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: