T$, which is an identity function. Since by textbook's theorem an identity function is always bijective, therefore $h: (S \cap T) \rightarrow T$ is indeed bijective.

(5) Hence there exists bijection $g: S \rightarrow T$, implying that $S \sim T$, as desired.

Thank you so much for your time and gracious help. ~MA

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