Let A and B be nonempty sets and $\displaystyle \pi_1 : A \times B \rightarrow A $ be the projection function $\displaystyle \pi_1 (x,y) = x $ for $\displaystyle (x,y) \epsilon A \times B $.

(a) For $\displaystyle C \subset A $, determine the inverse image set $\displaystyle \pi^{-1} (C) $.

(b) For $\displaystyle C \subset A, D \subset B $ and $\displaystyle D \neq \emptyset $, determine the image $\displaystyle \pi_1 (C \times D) $.