수학적으로 set이 어떤 개념인지 파악해보자.
1. Axiom of extension.
\[\forall x\,(x\in A\leftrightarrow x\in B)\leftrightarrow A=B\]2. Belonging. a binary “relation” $\in$.
hierarchical한 느낌이다: [$n$-dimension] $\in$ [($n+1$)-dimension].
3. Inclusion.
\[A\subset B \leftrightarrow \forall x\,(x\in A \rightarrow x\in B).\]여기서 두 set은 same-dimensional로 보여진다.
$A\subset B\wedge B\subset C \rightarrow A\subset C\ (transitive)$
4. Axiom of specification.
\[\forall A\,\forall p\,\exists B \,\forall x\,(x\in B\leftrightarrow x\in A\wedge \varphi (x,p)),\]denoted by $B=\lbrace x\in A:\varphi(x,p)\rbrace.$
given set이 있어야 carve out하여 new one을 만들 수 있다.
5.1. Axiom of pairing.
\[\forall a\,\forall b\,\exists B\,\forall x\,(x\in B\leftrightarrow \varphi(x,a,b)) \text{ where } \varphi(x,a,b):x=a\lor x=b,\]denoted by $B=\lbrace a,b\rbrace$; the set is called the unordered pair.
We may refer to the axiom as a pseudo-special case of axiom of specification in the sense that if there were a universe, then the axiom would follow as a special case.
5.2. Notation. 앞으로 이런 generating axiom들이 추가될 예정이므로 이쯤에서 새 notation을 도입하자: If $\varphi(x,p)$ is a condition such that $x$s that $\varphi(x,p)$ specifies constitute a set, then we denote that set by $\lbrace x:\varphi(x,p)\rbrace.$
6.1. Axiom of unions. $\forall A\,\exists B\,\forall x\,( \exists y\,(y\in A\wedge x\in y)\rightarrow x\in B).$ There exists a comprehensive set. And, applying axiom of specification, we get
\[\forall A\,\exists B\,\forall x\,( \exists y\,(y\in A\wedge x\in y)\leftrightarrow x\in B),\]denoted by $B=\bigcup A$; the union of $A$.
6.2. Notation. We introduce special notation for a pair: $X\cup Y=\bigcup\lbrace X,Y\rbrace$. It follows that $X\cup Y=\lbrace x:x\in X\lor x\in Y\rbrace$ by general definition.
6.3. Definition. Now we can generalize pairs: $\lbrace a,b,c\rbrace =\lbrace a\rbrace \cup \lbrace b\rbrace \cup \lbrace c\rbrace , \text{etc.}$
Proving strategy
Goal: To explicate that a sentence is trivial.
Strategy 1: If a sentence has disjunction, then split into the cases.
7.1. Definition of complement. Relative complement of $B$ in $A$ is the set $A-B$ defined by
\[A-B=\lbrace x\in A:x\notin B\rbrace.\]Dealing with sets which are subsets of $E$, we can define absolute complement. Often used symbol for absolute complement of $A$ is $A’$.
7.2. Theorem (De Morgan laws). Basically, they are about unions and intersections:
\[(A\cup B)'=A'\cap B',\ (A\cap B)'=A'\cup B'.\]8. Axiom of powers. $\forall E\, \exists \mathcal{P}\, \forall x\, (x\subset E\rightarrow x\in \mathcal{P})$. And, applying axiom of specification, we get
\[\forall E\, \exists \mathcal{P}\, \forall x\, (x\subset E\leftrightarrow x\in \mathcal{P}).\]The set $\mathcal{P}$ is called the power set of $E$; the dependence of $\mathcal{P}$ on $E$ is denoted by $\mathcal{P}(E)$.