Proving finite sets

**Jdg6057** · September 21st 2009, 08:56 AM

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

**Plato** · September 21st 2009, 09:11 AM

Originally Posted by Jdg6057

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

Suppose that $f:A \mapsto B$ is injective.
Note that means $x \in A,\,y \in A\;\& \,x \ne y\, \Rightarrow \,f(x) \ne f(y)$
Using that property; $\left( {\forall a \in A} \right)\left\{ {\{ f(a)\} } \right\}$ is collection of pair-wise disjoint subsets of $B$ .
If $B$ is finite then what can you say about $A$ ?

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