Question: Let \(f:A\to B\) be a function.

(a) Show that the corresponding direct image function preserves all unions.

(b) Show that the corresponding inverse image function preserves all unions, intersections, and set complements.

(c) Prove or disprove the following statement: the inverse image function

\[\overset{\leftarrow }{\mathop{f}}\,:{{2}^{B}}\to {{2}^{A}}\]

is inverse to the direct image function

\[\overset{\to }{\mathop{f}}\,:{{2}^{A}}\to {{2}^{B}}\]