Set Theory – Definition of the Infinite Cartesian Product

elementary-set-theoryproducts

(1) If $X$ and $Y$ are two sets, we define the Cartesian product $X \times Y$ as the set of ordered pairs $(x,y)$, such that $x \in X$ and $y \in Y$.

(2) On the other hand [Folland, Real Analysis, page 4], if $\{X_\alpha\}_{\alpha \in A}$ is infinite indexed family of sets, their Cartesian product
$$
\prod_{\alpha \in A}X_\alpha
$$
is defined as the set of maps $f: A \to \bigcup\limits_{\alpha \in A} X_\alpha$ such that $f(\alpha) \in X_\alpha$ for every $\alpha \in A$.

After saying this, Folland remarks:

it should be noted, and promptly forgotten, that when $A = \{1,2\}$, the previous definition of $X_1 \times X_2$ [that's (1) above] is set-theoretically different from the present definition of $\prod_1^2 X_j$ [that's (2) above]. Indeed, the latter concept depends on the mappings, which are defined in terms of the former one.

I am not grasping this remark. Specifically, here are my questions.

Question 1: How is (2) set-theoretically different from (1)? A simple illustrative example?

Question 2: If (1) is extended to infinite families, which definition would be stronger? A simple illustrative example?

Question 3: Why should this be "promptly forgotten"?

I'll probably have more questions depending on the type of answers I'll get to these.

Thanks!

Best Answer

  1. An order pair and a function $f : \{1, 2\} \rightarrow X_1 \cup X_2$ are not the same thing. Depending on your definition, some people define the order pair $(a,b)$ to be the set $\{a, \{a,b\}\}$. A function is a subset of $X_1 \times X_2$ satisfing the usual property of functions. Hence set-theoretically they are different. "Indeed, the latter concept depends on the mappings, which are defined in terms of the former one." means that functions are usually defined as a subset of $X \times Y$, i.e. subset of the set of order pairs. Hence Folland is just remarking that the definition of functions uses the notion of order pairs.

  2. Folland only defines $(1)$ for finite cartesian product. So in the context of Folland's book, it doesn't make any sense to ask which definition is stronger for infinite families. According to folland, for infinite families there is only one notion of cartesian product, and that is the second one.

  3. By being "prompty forgotten", is that although order pairs and certain functions from $\{1, 2\} \rightarrow X_1 \cup X_2$ are not set theoretically the same, there is a very nice bijection $\Phi$ between the two concepts. $\Phi((x_1,x_2)) = f_{(x_1,x_2)}$ where $f_{(x_1, x_2)}$ defined by $f_{(x_1,x_2)}(1) = x_1$ and $f_{(x_1,x_2)}(2) = x_2$.

(Note that its inverse would then be : for any $f : \{0,1\} \rightarrow X_1 \cup X_2$ with the property that $f(i) \in X_i$, $f$ is map to $(f(1), f(2))$. )

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