Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there a set theoretic construction of the natural numbers or integers such that the product of two numbers is their Cartesian product? What I mean is, e.g., if $25 = A$ and $2 = B$ then $50 = A\times B = \{(a,b)\mid a\in A \land b\in B\}$.

share|cite|improve this question
Almost certainly not, since from $X\times Y$ we can determine $X$ and $Y$, so you'd have trouble with $3\times 4=2\times 6$, or even $1\times 6=2\times 3$ – Thomas Andrews Jan 10 '13 at 16:34
@Thomas: In fact, we can determine $X$ and $Y$ from $X \times Y$ whenever $X$ and $Y$ are nonempty. But of course we can only use the empty set once. – Chris Eagle Jan 10 '13 at 16:36
Can someone clarify the question please? It looks like A and B are being used as numbers $and$ sets. A number is not a set. – Adam Rubinson Jan 10 '13 at 19:07
@AdamRubinson in this case, numbers are sets. That's what is meant by "set theoretic construction". – Massey Cashore Jan 10 '13 at 19:23
Alright fair enough. I have no idea about it then – Adam Rubinson Jan 10 '13 at 19:59
up vote 4 down vote accepted

If this property holds, then at most one number is represented by the empty set. Let $n,k$ be two numbers which are not empty sets.

We have that $n\times k=k\times n$. Therefore $(a,b)$ appears in both and so $a\in n$ implies that $a\in k$ and similarly $b\in k$ implies that $b\in n$. So $n=k$.

This means that all the numbers which are non-empty sets are equal, which is impossible.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.