Family of sets

A lot of system is to be a set whose elements are all subsets of a common ground set.

In the context of graph theory, a lot of system is called a hypergraph.

Formal definition

Is a basic amount given, ie every subset of the power set over a quantity system. In other words, is a set of sets, and each element of a subset of the.

Stability

A lot of system is called closed or stable with respect to a set operation (average, union, complement, etc. ), if the application of the operation to elements of re- supplies an element of. For example, average stable if true, that the intersection of two sets of lies again. In formulas:

Examples

The following mathematical objects are set systems with additional features. In formulating these properties often the stability with respect to certain set operations plays a role.

• Dynkin system
• Fréchet filter
• Envelope system
• Core system
• Matroid
• Algebra
• Amount of filter
• Amount semiring

• Amount of ring
• Partition
• Power set
• σ - algebra
• Topology (system of open sets of a topological space )
• Undirected graph
• Zermelosystem

Hypergraphs

In the context of graph theory, a lot of system is also called a hypergraph. The elements of the ground set are then called nodes and the elements of the system of sets are called hyperedges. One can get a hyperedge as a generalization of an edge present in an ordinary graphs just not two, but multiple nodes " connects " with each other simultaneously. In the example opposite is true:

In many applications of hypergraph the set of nodes is defined as finite and excluded the empty hyperedge.

Connects each hyperedge exactly two nodes, an undirected graph is present (more precisely, an undirected graph without multiple edges and without loops). The quantity system is then only work from 2- element subsets of the universal set. In the example opposite is true:

Axiomatic set theory

In the Zermelo -Fraenkel set theory there is only one type of objects, namely sets. To ensure that all elements of a set are themselves again amounts and the terms quantity and system match.

Example: Every natural number is identified in this context with the amount of their predecessors. This results in the following construction: