I have this question and it seems a tad redundant
If $A$ and $B$ are infinite sets, is it possible for there to be a 1-1 function from $A$ to $B$ and a 1-1 function from $B$ to $A$ without there being a 1-1 correspondence from $A$ to $B$?
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Thomas Andrews
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oliverjones
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1No, see the Schroder-Bernstein Theorem. – Plutoro Jul 01 '15 at 23:19
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1What seems redundant? The only word that can be deleted is "infinite". – Robert Israel Jul 01 '15 at 23:21
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1You’ll find the Cantor-Schröder-Bernstein theorem here, with more information here. – Brian M. Scott Jul 01 '15 at 23:22
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$1-1$ correspondence is probably taken to mean something different from a $1-1$ function. It probably means a $1-1$ and onto function and/or a function with a left-and-right inverse. – Thomas Andrews Jul 01 '15 at 23:30
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That's not a redundant question at all! If I've understood you right, you want to know whether there could be an injection $A \rightarrow B$ and an injection $B \rightarrow A$ without there being a bijection between $A$ and $B$.
Actually, this is not possible.
There is a theorem called the Schröder–Bernstein theorem that states: Whenever there is an injection $f:A \rightarrow B$ and an injection $g:B \rightarrow A$, then there exists a bijection $h:A \rightarrow B$.
Mathematics
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