How to write the Cartesian product of all elements of an uncountable set?

Question

Let $A$ be an uncountable set with arbitrary element $a\in A$. I want to define the Cartesian product of all its elements. In other words, all the vectors belonging to

\begin{gather} \underbrace{A\times\dots\times A}_{A} \end{gather}

Suppose $A=\{a_1,a_2\}$. Then, I am looking for the notation that generalises the following expression to the case in which the set $A$ is uncountable:

\begin{gather} A\times A\equiv\{(a_1,a_1),(a_1,a_2),(a_2,a_1),(a_2,a_2)\} \end{gather}

I have thought of two possibilities, but both seem unsatisfactory:

1. To define $\prod_{a\in A}i(a)$, where $i:A\to A$ is the identity function (i.e., $i(a)=a$ for all $a\in A$);
2. To simply define $\prod_{a\in A}a$.

Any help will be much appreciated.

Thank you all.

Answer 1

I’m not exactly sure what you want. From your text I suppose you want $$ \prod_{a\in A} a$$ from your formulae I get you want $$ \prod_{A}A = A^A $$ The latter can be understood as all functions from $A$ to $A$. The first one can be understood as all maps $f$ from $A$ to $\bigcup A$ so that for $a\in A$ we have $f(a)\in a$.

Answer 2

Since I'm not able to comment yet, I'll post as an answer. It is $\prod_{A}A$ - and not $\prod_{a\in A}a$ - the set you're pointing out. Cartesian products are defined between sets, not elements of a set. Note that in your example $A$ is in fact finite. When $A$ is uncountable, all the "$A$-indexed vectors", with coordinates in $A$, would be denoted by $$\prod_{A}A$$