how to show the identiy of direct image

Question

This question in the test I wad writing yesterday but I did not attempt it because I didn't know what to do. The question reads: show that if $E\cap D=\{ \}$,then $f(E)=\{ \}$. Where $D$ is the domain of $f$ and and $\cap$ between $E$ and $D$ means intersection.where in both equalities the results are empty sets. I have learnt how to prove by arbitrary element when the image of the domain does not become an empty set. Can someone- help me here?

Answer 1

The only way for the notation $f(E)$ to make sense is for $E$ to be a subset of $f$'s domain $D$. In that case $E = E \cap D = \emptyset$, hence $f(E) = \emptyset$.