\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
Ordered Pairs
Set Powers
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Monotone Classes
Subset Algebras
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Subset Systems


We are often interested in a set of subsets of a given set.


Let $A$ be a non-empty set. A subset system (or set system) is a pair $(A, \mathcal{A} )$ in which $\mathcal{A} \subset \powerset{A}$. In this common case we call the first set the base set and the second set the distinguished subsets. A subset of $B \subset A$ which is not distinguished (i.e., $B \not\in \mathcal{A} $) is called undistinguished.

Let $A$ be a nonempty set. Let $\mathcal{A} $ be $\powerset{A}$. Then $(A, \mathcal{A} )$ is a subset system.

Other terminology

Other terminology refers to $(U, \mathcal{F} )$ as a set system when $U$ is a nonempty finite set and $\mathcal{F} $ is a family of subsets of $U$. Set systems are also known as hypergraphs.

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