Partitioning of a Set

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

• P_i does not contain the empty set.

                          [ P_i ≠ { ∅ } for all 0 < i ≤ n ]

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

                         [ P₁ ∪ P₂ ∪ ... ∪ P_n = S ]

• The intersection of any two distinct sets is empty.

                        [ P_a ∩ P_b = { ∅ }, 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 B_n 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

Mahesh Parahar

Updated on: 23-Aug-2019

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