# Definition:Set of Sets

## Contents

## Definition

A **set of sets** is a set, whose elements are themselves all sets.

Those elements can themselves be assumed to be subsets of some particular fixed set which is frequently referred to as the universe.

## Also known as

Many sources (perhaps still feeling the wrath of the cane from schoolteachers of English) feel uncomfortable about referring to a **set of sets** and use a synonym instead.

Thus you will find terms such as **collection of sets**, **family of sets**, **assembly of sets** - it goes on and on.

- Beware the following

Take care when you see **class of sets**, because in modern set theory a class is a subtly different object from a set.

Also note that some books on, for example, topology and analysis will use the word **family of sets** to mean **set of sets**, whereas the technically accurate definition for (indexed) family in recent times is a subtly different concept.

A **system of sets**, or a **set system**, is also defined as a set whose elements are themselves all sets, but the implication here is that the sets in question are augmented by the various operations, thus turning such sets of sets into algebraic structures.

## Examples

### Set of Arbitrary Sets

Let:

\(\displaystyle A\) | \(=\) | \(\displaystyle \set {1, 2, 3, 4}\) | |||||||||||

\(\displaystyle B\) | \(=\) | \(\displaystyle \set {a, 3, 4}\) | |||||||||||

\(\displaystyle C\) | \(=\) | \(\displaystyle \set {2, a}\) |

Let $\mathscr S = \set {A, B, C}$.

Then:

- $\mathscr S = \set {\set {1, 2, 3, 4}, \set {a, 3, 4}, \set {2, a} }$

Note that none of $a, 1, 2, 3, 4$ are elements of $S$.

### Set of Initial Segments

Let $\Z$ denote the set of integers.

Let $\map \Z n$ denote the initial segment of $\Z_{> 0}$:

- $\map \Z n = \set {1, 2, \ldots, n}$

Let $\mathscr S := \set {\map \Z n: n \in \Z_{> 0} }$

That is, $\mathscr S$ is the set of all initial segments of $\Z_{> 0}$.

Then:

- $\mathscr S := \set {\set 1, \set {1, 2}, \set {1, 2, 3}, \ldots}$

and we have that:

- $\mathscr S \subsetneq \powerset \Z$

where $\powerset \Z$ denotes the power set of $\Z$.

## Also see

### Monotone Class

Let $X$ be a set, and let $\mathcal P \left({X}\right)$ be its power set.

Let $\mathcal M \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Then $\mathcal M$ is said to be a **monotone class (on $X$)** if and only if for every countable, nonempty, index set $I$, it holds that:

- $\displaystyle \left({A_i}\right)_{i \in I} \in \mathcal M \implies \bigcup_{i \mathop \in I} A_i \in \mathcal M$
- $\displaystyle \left({A_i}\right)_{i \in I} \in \mathcal M \implies \bigcap_{i \mathop \in I} A_i \in \mathcal M$

that is, if and only if $\mathcal M$ is closed under countable unions and intersections.

### Semiring of Sets

A **semiring of sets** or **semi-ring of sets** is a system of sets $\mathcal S$, subject to:

- $(1):\quad \varnothing \in \mathcal S$
- $(2):\quad A, B \in \mathcal S \implies A \cap B \in \mathcal S$; i.e., $\mathcal S$ is $\cap$-stable
- $(3):\quad$ If $A, A_1 \in \mathcal S$ such that $A_1 \subseteq A$, then there exists a finite sequence $A_2, A_3, \ldots, A_n \in \mathcal S$ such that:
- $(3a):\quad \displaystyle A = \bigcup_{k \mathop = 1}^n A_k$
- $(3b):\quad$ The $A_k$ are pairwise disjoint

Alternatively, criterion $(3)$ can be replaced by:

- $(3'):\quad$ If $A, B \in \mathcal S$, then there exists a finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \mathcal S$ such that $\displaystyle A \setminus B = \bigcup_{k \mathop = 1}^n A_k$.

### Ring of Sets

A **ring of sets** $\mathcal R$ is a system of sets with the following properties:

\((RS \, 1_1)\) | $:$ | Non-Empty: | \(\displaystyle \mathcal R \ne \varnothing \) | |||||

\((RS \, 2_1)\) | $:$ | Closure under Intersection: | \(\displaystyle \forall A, B \in \mathcal R:\) | \(\displaystyle A \cap B \in \mathcal R \) | ||||

\((RS \, 3_1)\) | $:$ | Closure under Symmetric Difference: | \(\displaystyle \forall A, B \in \mathcal R:\) | \(\displaystyle A * B \in \mathcal R \) |

### Algebra of Sets

Let $X$ be a set.

Let $\powerset X$ be the power set of $X$.

Let $\mathcal R \subseteq \powerset X$ be a set of subsets of $X$.

Then $\mathcal R$ is an **algebra of sets over $X$** if and only if the following conditions hold:

\((AS \, 1)\) | $:$ | Unit: | \(\displaystyle X \in \mathcal R \) | |||||

\((AS \, 2)\) | $:$ | Closure under Union: | \(\displaystyle \forall A, B \in \mathcal R:\) | \(\displaystyle A \cup B \in \mathcal R \) | ||||

\((AS \, 3)\) | $:$ | Closure under Complement Relative to $X$: | \(\displaystyle \forall A \in \mathcal R:\) | \(\displaystyle \relcomp X A \in \mathcal R \) |

### Sigma-Ring

A **$\sigma$-ring** is a ring of sets which is closed under countable unions.

That is, a ring of sets $\mathcal R$ is a **$\sigma$-ring** if and only if:

- $\displaystyle A_1, A_2, \ldots \in \mathcal R \implies \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R$

### Delta-Ring

A **delta-ring** (which can conveniently be written **$\delta$-ring**) is a ring of sets which is closed under countable intersections.

That is, a ring of sets $\mathcal R$ is a **delta-ring** iff:

- $\displaystyle A_1, A_2, \ldots \in \mathcal R \implies \bigcap_{n=1}^\infty A_n \in \mathcal R$

### Sigma-Algebra

Let $X$ be a set.

A **$\sigma$-algebra** $\mathcal R$ over $X$ is a system of subsets of $X$ with the following properties:

\((SA \, 1)\) | $:$ | Unit: | \(\displaystyle X \in \mathcal R \) | |||||

\((SA \, 2)\) | $:$ | Closure under Complement: | \(\displaystyle \forall A \in \mathcal R:\) | \(\displaystyle \complement_X \left({A}\right) \in \mathcal R \) | ||||

\((SA \, 3)\) | $:$ | Closure under Countable Unions: | \(\displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots:\) | \(\displaystyle \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R \) |

### Delta-Algebra

A **delta-algebra** is a delta-ring with a unit.

Thus, a **delta-algebra** is an algebra of sets which is closed under countable intersections.

### Borel Algebra

A **Borel algebra** is sometimes used as another name for a $\sigma$-algebra or $\delta$-algebra.

Can be confused with Borel $\sigma$-Algebra, which is a specific kind of $\sigma$-algebra.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.8$. Sets of sets - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 2$. Sets of sets - 1999: András Hajnal and Peter Hamburger:
*Set Theory*... (previous) ... (next): $1$. Notation, Conventions: $7$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts