[Naive Set Theory] Section 6 Ordered Pairs

Naive Set Theory

Section 6 Ordered Pairs


前置要点 Highlights

P. 22 Para. 1: Even without a precise definition of what this means, we can do something set-theoretically intelligent with it. We can, namely, consider, for each particular spot in the ordering, the set of all those elements that occur at or before that spot; we obtain in this way the sets {c} {c, b} {c, b, d} {c, b, d, a}.

  • 将会用集合去定义有序对,并且基本思路是逐一确定位置.

阐明 Exposition

  • 有序对 (ordered pairs)

    • 考虑一个简单情况,即元素$a,b$的有序对,其中第一维是 $a$,第二维是$b$,该有序对记作$(a,b)$被定义为$(a,b)=\{\{a\},\{a,b\}\}$.
  • 笛卡儿积 (Cartesian product)

    • 二元形式的笛卡儿积被表示为$A\times B = \{x:x=(a,b), a\in A\land b\in B\}$.
    • $R$是一些有序对的集合,那么必存在集合$A,B$,使得$R\subset A\times B$.
  • 投影
    • $A=\{a:\exists b,(a,b)\in R\}$$R$在第一维上的投影.
    • $B=\{b:\exists a,(a,b)\in R\}$$R$在第二维上的投影.

习题 Exercise

  • 证明 $(A\cup B)\times X = (A\times X)\cup(B\times X)$.
    &\forall x \in (A\cup B)\times X, x = (a,b) ,\\
    & a\in A\cup B, b\in X\\
    \Leftrightarrow & a\in A \lor a\in B, b\in X\\
    \Leftrightarrow & (a\in A\land b\in X)\lor (a\in B\land b\in X)\\
    \Leftrightarrow & (a,b)\in A\times X\lor (a,b)\in B\times X\\
    \Leftrightarrow & (a,b)\in (A\times X)\cup (B\times X)\\
    & \mathrm{Q.E.D.}
  • 证明$(A\cap B)\times (X\cap Y) = (A\times X)\cap(B\times X)$.
    &\forall x \in (A\cap B)\times (X\cap Y), x = (a,b) ,\\
    & a\in A\cap B, b\in X\cap Y\\
    \Leftrightarrow & a\in A\land a\in B \land b\in X \land b\in Y\\
    \Leftrightarrow & (a\in A\land b\in X)\land(a\in B\land b\in Y)\\
    \Leftrightarrow & (a,b)\in (A\times X)\land(a,b)\in(B\times Y)\\
    \Leftrightarrow & (a,b)\in (A\times X)\cap (B\times Y)\\
    & \mathrm{Q.E.D.}
  • 证明$(A-B)\times X =(A\times X) - (B\times X)$.
    &\forall x \in (A-B)\times X, x = (a,b) ,\\
    & a\in A-B, b\in X\\
    \Leftrightarrow & (a\in A\land a\notin B)\land b\in X\\
    \Leftrightarrow & (a\in A\land b\in X)\land(a\notin B\land b\in X)\\
    \Leftrightarrow & (a,b)\in A\times X\land (a,b)\notin B\times X\\
    \Leftrightarrow & (a,b)\in (A\times X)-(B\times X)\\
    & \mathrm{Q.E.D.}
  • 证明$A=\varnothing\lor B=\varnothing \Leftrightarrow A\times B=\varnothing$.
  • 证明$A\subset X\land B\subset Y \Leftrightarrow A\times B\subset X\times Y$.
    &\forall x \in A\times B, x = (a,b) ,\\
    & a\in A, b\in B, A\subset X, B\subset Y\\
    \Rightarrow & a\in X, b\in Y\\
    \Leftrightarrow & (a,b)\in X\times Y\\
    \Leftrightarrow & A\times B\subset X\times Y \\
    & \mathrm{Q.E.D.}