Need help with a proof dealing with functions

**steph3824** · November 19th 2009, 07:10 AM

Let f:A--->B be a function. Prove that f is surjective if and only if f^(-1)(W) does not equal the empty set for all nonempty sets W of B.

I really do not know what to do with this. Thanks for your help everyone

**Plato** · November 19th 2009, 07:37 AM

Originally Posted by steph3824

Let f:A--->B be a function. Prove that f is surjective if and only if f^(-1)(W) does not equal the empty set for all nonempty sets W of B.

Because $f$ is sujective $\left( {\forall b \in B} \right)\left( {\exists a \in A} \right)\left[ {f(a) = b} \right]$ . Is it possible for $f^{-1}(\{b\})$ to be empty?

Likewise if $\left( {\forall b \in B} \right)\left[ {f^{ - 1} (\{ b\} ) \ne \emptyset } \right]$ must $f$ be surjective?

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