# 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 . ${\ displaystyle X}$ ${\ displaystyle {\ mathcal {S}}}$ ${\ displaystyle {\ mathcal {P}} (X) = \ {A \ mid A \ subseteq X \}}$ ${\ displaystyle X}$${\ displaystyle {\ mathcal {S}}}$${\ displaystyle {\ mathcal {S}}}$${\ displaystyle X}$

## 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 ${\ displaystyle S}$${\ displaystyle {\ mathcal {S}}}$${\ displaystyle {\ mathcal {S}}}$

• ${\ displaystyle \ cap}$-stable (average stable) or a π-system , if applies;${\ displaystyle A, B \ in {\ mathcal {S}} \ Rightarrow A \ cap B \ in {\ mathcal {S}}}$
• ${\ displaystyle \ cup}$-stable (union stable), if applies;${\ displaystyle A, B \ in {\ mathcal {S}} \ Rightarrow A \ cup B \ in {\ mathcal {S}}}$
• σ- -stable or a δ-system , if for countable infinitely many sets is also in again ;${\ displaystyle \ cap}$${\ displaystyle A_ {i} \ in {\ mathcal {S}}, \; i \ in \ mathbb {N},}$${\ displaystyle \ bigcap _ {i \ in \ mathbb {N}} A_ {i}}$${\ displaystyle {\ mathcal {S}}}$
• σ- -stable or briefly a σ-system, if for countable infinitely many sets is also in again ;${\ displaystyle \ cup}$${\ displaystyle A_ {i} \ in {\ mathcal {S}}, \; i \ in \ mathbb {N},}$${\ displaystyle \ bigcup _ {i \ in \ mathbb {N}} A_ {i}}$${\ displaystyle {\ mathcal {S}}}$
• ${\ displaystyle \ setminus}$-stable (differential stable), if applies;${\ displaystyle A, B \ in {\ mathcal {S}} \ Rightarrow A \ setminus B \ in {\ mathcal {S}}}$
• complementary stable if applies.${\ displaystyle A \ in {\ mathcal {S}} \ Rightarrow A ^ {c} \ in {\ mathcal {S}}}$

## 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.

 Mass association Monotonous class Partition Power set σ-algebra σ ring Topology (system of open sets of a topological space ) Undirected graph Zermelosystem
A hypergraph with 7 nodes and 4 hyper-edges
An undirected graph with 6 nodes and 7 edges

## 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 .${\ displaystyle = \ {v_ {1}, v_ {2}, v_ {3}, v_ {4}, v_ {5}, v_ {6}, v_ {7} \}}$
Set of hyperedges , where ${\ displaystyle = \ {e_ {1}, e_ {2}, e_ {3}, e_ {4} \}}$
Hyper edge ,${\ displaystyle e_ {1} = \ {v_ {1}, v_ {2}, v_ {3} \}}$
Hyper edge ,${\ displaystyle e_ {2} = \ {v_ {2}, v_ {3} \}}$
Hyper edge ,${\ displaystyle e_ {3} = \ {v_ {3}, v_ {5}, v_ {6} \}}$
Hyperedge .${\ displaystyle e_ {4} = \ {v_ {4} \}}$

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 = ,${\ displaystyle \ {1,2,3,4,5,6 \}}$
Set system = .${\ displaystyle \ {\ {1.2 \}, \ {1.5 \}, \ {2.3 \}, \ {2.5 \}, \ {3.4 \}, \ {4.5 \}, \ {4,6 \} \}}$

## 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:

${\ displaystyle 0 = \ emptyset}$(the empty set ),
${\ displaystyle 1 = \ {0 \} = \ {\ emptyset \}}$,
${\ displaystyle 2 = \ {0,1 \} = \ {\ emptyset, \ {\ emptyset \} \}}$,
${\ displaystyle 3 = \ {0,1,2 \} = \ {\ emptyset, \ {\ emptyset \}, \ {\ emptyset, \ {\ emptyset \} \} \}}$,
${\ displaystyle 4 = \ {0,1,2,3 \} = \ dotsc}$,
${\ displaystyle \ vdots}$