## Set comprehension when the condition is false

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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]

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If $A = \emptyset$ is empty, then A x B is empty for any B. Note that the empty set is NOT equal to {$\emptyset$} – Jason DeVito Mar 6 2010 at 0:12

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I believe you're confusing "the empty product" with "the product of empty sets." The former is a singleton but the latter is empty. – François G. Dorais♦ Mar 6 2010 at 0:35

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I don't think this question is suitable for MO. – Andrea Ferretti Mar 6 2010 at 1:19

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The first equality in your equation is true, but for reasons probably too confusing for your own good, the second equality, on the other hand, is false. – Mariano Suárez-Alvarez Mar 6 2010 at 3:26

## 1 Answer

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.

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I now see where is my confusion. Thanks – M.S. Mar 6 2010 at 20:25
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