List of set identities and relations

This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
The binary operations of set union and intersection satisfy many identities. Several of these identities or "laws" have well established names.

Notation

Throughout this article, capital letters will denote sets. On the left hand side of an identity, typically,

will be the leftmost set,
will be the middle set, and
will be the rightmost set.

This is to facilitate applying identities to expressions that are complicated or use the same symbols as the identity.
For example, the identity
may be read as:

Elementary set operations

For sets and define:
and
where the is sometimes denoted by and equals:
One set is said to another set if Sets that do not intersect are said to be.
The power set of is the set of all subsets of and will be denoted by
Universe set and complement notation
The notation
may be used if is a subset of some set that is understood.
It is emphasized that the definition of depends on context. For instance, had been declared as a subset of with the sets and not necessarily related to each other in any way, then would likely mean instead of
If it is needed then unless indicated otherwise, it should be assumed that denotes the universe set, which means that all sets that are used in the formula are subsets of
In particular, the complement of a set will be denoted by where unless indicated otherwise, it should be assumed that denotes the complement of in

One subset involved

Assume
Identity:
Definition: is called a left identity element of a binary operator if for all and it is called a right identity element of if for all A left identity element that is also a right identity element if called an identity element.
The empty set is an identity element of binary union and symmetric difference and it is also a right identity element of set subtraction
but is not a left identity element of since
so if and only if
Idempotence and Nilpotence :
Domination/Absorbing element:
Definition: is called a left absorbing element of a binary operator if for all and it is called a right absorbing element of if for all A left absorbing element that is also a right absorbing element if called an absorbing element. Absorbing elements are also sometime called annihilating elements or zero elements.
A universe set is an absorbing element of binary union The empty set is an absorbing element of binary intersection and binary Cartesian product and it is also a left absorbing element of set subtraction
but is not a right absorbing element of set subtraction since
where if and only if
Double complement or involution law:

Two sets involved

In the left hand sides of the following identities, is the eft most set and is the ight most set.
Assume both are subsets of some universe set

Formulas for binary set operations, and

In the left hand sides of the following identities, is the eft most set and is the ight most set. Whenever necessary, both and should be assumed to be subsets of some universe set, so that

De Morgan's laws

De Morgan's laws state that for

Commutativity

Unions, intersection, and symmetric difference are commutative operations:
Set subtraction is not commutative. However, the commutativity of set subtraction can be characterized: from it follows that:
Said differently, if distinct symbols always represented distinct sets, then the true formulas of the form that could be written would be those involving a single symbol; that is, those of the form:
But such formulas are necessarily true for binary operation, and so in this sense, set subtraction is as diametrically opposite to being commutative as is possible for a binary operation.
Set subtraction is also neither left alternative nor right alternative; instead, if and only if if and only if
Set subtraction is quasi-commutative and satisfies the Jordan identity.

Other identities involving two sets

Absorption laws:
Other properties
Intervals:

Subsets ⊆ and supersets ⊇

The following statements are equivalent for any

- Definition of : if then
and are disjoint

The following statements are equivalent for any

There exists some

Set equality

The following statements are equivalent:

If then if and only if
Uniqueness of complements: If then

Empty set

A set is [Axiom of Axiom of empty set|empty set|empty] if the sentence is true, where the notation is shorthand for
If is any set then the following are equivalent:

is not empty, meaning that the sentence is true.
is inhabited, meaning:
- In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty might not have an inhabitant.
for some set

If is any set then the following are equivalent:

is empty, meaning:
for every set
for every set
for some/every set

Given any the following are equivalent:

Moreover,

Meets, Joins, and lattice properties

Inclusion is a partial order:
Explicitly, this means that inclusion which is a binary operation, has the following three properties:

The following proposition says that for any set the power set of ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra.
Existence of a least element and a greatest element:
Joins/supremums exist:
The union is the join/supremum of and with respect to because:

and and
if is a set such that and then

The intersection is the join/supremum of and with respect to
Meets/infimums exist:
The intersection is the meet/infimum of and with respect to because:

if and and
if is a set such that and then

The union is the meet/infimum of and with respect to
Other inclusion properties:

If then
If and then

Three sets involved

In the left hand sides of the following identities, is the eft most set, is the iddle set, and is the ight most set.

Precedence rules

There is no universal agreement on the order of precedence of the basic set operators.
Nevertheless, many authors use precedence rules for set operators, although these rules vary with the author.
One common convention is to associate intersection with logical conjunction (and) and associate union with logical disjunction (or) and then transfer the precedence of these logical operators to these set operators, thereby giving precedence over
So for example, would mean since it would be associated with the logical statement and similarly, would mean since it would be associated with
Sometimes, set complement is also associated with logical complement (not) in which case it will have the highest precedence.
More specifically, is rewritten so that for example, would mean since it would be rewritten as the logical statement which is equal to
For another example, because means which is equal to both and, the formula would refer to the set
moreover, since this set is also equal to .
However, because set subtraction is not associative a formula such as would be ambiguous; for this reason, among others, set subtraction is often not assigned any precedence at all.
Symmetric difference is sometimes associated with exclusive or (xor), in which case if the order of precedence from highest to lowest is then the order of precedence for the set operators would be
There is no universal agreement on the precedence of exclusive disjunction with respect to the other logical connectives, which is why symmetric difference is not often assigned a precedence.

