Definition:Set of Sets

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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: $1$

Let:

\(\ds A\) \(=\) \(\ds \set {1, 2, 3, 4}\)
\(\ds B\) \(=\) \(\ds \set {a, 3, 4}\)
\(\ds C\) \(=\) \(\ds \set {2, a}\)


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 Arbitrary Sets: $2$

Let $A$ be the set of (strictly) positive odd integers less than $8$.

Let $B$ be the set of (strictly) positive even integers less than $8$.

Then:

\(\ds A\) \(=\) \(\ds \set {1, 3, 5, 7}\)
\(\ds B\) \(=\) \(\ds \set {2, 4, 6}\)


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

Then:

$\mathscr S = \set {\set {1, 3, 5, 7}, \set {2, 4, 6} }$


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 $\powerset X$ be its power set.

Let $\MM \subseteq \powerset X$ be a collection of subsets of $X$.


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

$\ds \family {A_i}_{i \mathop \in I} \in \MM \implies \bigcup_{i \mathop \in I} A_i \in \MM$
$\ds \family {A_i}_{i \mathop \in I} \in \MM \implies \bigcap_{i \mathop \in I} A_i \in \MM$

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


Semiring of Sets

Let $\SS$ be a system of sets.


$\SS$ is a semiring of sets or semi-ring of sets if and only if $\SS$ satisfies the semiring of sets axioms:

\((1)\)   $:$   \(\ds \O \in \SS \)      
\((2)\)   $:$   $\cap$-stable      \(\ds \forall A, B \in \SS:\) \(\ds A \cap B \in \SS \)      
\((3)\)   $:$     \(\ds \forall A, A_1 \in \SS : A_1 \subseteq A:\) $\exists n \in \N$ and pairwise disjoint sets $A_2, A_3, \ldots, A_n \in \SS : \ds A = \bigcup_{k \mathop = 1}^n A_k$      


Ring of Sets

A system of sets $\RR$ is a ring of sets if and only if $\RR$ satisfies the ring of sets axioms:

\((\text {RS} 1_1)\)   $:$   Non-Empty:    \(\ds \RR \ne \O \)      
\((\text {RS} 2_1)\)   $:$   Closure under Intersection:      \(\ds \forall A, B \in \RR:\) \(\ds A \cap B \in \RR \)      
\((\text {RS} 3_1)\)   $:$   Closure under Symmetric Difference:      \(\ds \forall A, B \in \RR:\) \(\ds A \symdif B \in \RR \)      


Algebra of Sets

Let $S$ be a set.

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

Let $\RR \subseteq \powerset S$ be a set of subsets of $S$.


$\RR$ is an algebra of sets over $S$ if and only if $\RR$ satisfies the algebra of sets axioms:

\((\text {AS} 1)\)   $:$   Unit:    \(\ds S \in \RR \)      
\((\text {AS} 2)\)   $:$   Closure under Union:      \(\ds \forall A, B \in \RR:\) \(\ds A \cup B \in \RR \)      
\((\text {AS} 3)\)   $:$   Closure under Complement Relative to $S$:      \(\ds \forall A \in \RR:\) \(\ds \relcomp S A \in \RR \)      


Sigma-Ring

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


That is, a ring of sets $\Sigma$ is a $\sigma$-ring if and only if:

$\ds A_1, A_2, \ldots \in \Sigma \implies \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma$


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 $\RR$ is a delta-ring if and only if:

$\ds A_1, A_2, \ldots \in \RR \implies \bigcap_{n \mathop = 1}^\infty A_n \in \RR$


Sigma-Algebra

Let $X$ be a set.

Let $\Sigma$ be a system of subsets of $X$.


$\Sigma$ is a $\sigma$-algebra over $X$ if and only if $\Sigma$ satisfies the sigma-algebra axioms:

\((\text {SA 1})\)   $:$   Unit:    \(\ds X \in \Sigma \)      
\((\text {SA 2})\)   $:$   Closure under Complement:      \(\ds \forall A \in \Sigma:\) \(\ds \relcomp X A \in \Sigma \)      
\((\text {SA 3})\)   $:$   Closure under Countable Unions:      \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) \(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \)      


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

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