Question

The Cartesian product of two empty sets is the singleton set $\{ () \}$ containing the empty tuple. So, given a set $A$ which is empty, $A \times A $ is defined as: $$ A \times A = \{ (a,a) \mid a \in A \} = \{ () \} $$ Now, does that mean that $()$ satisfies the condition $a \in A$? And if so, why don't we include the empty tuple in the Cartesian product of non-empty sets?

(It would be nice if you point out which concept I mis-understand: the set comprehension, or the tuple.)

Thanks in advance.

[edit: I should add the following link: Wikipedia: Empty_product#Nullary_Cartesian_product]

Answer 1

You're misinterpreting the cartesian product. Your link to La Wik describes the cartesian product of no sets, i.e. the zero-th cartesian power of any set $A$. This is isomorphic to the space of functions from the empty set to $A$, which contains the empty function.

The cartesian product of two empty sets is the cartesian square of the empty set, which is empty.