Understanding the definition of cartesian product for an arbitrary number of sets My book defines the cartesian product of an arbitrary number of sets this way:
It first defines a $J$-tuple of elements of $X$, by saying: let $J$ be an index set. Given a set $X$, we define a $J$-tuple of elements of $X$ to be a function $x:J\to X$. If $\alpha$ is an element of $J$, we often denote the value of $x$ at $\alpha$ by $x_{\alpha}$ , we call it the $\alpha$th coordinate of $x$, and we often denote the function $x$ itself by the symbol
$$(x_\alpha)_{\alpha\in J}$$
First of all, how can he talk about an $\alpha$th coordinate of $x$, when $x$ is just the image of $\alpha$?
Then, the cartesian product is defined this way:

Let $\{A_\alpha\}_{\alpha\in J}$ be an indexed family of sets; let $X\supseteq \cup_{\alpha\in J} A_\alpha$. The cartesian product of this indexed family, denoted by $\Pi_{\alpha\in J} A_\alpha$ is defined to be the
  set of all $J$-tuples $(x_\alpha)_{\alpha\in J}$ of elements of $X$
  such that $X_\alpha\in A_\alpha$ for each $\alpha \in J$. That is, it
  is the set of all functions $$x:J\to \cup_{\alpha\in J}A_\alpha$$
such that $x(\alpha)\in A_\alpha$ for each $\alpha\in J$

Could somebody give me a concrete example, using this definition, of a cartesian product that is simply $X\times Y$? I really didn't understand this definition.
 A: Let $X = \mathbb R $.  Let $J=\{1,2\} $.
Let $x :J\rightarrow \mathbb R$ be the function $x (1)=23.5;x (2)=16.7$.
For notation purposes we write $x_1=x (1)=23.5$ and $x_2 = x (2)=16.7$.  And for notation purposes we write $x = (23.5, 16.7)$.
We define $\mathbb R \times \mathbb R = \{x:J\rightarrow \mathbb R\} = \{\{x(1),x (2)\}| x:J\rightarrow \mathbb R\}=\{(x_1,x_2)|x_1,x_2\in \mathbb R\} $.
In short $\mathbb R \times \mathbb R $ is the set of ordered pairs.  But we have to define what on "ordered" pair is.  Well, in set vocabulary you "order" something by having an indexing function act on elements of a universe set.
It seems very abstract and strange at first but .... it works.
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Going to make a more detailed solution:
Let $U = \mathbb R$.  Let $X = \mathbb Q$.  Let $Y = \mathbb N$. Let $J = \{1,2\}$.
Let $x:J \rightarrow \{3, 22/7\}$ be the function $x: 1\rightarrow 22/7;2\rightarrow 3$.  For notation purposes we can write $x = \{x(1),x(2)\} = (x_1, x_2) = (22/7, 3)$.
Let $y:J \rightarrow \{5/9, 6\}$ be the function $y: 1\rightarrow 5/9;2\rightarrow 6$.  For notation purposes we can write $y = \{y(1),y(2)\} = (y_1, y_2) = (5/9, 6)$.
Let $\{22/7,5/9\} = \{x(1),y(1)\} \subset A_1 \subset \mathbb R$ for some set $A_1$ which I will define later.Let $\{3,6\} = \{x(2),y(2)\} \subset A_2 \subset \mathbb R$ for some set $A_2$ which I will define later.
Let $V = \{x|J \rightarrow \mathbb R; x(1) \in \mathbb Q; x(2) \in \mathbb N\}$.  
We can notice that each $x \in V$ is a function $x:J \rightarrow \mathbb Q \cup \mathbb N= \mathbb Q \subset \mathbb R$ but there isn't really any reason to note it.
Let $A_1 = \{x(1)|x \in V\}$ and $A_2 = \{x(2)| x \in V\}$.  Notice $A_1 = \mathbb Q$ and $A_2 = \mathbb N$.  And notice $x:J \rightarrow A_1 \cup A_2 \subset \mathbb R$ which is slightly more pertinent when view as the union of the collected images of the element 1 and the collected images of the element 2.
Back to  $V = \{x|J \rightarrow \mathbb R; x(1) \in \mathbb Q; x(2) \in \mathbb N\}= \{x|J \rightarrow \mathbb R; x(1) \in A_1=\{\text{the collected images of } x(1) \forall x\in V\}=\mathbb Q; x(2)  \in A_2=\{\text{the collected images of }x(2) \forall x\in V\}=\mathbb N\}$ 
We define $V$ as $\mathbb Q \times \mathbb N$.
Now for example, we know that somewhere in $\mathbb Q \times \mathbb N$ there is an element $z$ where $z:\{1,2\}\rightarrow \mathbb Q \cup \mathbb N$ where $z = (89/92,47)$ where $(89/92,47)$, if viewed as a function, maps $ \{1,2\} \rightarrow \{89/92,47\}$ via $(89/92,47):1\rightarrow 89/92; 2\rightarrow 47$.
Okay, I admit it is really weird to think of it in those terms when it's so much easier to think of it as " $89/92$ attached to $47$ to make an ordered pair" but this way will work to be set theory consistent, and as Math100 pointed out in the comments, to allow for a definition for uncountable-tuples.
