CSS

群论

[1] 设$\varphi$是集合$A$到$B$的一个映射,$C$和$D$分别为$A$和$B$的非空子集.证明:
a) $\varphi^{-1}(\varphi(C))\supseteq C$,且当$\varphi$为单射时等号成立;
b) $\varphi(\varphi^{-1}(D))\subseteq D$,且当$\varphi$为满射时等号成立.

a) 任取$x\in C$,$\varphi(x)\in\varphi(C)$,
$x\in\varphi^{-1}(\varphi(C))$,$\varphi^{-1}(\varphi(C))\supseteq S$.
若$\varphi$为单射,
任取$y\in\varphi^{-1}(\varphi(C))$,$\varphi(y)\in\varphi(C)$
故有$x\in C$,$\varphi(x)=\varphi(y)$.
由于$\varphi$是单射,
$x=y\in S$,
$\varphi^{-1}(\varphi(C))=C$

b) 任取$y\in\varphi(\varphi^{-1}(D))$,
存在$x\in\varphi^{-1}(D)$,$\varphi(x)=y$,
同时$\varphi(x)\in D$,$y\in D$,
$\varphi(\varphi^{-1}(D))\subseteq D$
若$\varphi$为满射,
$\forall z\in D$,$\exists w\in\varphi^{-1}(D),\varphi(w)=z$,
$\varphi(w)\in\varphi(\varphi^{-1}(D))$,
$\varphi(\varphi^{-1}(D))=D$

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