partition of a set

A partition of a set \[X\] is a set of non-empty subsets of \[X\] such that every element \[x\] \[X\] is in exactly one of these subsets. Equivalently, a family of sets \[P\] is a partition of \[X\] if and only if all of the following conditions hold:

• The family \[P\] does not contain the empty set, \[\emptyset\notin P\]
  
• The union of the sets in \[P\] is equal to \[X\], \[\bigcup_{A\in P}A=X\]
  
• The intersection of any two distinct sets in \[P\] is empty, i.e. for all \[A\] and \[B\] in \[P\], \[A\] is not equal to \[B\], \[(\forall A,B\in P)A\ne B\implies A\cap B=\emptyset\]