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