Quantity system
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 .
stability
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 πsystem , if applies;
 stable (union stable), if applies;
 σ stable or a δsystem , if for countable infinitely many sets is also in again ;
 σ stable or briefly a σsystem, if for countable infinitely many sets is also in again ;
 stable (differential stable), if applies;
 complementary stable if applies.
Examples
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.

Hypergraph
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 2element subsets of the basic set. In the example opposite, the following applies:
 Basic amount = ,
 Set system = .
Axiomatic set theory
In ZermeloFraenkel 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 ),
 ,
 ,
 ,
 ,
literature
 Oliver Deiser: Introduction to set theory . Springer, 2004, ISBN 9783540204015