数学 – 代数学 - 群 - 写像(包含関係、逆像)

代数系入門

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代数系入門 (松坂 和夫(著)、岩波書店)の第2章(群)、1(写像)、問題1.を取り組んでみる。

問題1.

1. 
   $\begin{array}{l}x\in f\left(U\cup W\right)\\ \iff \\ x\in f\left(U\right)\vee x\in f\left(W\right)\\ \iff \\ x\in f\left(U\right)\cup f\left(W\right)\end{array}$
   
   $\begin{array}{l}x\in f\left(U\cap W\right)\\ \Rightarrow \\ x\in f\left(U\right)\wedge x\in f\left(W\right)\\ \Rightarrow \\ x\in f\left(U\right)\cap f\left(W\right)\end{array}$
2. 
   $\begin{array}{l}x\in f^{-1}\left(U\text{'}\cup W\text{'}\right)\\ \iff \\ f\left(x\right)\in U\text{'}\cup W\text{'}\\ \iff \\ f\left(x\right)\in U\text{'}\vee f\left(x\right)\in W\text{'}\\ \iff \\ x\in f^{-1}\left(U^{\prime }\right)\vee x\in f^{-1}\left(W\text{'}\right)\\ \iff \\ x\in f^{-1}\left(U\text{'}\right)\cup f^{-1}\left(W\text{'}\right)\\ \\ x\in f^{-1}\left(U\text{'}\cap W\text{'}\right)\\ \iff \\ f\left(x\right)\in U\text{'}\cap W\text{'}\\ \iff \\ f\left(x\right)\in U\text{'}\wedge f\left(x\right)\in W\text{'}\\ \iff \\ x\in f^{-1}\left(U\text{'}\right)\wedge x\in f^{-1}\left(W\text{'}\right)\\ \iff \\ x\in f^{-1}\left(U\text{'}\right)\cap f^{-1}\left(W\text{'}\right)\end{array}$
3. 
   $\begin{array}{l}x\in U\\ \Rightarrow \\ f\left(x\right)\in f\left(U\right)\\ \Rightarrow \\ x\in f^{-1}\left(f\left(U\right)\right)\end{array}$
4. 
   $\begin{array}{l}x\in f\left(f^{-1}\left(U\text{'}\right)\right)\\ \Rightarrow \\ x\in U\text{'}\end{array}$