Set comprehension when the condition is false - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:35:27Z http://mathoverflow.net/feeds/question/17247 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17247/set-comprehension-when-the-condition-is-false Set comprehension when the condition is false M.S. 2010-03-06T00:07:56Z 2010-03-06T01:29:01Z <p>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?</p> <p>(It would be nice if you point out which concept I mis-understand: the set comprehension, or the tuple.)</p> <p>Thanks in advance.</p> <p>[edit: I should add the following link: <a href="http://en.wikipedia.org/wiki/Empty_product#Nullary_Cartesian_product" rel="nofollow">Wikipedia: Empty_product#Nullary_Cartesian_product</a>]</p> http://mathoverflow.net/questions/17247/set-comprehension-when-the-condition-is-false/17253#17253 Answer by mathy for Set comprehension when the condition is false mathy 2010-03-06T01:29:01Z 2010-03-06T01:29:01Z <p>You're misinterpreting the cartesian product. Your link to La Wik describes the cartesian product of <i>no sets</i>, 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.</p> <p>The cartesian product of two empty sets is the cartesian square of the empty set, which is empty.</p>