# Infinite Cartesian Products

Yao Yao on July 24, 2018

Wikipedia: Sequence

Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first n natural numbers (for a sequence of finite length $n$)

Axiom 3.10 (Power set axiom). Let $X$ and $Y$ be sets. Then there exists a set, denoted $Y^X$, which consists of all the functions from $X$ to $Y$, thus

$f \in Y^X \Rightarrow f \text{ is a function with domain } X \text{ and range } Y$
• 亦即 $Y^X = \lbrace f \mid f \text{ is a function with domain } X \text{ and range } Y \rbrace$
• 注意是 “以 $Y$ 为 range” 而不是 “以 $Y$ 为 codomain”

Definition 8.4.1 (Infinite Cartesian products). Let I be an index set (possibly infinite), and for each $i \in I$ let $X_i$ be a set. We then define the Cartesian product $\underset{i \in I}{\prod} X_i$ to be the set

$\underset{i \in I}{\prod} X_i = \lbrace (x_i)_{i \in I} \in (\underset{j \in I}{\cup} X_j)^I \mid \forall i \in I, x_i \in X_i \rbrace$

• $I = \lbrace 1,2,3 \rbrace$
• $\underset{j \in I}{\cup} X_j = \lbrace a_1, a_2, a_3 \rbrace$
• $(\underset{j \in I}{\cup} X_j)^I$ 是所有 $f: \lbrace 1,2,3 \rbrace \rightarrow \lbrace a_1, a_2, a_3 \rbrace$ 的集合
• 条件 $\forall i \in I, x_i \in X_i$ 其实应该理解为 $\forall i \in I, f(i) \in X_i$。这样的 $f$ 只有一个，即 $f(1) = x_1 = a_3, f(2) = x_2 = a_3, f(3) = x_3 = a_1$
• 这个 $f$ 亦即 sequence $(a_3, a_2, a_1)$