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\title[Math 250 Handout 10 - Preimages]{Math 250 Handout 10  - Preimages}\maketitle


  Let $f\colon A \to B$ be a function, let $W \subset A$, and let $X,Y \subset B$. Prove
  or disprove each of the following:  
\begin{enumerate}
  \item $f^{-1}(X \cup Y) \subset f^{-1}(X) \cup f^{-1}(Y)$.
  \item $f^{-1}(X \cup Y) \supset f^{-1}(X) \cup f^{-1}(Y)$.
  \item $f^{-1}(X \cap Y) \subset f^{-1}(X) \cap f^{-1}(Y)$.
  \item $f^{-1}(X \cap Y) \supset f^{-1}(X) \cap f^{-1}(Y)$.
  \item $W \subset f^{-1}(f(W))$.
  \item $W \supset f^{-1}(f(W))$.
  \item (HW) $X \subset f(f^{-1}(X))$.
  \item (HW) $X \supset f(f^{-1}(X))$.
\end{enumerate}  

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