Let f be a one-to-one function from A to B, with B being finite. Prove that A is finite.

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- Sep 21st 2009, 08:56 AMJdg6057Proving finite sets
Let f be a one-to-one function from A to B, with B being finite. Prove that A is finite.

- Sep 21st 2009, 09:11 AMPlato
Suppose that $\displaystyle f:A \mapsto B$ is injective.

Note that means $\displaystyle x \in A,\,y \in A\;\& \,x \ne y\, \Rightarrow \,f(x) \ne f(y)$

Using that property; $\displaystyle \left( {\forall a \in A} \right)\left\{ {\{ f(a)\} } \right\}$ is collection of pair-wise disjoint subsets of $\displaystyle B$.

If $\displaystyle B$ is finite then what can you say about $\displaystyle A$?