# Proving finite sets

• Sep 21st 2009, 09:56 AM
Jdg6057
Proving 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, 10:11 AM
Plato
Quote:

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 $A$?