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Given an indexed family of sets $(X_i)_{i\in I}$, a canonical definition of Cartesian product is: $$ \prod_{i \in I} X_i = \left\{f\ \Big|\ f: I \longrightarrow \bigcup_{i \in I} X_i\ \wedge (\forall i\in I)\big(f(i) \in X_i\big)\right\} $$

I wonder if this definition is correct as well: $$ \prod_{i \in I} X_i = \left\{t\ \Big|\ t=(x_i)_{i\in I} \wedge (\forall i\in I)\big(x_i \in X_i\big)\right\} $$

Setting $\mathcal{X}=\bigcup_{i \in I} X_i$, the domain of each family $t$ (in the second notation) is not explicitly set, but values $t(i)$, that is $x_i$, should be in $X_i$ and so only those functions $t$ compatible with $\mathcal{X}$ are acceptable.
Also, the cardinality of $\mathsf{Rng} f$ (in the first notation) is usually smaller than the cardinality of $\mathcal{X}$ and the same as $I$. That is not possible for $t$, since families are surjections. So each $t$ should have a distinct domain, with the cardinality of $I$.

Update

I reply here to a comment regarding indexed families.
I define an indexed family as an alias for surjective function.

Given a surjective function: \begin{align} x\colon I &\longrightarrow X \\ i &\mapsto x_i = x(i), \end{align} it can be denoted also with: $$(x_i)_{i\in I}$$ The range of the function $x$ might be denoted with: $$\{x_i\}_{i\in I}$$ or $\{ x_i \ \big|\ i \in I \} $. This is the standard definition I found in several books, e.g. Tourlakis, "Lectures in Logic and Set Theory" or Wikipedia.

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  • $\begingroup$ What does $(x_i)_{i\in I}$ mean? You refer to it as a “family” and as a function, and you say “families are surjections.” I think it’s safer to define things in terms of more commonly-understood objects, like functions (in the top definition). $\endgroup$ – Steve Kass Jun 30 '18 at 21:48
  • $\begingroup$ Hi Antonio, the axiom of choice can be rewritten as "the Cartesian product of nonempty sets is a nonempty set itself." Hence, the first equality holds. The second equality is plainly odd. $\endgroup$ – Will M. Jun 30 '18 at 21:53
  • $\begingroup$ @SteveKass please, see the update on families $\endgroup$ – antonio Jun 30 '18 at 22:24
  • $\begingroup$ @WillM. The CP is a set whose elements are functions from $I$ to the bigunion. The indexed families in the second notation are functions from $I$ to a non-specified range, but the condition $x_i \in X_i$ should work as an implicit definition of the range. $\endgroup$ – antonio Jun 30 '18 at 22:30
  • $\begingroup$ @antonio a formula $\{x \mid p(x)\}$ is always assumed to be of the form $\{x \in S \mid p(x)\}$ where $S$ is a set. In other words, it is kind of a requirement of set theory that sets defined by "relations" are always subsets of a known set. $\endgroup$ – Will M. Jun 30 '18 at 22:39
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$t = (x_i)_{i \in I}$ already means that $t$ is a function defined on $I$. A family of sets indexed by $I$ is a function on $I$. The right codomain can be deduced.

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  • $\begingroup$ So, I understand this means the second definition is valid. $\endgroup$ – antonio Jul 1 '18 at 10:07
  • $\begingroup$ @antonio sort of. The first one is much better, being more explicit. $\endgroup$ – Henno Brandsma Jul 1 '18 at 10:29

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