Editing Bell number (section)

===Set partitions===
{{main article|Partition of a set}}
[[File:Bell numbers subset partial order.svg|thumb|right|Partitions of sets can be arranged in a partial order, showing that each partition of a set of size ''n'' "uses" one of the partitions of a set of size&nbsp;''n''&nbsp;−&nbsp;1.]]
[[File:Set partitions 5; circles.svg|thumb|The 52 partitions of a set with 5 elements]]
In general, <math>B_n</math> is the number of partitions of a set of size <math>n</math>. A partition of a set <math>S</math> is defined as a family of nonempty, pairwise disjoint subsets of <math>S</math> whose union is <math>S</math>. For example, <math>B_3 = 5</math> because the 3-element set <math>\{a,b,c\}</math> can be partitioned in 5 distinct ways:

:<math>\{ \{a\}, \{b\}, \{c\} \},</math>
:<math>\{ \{a\}, \{b, c\} \},</math>
:<math>\{ \{b\}, \{a, c\} \},</math>
:<math>\{ \{c\}, \{a, b\} \},</math>
:<math>\{ \{a, b, c\} \}.</math>

As suggested by the set notation above, the ordering of subsets within the family is not considered; [[Weak ordering|ordered partitions]] are counted by a different sequence of numbers, the [[ordered Bell number]]s. <math>B_0</math> is 1 because there is exactly one partition of the [[empty set]]. This partition is itself the empty set; it can be interpreted as a family of subsets of the empty set, consisting of zero subsets. It is [[vacuous truth|vacuously true]] that all of the subsets in this family are non-empty subsets of the empty set and that they are pairwise disjoint subsets of the empty set, because there are no subsets to have these unlikely properties.

The partitions of a set [[bijection|correspond one-to-one]] with its [[equivalence relation]]s. These are [[binary relation]]s that are [[Reflexive relation|reflexive]], [[Symmetric relation|symmetric]], and [[Transitive relation|transitive]]. The equivalence relation corresponding to a partition defines two elements as being equivalent when they belong to the same partition subset as each other. Conversely, every equivalence relation corresponds to a partition into [[equivalence class]]es.<ref>{{cite book | last = Halmos | first = Paul R. | mr = 0453532 | pages = 27–28 | publisher = Springer-Verlag, New York-Heidelberg | series = Undergraduate Texts in Mathematics | title = Naive set theory | url = https://books.google.com/books?id=jV_aBwAAQBAJ&pg=PA27 | year = 1974| isbn = 9781475716450 }}</ref> Therefore, the Bell numbers also count the equivalence relations.