数学 – 集合と写像 - 写像に関する諸概念(写像による像および原像, 和集合)

学習環境

集合・位相入門(松坂和夫(著)、岩波書店)の第1章(集合と写像)、4(写像に関する諸概念)、問題1.を解いてみる。

問題1.

(4.2)

$\begin{array}{l} P_{1} \subset P_{1} \cup P_{2} \\ f (P_{1}) \subset f (P_{1} \cup P_{2}) \\ f (P_{2}) \subset f (P_{1} \cup P_{2}) \\ f (P_{1}) \cup f (P_{2}) \subset f (P_{1} \cup P_{2}) \\ y \in f (P_{1} \cup P_{2}) \\ \Rightarrow \exists x \in P_{1} \cup P_{2} [f (x) = y] \\ \Rightarrow \exists x_{1} \in P_{1} [f (x_{1}) = y] \lor \exists x_{2} \in P_{2} [f (x_{2}) = y] \\ \Rightarrow y \in f (P_{1}) \lor y \in f (P_{2}) \\ \Rightarrow y \in f (P_{1}) \cup f (P_{2}) \\ f (P_{1} \cup P_{2}) \subset f (P_{1}) \cup f (P_{2}) \\ f (P_{1} \cup P_{2}) = f (P_{1}) \cup f (P_{2}) \end{array}$
(4.2)'

$\begin{array}{l} Q_{1} \subset Q_{1} \cup Q_{2} \\ f^{- 1} (Q_{1}) \subset f^{- 1} (Q_{1} \cup Q_{2}) \\ Q_{2} \subset Q_{1} \cup Q_{2} \\ f^{- 1} (Q_{2}) \subset f^{- 1} (Q_{1} \cup Q_{2}) \\ f^{- 1} (Q_{1}) \cup f^{- 1} (Q_{2}) \subset f^{- 1} (Q_{1} \cup Q_{2}) \\ x \in f^{- 1} (Q_{1} \cup Q_{2}) \\ \Rightarrow f (x) \in Q_{1} \cup Q_{2} \\ \Rightarrow f (x) \in Q_{1} \lor f (x) \in Q_{2} \\ \Rightarrow x \in f^{- 1} (Q_{1}) \lor x \in f^{- 1} (x) \in Q_{2} \\ \Rightarrow x \in f^{- 1} (Q_{1}) \cup f^{- 1} (Q_{2}) \\ f^{- 1} (Q_{1} \cup Q_{2}) \subset f^{- 1} (Q_{1}) \cup f^{- 1} (Q_{2}) \\ f^{- 1} (Q_{1} \cup Q_{2}) = f^{- 1} (Q_{1}) \cup f^{- 1} (Q_{2}) \end{array}$

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