[선형대수학] 극대 부분 집합, 체인, 조른의 보조정리, 선택공리 (Maximal Subset, Chain, Zorn's Lemma, Axiom of Choice)

$\underline{Def}$ (극대 부분 집합)

Let $\mathcal{F}$ be a set of sets. A set $M \in \mathcal{F}$ is called maximal if $\nexists K \in \mathcal{F}$ such that $M \subsetneq K$; that is, $A \in \mathcal{F}$ & $M \subset A \Rightarrow A=M$.

 

$\underline{Def}$

A set $\mathcal{C}$ of sets is called chain if $\forall A, B \in \mathcal{C},\; A \subset B$ or $B \subset A$.

 

$\underline{Maximal\;Principle}$ (Zorn's lemma)

Let $\mathcal{F}$ be a set of set. If for each chain $\mathcal{C} \subset \mathcal{F},\, \exists L_{\mathcal{C}} \in \mathcal{F}$ such that $A \subset L_{\mathcal{C}}\, \forall A \in \mathcal{C}$, then $\mathcal{F}$ contains at least one maximal set $M$.

 

$\underline{Def}$

Let $\mathcal{F}$ be a set of non-empty sets.

A function $\displaystyle f \colon \mathcal{F} \rightarrow \underset{A \in \mathcal{F}}{\cup}$ such that $f(A) \in A\; \forall A \in \mathcal{F}$ is called a choice function of $\mathcal{F}$

 

$\underline{Axiom\;of\;choic}$ (선택 공리)

For any set $\mathcal{F}$ of non-empyt sets, there exists a choice function $f$ on $\mathcal{F}$.