# Partitioning of a Set

MathematicsComputer EngineeringMCA

Partition of a set, say S, is a collection of n disjoint subsets, say P1, P1, ... Pn that satisfies the following three conditions −

• Pi does not contain the empty set.

[ Pi ≠ { ∅ } for all 0 < i ≤ n ]

• The union of the subsets must equal the entire original set.

[ P1 ∪ P2 ∪ ... ∪ Pn = S ]

• The intersection of any two distinct sets is empty.

[ Pa ∩ Pb = { ∅ }, for a ≠ b where n ≥ a, b ≥ 0 ]

Example

Let S = { a, b, c, d, e, f, g, h }

One probable partitioning is { a }, { b, c, d }, { e, f, g, h }

Another probable partitioning is { a, b }, { c, d }, { e, f, g, h }

## Bell Numbers

Bell numbers give the count of the number of ways to partition a set. They are denoted by Bn where n is the cardinality of the set.

Example

Let S = { 1, 2, 3}, n = |S| = 3

The alternate partitions are −

1. ∅ , { 1, 2, 3 }

2. { 1 } , { 2, 3 }

3. { 1, 2 } , { 3 }

4. { 1, 3 } , { 2 }

5. { 1 } , { 2 } , { 3 }

Hence B3 = 5

Published on 23-Aug-2019 13:18:18