Let $$Y=\prod_{X\in \emptyset}X$$

What is $Y$? The empty set? The singleton set?

EDIT: Here is another way of reading the conclusion in the comments so far:

$Y$ is the set $$Y=\{(x_1,...,x_n) | x_1\in X_1,...,x_n\in X_n, \text{ for } X_1,...,X_n\in \emptyset\}$$

But the only tuple satisfying this is $()$, i.e. the empty tuple. Hence $Y$ is the singleton. Is this argument correct?

  • $\begingroup$ It has one element, and that element is the function whose domain is the empty set. $\endgroup$ – bof Apr 28 '19 at 5:48
  • $\begingroup$ @bof, can you give me a function whose domain is the empty set? It seems to me that no such function exists. $\endgroup$ – user56834 Apr 28 '19 at 5:58
  • $\begingroup$ @user56834 In set theory, a function $f : X \to Y$ is just a particular type of subset of $X \times Y$. So, a function whose domain and codomain are empty, if it existed, would be some subset of $\emptyset \times \emptyset = \emptyset$. But there is only one such subset: $\emptyset$. So that's the function they were referring to. $\endgroup$ – 0XLR Apr 28 '19 at 6:04
  • $\begingroup$ @ZeroXLR, see my edit. is this correct? $\endgroup$ – user56834 Apr 28 '19 at 7:31
  • $\begingroup$ You should not think of elements of the general cartesian product as “tuples” (except in the special case of elements of a cartesian product of two sets, which is needed to define functions). It makes no sense for arbitrary index sets. The elements of the cartesian product are functions with domain the index set and codomain the union of the sets in. the family. Writing it as a tuple just re-phrases the problem, it does not solve it. $\endgroup$ – Arturo Magidin Apr 28 '19 at 17:10

By definition, $\prod_{i\in I}X_i$ is the collection of all functions $f\colon I \to \cup_{i\in I}X_i$ such that for all $i\in I$, $f(i)\in X_i$.

Thus, by definition, your $Y$ is the collection of all functions $f\colon \varnothing \to \cup_{X\in\varnothing}X$ such that for all $X\in\varnothing$, $f(X)\in X$. But $\cup_{X\in\varnothing}X = \varnothing$. So you are looking for all functions $f\colon \varnothing\to\varnothing$ which satisfy a vacuous property.

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    $\begingroup$ There is exactly one function from $\emptyset$ to $\emptyset$ so we get a singleton set. Just like the empty product in a group is the unit element. $\endgroup$ – Henno Brandsma Apr 28 '19 at 6:17
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    $\begingroup$ @HennoBrandsma, see my edit. Is this correct? $\endgroup$ – user56834 Apr 28 '19 at 7:31
  • $\begingroup$ @user56834 on what grounds is an empty tuple a singleton? Seems to replace the problem by another one. $\endgroup$ – Henno Brandsma Apr 28 '19 at 7:45

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