halmazfelosztást

Halmazfelosztás, also known as set partition, is a fundamental concept in combinatorics and set theory. A partition of a set S is a collection of non-empty subsets of S such that every element in S is included in exactly one of these subsets. These subsets are called the blocks or parts of the partition. The union of all the blocks must equal the original set, and the intersection of any two distinct blocks must be the empty set.

For example, if we have the set S = {1, 2, 3}, its possible partitions are:

{{1, 2, 3}}

{{1, 2}, {3}}

{{1, 3}, {2}}

{{2, 3}, {1}}

{{1}, {2}, {3}}

The number of partitions of a set of size n is given by the Bell number, denoted by B_n. Bell numbers grow rapidly with n. The study of set partitions is crucial for understanding various combinatorial objects and problems.

Related concepts include Stirling numbers of the second kind, which count the number of partitions of a set of n elements into exactly k non-empty subsets. Halmazfelosztás has applications in areas such as probability, statistics, and computer science.