Quantity system

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In mathematics, a system of sets is a set whose elements are all subsets of a common basic set.

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

Formal definition

If a basic set is given, then every subset of the power set is called a set system over . In other words: is a set of sets and each element of is a subset of .


A set system is said to be closed or stable with respect to a set operation ( intersection , union , complement, etc.) if the application of the operation to elements of yields an element of again . Set systems are often named in terms of stable operations. This is the name of a set system, for example

  • -stable (average stable) or a π-system , if applies;
  • -stable (union stable), if applies;
  • σ- -stable or a δ-system , if for countable infinitely many sets is also in again ;
  • σ- -stable or briefly a σ-system, if for countable infinitely many sets is also in again ;
  • -stable (differential stable), if applies;
  • complementary stable if applies.


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

A hypergraph with 7 nodes and 4 hyper-edges
An undirected graph with 6 nodes and 7 edges


In the context of graph theory, a set system is also referred to as a hypergraph. The elements of the basic set are then called nodes and the elements of the system of sets are called hyperedges . A hyperedge can be thought of as a generalization of an edge in an ordinary graph that “connects” not two but several nodes at the same time. In the example opposite, the following applies:

Set of knots .
Set of hyperedges , where
Hyper edge ,
Hyper edge ,
Hyper edge ,
Hyperedge .

In many use cases of hypergraphs, the set of nodes is defined as finite and the empty hyperedge is excluded.

If each hyperedge connects exactly 2 nodes, then there is an undirected graph (more precisely: an undirected graph without multiple edges and without loops ). The set system then only consists of 2-element subsets of the basic set. In the example opposite, the following applies:

Basic amount = ,
Set system = .

Axiomatic set theory

In Zermelo-Fraenkel set theory there is only one type of object, namely sets. Thus all elements of a set are themselves sets again, and the terms set and system of sets coincide.

Example: Every natural number is identified in this context with the set of its predecessors. This results in the following structure:

(the empty set ),