# How to denote an unordered product of two sets?

I would like to denote the following set:

$$\bigl\{\{a,b\}:a\in A\text{ and }b\in B\bigr\}$$

I have found this (Notation for unordered product of sets) but I do not know how this is answer my question. The answer says to use the notation $\binom{V}{2}$ but I have two sets $A$ and $B$ not only one $V$.

Thanks.

• I don't know of a symbol for this. It's probably fine if you make up your own as long as the definition is clear. – Jair Taylor Feb 15 '16 at 3:20
• why not use the description you gave: $\{\{a,b\}: a\in A, b\in B\}$? – Jorge Fernández Hidalgo Feb 15 '16 at 3:20
• To begin with, what if $A\cap B\neq \emptyset$? If $x\in A\cap B$, you wouldn't write $\{x,x\}$ as a set, as that would simply be the set $\{x\}$. – JMoravitz Feb 15 '16 at 3:20
• @JMoravitz I don't see any harm in this. – Stefan Mesken Feb 15 '16 at 3:22
• Thank you all, I will just use this one $\{\{a,b\},a\in A\text{ and } b\in B\}$ then. – Chiba Feb 15 '16 at 3:34

Assuming that $A$ is disjoint from $B$, you could denote this as:
$$\left\{\{a,b\}~:~a\in A,b\in B\right\}=\binom{A\cup B}{2}\setminus \left(\binom{A}{2}\cup \binom{B}{2}\right)$$
I.e. the family of all subsets of size two of $A\cup B$ such that the elements are neither both from $A$ nor are both from $B$, using the notation that $\binom{X}{2} = \{E~:~E\subseteq X,~|E|=2\}$, the family of all subsets of size two of $X$.
• If you wish to fix the above for use with sets that are not necessarily disjoint, you could try playing with unioning the above with things like $\binom{A}{2}\setminus \binom{A\setminus B}{2}$ to add back in the pairs which were accidentally removed. – JMoravitz Feb 15 '16 at 3:35