If $U=\{1,2,3\}$, is $\{U\}$ a partition of $U$?

Question

From my understanding, a partition $F$ of a set $U$ means that $F$ is pairwise disjoint, and $\bigcup F=U$, and empty set is not in $F$. So if $U=\{1,2,3\}$, would $F =\{\{1,2,3\}\}$ counts as a partition? because it is the only set in the set, I don't have to check for its union or anything.

I do understand that $F=\{\{1\},\{2,3\}\}$ would be a partition.

Answer 1

Yes, if $U$ is a nonempty set then $F:=\{U\}$ is a partition of it.

"I don't have to check for its union or anything": yes you have to, but the three properties are easy to check:

• $\bigcup F=U$ $\checkmark$
• $\forall X\in F,\;X\ne\varnothing$, i.e. $U\ne\varnothing$ $\checkmark$
• $\forall X,Y\in F,\;(X\ne Y\implies X\cap Y=\varnothing)$ (vacuously true indeed) $\checkmark$.