Elementary Classical Analysis, Marsden, 2nd ed, Introduction, Problem 3

Problem

Let $f:A \rightarrow B$ be a function, $C_1,C_2 \subset B$, and $D_1,D_2 \subset A$. Prove

(a) $f^{-1}(C_1 \cup C_2) = f^{-1}(C_2) \cup f^{-1}(C_2)$

(b) $f(D_1 \cup D_2) = f(D_1) \cup f(D_2)$

(c) $f^{-1}(C_1 \cap C_2) = f^{-1}(C_1) \cap f^{-1}(C_2)$

(d) $f(D_1 \cap D_2) \subset f(D_1) \cap f(D_2)$


Answer

(a) $x \in f^{-1}(C_1 \cup C_2)$  $\Leftrightarrow$  there exists $y \in C_1 \cup C_2$ such that $y = f(x)$  $\Leftrightarrow$  there exists $y_1 \in C_1$ such that $y_1 = f(x)$ or there exists $y_2 \in C_2$ such that $y_2 = f(x)$  $\Leftrightarrow$  $x \in f^{-1}(C_1)$ or $x \in f^{-1}(C_2)$  $\Leftrightarrow$  $x \in f^{-1}(C_1) \cup f^{-1}(C_2)$

(b) $y \in f(D_1 \cup D_2)$  $\Leftrightarrow$  $y=f(x)$ for some $x \in D_1 \cup D_2$  $\Leftrightarrow$  $y=f(x_1)$ for some $x_1 \in D_1$ or $y=f(x_2)$ for some $x_2 \in D_2$  $\Leftrightarrow$  $y \in f(D_1)$ or $y \in f(D_2)$  $\Leftrightarrow$  $y \in f(D_1) \cup f(D_2)$

(c) $x \in f^{-1}(C_1 \cap C_2)$  $\Rightarrow$  there exists $y \in C_1 \cap C_2$ such that $y = f(x)$  $\Rightarrow$  $x \in f^{-1}(C_1)$ and $x \in f^{-1}(C_2)$  $\Rightarrow$  $x \in f^{-1}(C_1) \cap f^{-1}(C_2)$

$x \in f^{-1}(C_1) \cap f^{-1}(C_2)$  $\Rightarrow$  $x \in f^{-1}(C_1)$ and $x \in f^{-1}(C_2)$  $\Rightarrow$  there exists $y_1 \in C_1$ and $y_2 \in C_2$ such that $y_1 = f(x) = y_2$  $\Rightarrow$  there exists $y \in C_1 \cap C_2$ such that $y = f(x)$  $\Rightarrow$  $x \in f^{-1}(C_1 \cap C_2)$

(d) $y \in f(D_1 \cap D_2)$  $\Rightarrow$  $y=f(x)$ for $x \in D_1 \cap D_2$  $\Rightarrow$  $y \in f(D_1)$ and $y \in f(D_2)$  $\Rightarrow$  $y \in f(D_1) \cap f(D_2)$

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