What is the cross-product of the null set with another set?

Question

Supposing that, for example, we have two sets $A$ and $B$ where $\;A = \varnothing \;$ and $\,B = \{a,b\}$.

What is the result of the cross product of those sets?

My first intuition would be to say that the resulting set would be the empty set as well since you can't set up any ordered pairs of $B$.

Am I correct?

This is just one amongst the many duplicates this question have on this site. If one is so inclined it is likely to find at least two more duplicates. — Asaf Karagila, Mar 05 '13 at 00:28
As a result the only relation on $A\times B$ is the so-called empty function with codomain $B$. — yearning4pi, Mar 05 '13 at 00:38
@RahulNarain: And in the comments, and related links, of that duplicate there are more duplicates... — Asaf Karagila, Mar 05 '13 at 00:39

amWhy · Accepted Answer · 2013-03-05T00:48:46.107

8

Yes, you're correct: For all sets $B$, if $A = \varnothing$, then $$A \times B = \varnothing \times B = B\times A = B\times \varnothing = \varnothing$$ and for the reason you argue: there exist no ordered pairs in $\varnothing \times B$, by the definition of the Cartesian Product.

edited Mar 05 '13 at 00:48

answered Mar 05 '13 at 00:29

amWhy

209,954

score 6 · Answer 2 · answered Mar 05 '13 at 00:39

Yes, you are correct.

The cartesian product is defined as $$A \times B = \{(a,b) \mid a\in A, b\in B\}.$$

In the case that one or both of the sets $A$ and $B$ are empty, there is no single index pair $(a,b)$ such that $a\in A$ and $b\in B$, which means $A\times B = \emptyset$.

What is the cross-product of the null set with another set?

2 Answers2