Partition of a set

Partition of a set --- разбиение множества.

A partition of a nonempty set [math]\displaystyle{ S }[/math] is a collection of pairwise disjoint nonempty subsets, whose union is [math]\displaystyle{ S }[/math]. If two partitions [math]\displaystyle{ \{A_{i}\} }[/math] and [math]\displaystyle{ \{B_{j}\} }[/math] of the same set are such that each [math]\displaystyle{ A_{i} }[/math] is a subset of some [math]\displaystyle{ B_{j} }[/math], then we say that the partition [math]\displaystyle{ \{A_{i}\} }[/math] is finer than the partition [math]\displaystyle{ B_{j} }[/math], and that [math]\displaystyle{ \{B_{j}\} }[/math] is coarser than [math]\displaystyle{ \{A_{i}\} }[/math].