# What if union of disjoint sets results in universal set?

I have a question related to set theory.

If $A_1,A_2,A_3\dots, A_n$ belongs to universal set $U$, and if all of the sets are disjoint i.e. $A_i \cap A_j = \emptyset$ for all $i$ and $j$.

And If their union equals to Universal set i.e. $A_1 \cup A_2 \cup \dots A_n = U$ = Universal set.

What is such situation called?

## migrated from mathoverflow.netMar 16 '14 at 20:20

This question came from our site for professional mathematicians.

• It's called a Partition! – Bjørn Kjos-Hanssen Mar 16 '14 at 19:12
• @Bjørn: If none of the sets is empty, of course. – Asaf Karagila Mar 16 '14 at 20:35
• Thanks a lot for answer.Sorry couldnt acknowledge answer as I was not well.My hearty thanks and sincerest apologies to naslundx,bjorn and asaf. – Pruthviraj Chauhan Apr 15 '14 at 15:13

If every $A_i$ is nonempty, you have described a partition of the set $U$.
The definition holds for any set, not just for the current universal set. If the union of disjoint (non-empty) sets equal any set $X$, we say in the same way that we have a partition of $X$.
• You have to require that none of the $A_i$ are empty, though. – Asaf Karagila Mar 16 '14 at 20:35