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Proof:

Suppose $X$ and $Y$ are any infinite sets.

Then

$$ \exists f:X \rightarrow X \text{ such that}\; f \;\text{ is injective }\;\land\; f(X) \neq X,\;\;\text{and}$$ $$\exists g:Y \rightarrow Y \text{ such that}\;\; g \text{ is injective}\;\;\land\;\;g(Y) \neq Y.$$

I'm sure that it's simple, but I don't see what I should do after this.

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    Don't be afraid to use more words to make your argument; if you need to have a proof in mostly symbolic notation, you can translate to logical notation once you tackle the proof in words expressing what you need to express.2012-12-11

2 Answers 2