This question is from the book Abstract Algebra: Theory and Applications by Thomas W. Judson.
Chapter 1 – Question 24
Let $f : X \to Y$ be a map with $A_1, A_2 \subset X$ and $B_1, B_2 \subset Y$.
(a) Prove $f(A_1 \cup A_2) = f(A_1) \cup f(A_2)$.
(b) Prove $f(A_1 \cap A_2) \subset f(A_1) \cap f(A_2)$. Give an example in which equality fails.
(c) Prove $f^{-1}(B_1 \cup B_2) = f^{-1}(B_1) \cup f^{-1}(B_2)$, where $f^{-1}(B) = \{x \in X : f(x) \in B\}$.
(d) Prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$.
(e) Prove $f^{-1}(Y \setminus B_1) = X \setminus f^{-1}(B_1)$.