Associativity

Definition: A binary operator is called associative if always holds.
The following set operators are associative:
For set subtraction, instead of associativity, only the following is always guaranteed:
where equality holds if and only if . Thus
if and only if
where the only difference between the left and right hand side set equalities is that the locations of have been swapped.

Distributivity

Definition: If are binary operators then if
while if
The operator if it both left distributes and right [|distributes over]
In the definitions above, to transform one side to the other, the innermost operator becomes the outermost operator and the outermost operator becomes the innermost operator.
Right distributivity:
Left distributivity:

Distributivity and symmetric difference

Intersection distributes over symmetric difference:
Union does not distribute over symmetric difference because only the following is guaranteed in general:
Symmetric difference does not distribute over itself:
and in general, for any sets, might not be a subset, nor a superset, of .

Distributivity and set subtraction

Failure of set subtraction to left distribute:
Set subtraction is distributive over itself. However, set subtraction is left distributive over itself because only the following is guaranteed in general:
where equality holds if and only if which happens if and only if
For symmetric difference, the sets and are always disjoint.
So these two sets are equal if and only if they are both equal to
Moreover, if and only if
To investigate the left distributivity of set subtraction over unions or intersections, consider how the sets involved in De Morgan's laws are all related:
always holds but equality is not guaranteed in general.
Equality holds if and only if which happens if and only if
This observation about De Morgan's laws shows that is left distributive over or because only the following are guaranteed in general:
where equality holds for one of the above two inclusion formulas if and only if
The following statements are equivalent:

that is, left distributes over for these three particular sets
that is, left distributes over for these three particular sets
and

Quasi-commutativity:
always holds but in general,
However, if and only if if and only if
Set subtraction complexity: To manage the many identities involving set subtraction, this section is divided based on where the set subtraction operation and parentheses are located on the left hand side of the identity. The great variety and complexity of formulas involving set subtraction is in part due to the fact that unlike and set subtraction is neither associative nor commutative and it also is not left distributive over or even over itself.

Two set subtractions

Set subtraction is associative in general:
since only the following is always guaranteed:

One set subtraction

'Set subtraction on the left'', and parentheses on the '''

Set subtraction on the left, and parentheses on the
where the above two sets that are the subjects of De Morgan's laws always satisfy

'Set subtraction on the right'', and parentheses on the '''

Three operations on three sets

Operations of the form :

Other simplifications

Other properties:

If then
If then
if and only if for any belongs to of the sets

Symmetric difference ∆ of finitely many sets

Given finitely many sets something belongs to their symmetric difference if and only if it belongs to an odd number of these sets. Explicitly, for any if and only if the cardinality is odd..
Consequently, the symmetric difference of three sets satisfies:

Cartesian products of finitely many sets

Binary distributes over and and and

The binary Cartesian product ⨯ distributes over unions, intersections, set subtraction, and symmetric difference:
But in general, ⨯ does not distribute over itself:

Difference of finite

and

Symmetric difference and finite

In general, need not be a subset nor a superset of

Arbitrary families of sets

Let and be indexed families of sets. Whenever the assumption is needed, then all indexing sets, such as and are assumed to be non-empty.

Definitions

A or a refers to a set whose elements are sets.
An is a function from some set, called its, into some family of sets.
An indexed family of sets will be denoted by where this notation assigns the symbol for the indexing set and for every index assigns the symbol to the value of the function at
The function itself may then be denoted by the symbol which is obtained from the notation by replacing the index with a bullet symbol explicitly, is the function:
which may be summarized by writing
Any given indexed family of sets can be canonically associated with its image/range .
Conversely, any given family of sets may be associated with the -indexed family of sets which is technically the identity map
However, this is a bijective correspondence because an indexed family of sets is required to be injective, which in particular means that it is possible for distinct indexed families of sets to be associated with the same family of sets.
Arbitrary unions defined
If then which is somethings called the .
If is a family of sets then denotes the set:
Arbitrary intersections defined
If then
If is a family of sets then denotes the set:
Nullary intersections
If then
where every possible thing in the universe vacuously satisfied the condition: "if then ". Consequently, consists of in the universe.
So if and:

if you are working in a model in which there exists some universe ' then
otherwise, if you are working in a model in which "the class of all things " is not a set then is because consists of, which makes a proper class and a set.

A consequence of this is the following assumption/definition:
Some authors adopt the so called ', which is the convention that an empty intersection of sets is equal to some canonical set. In particular, if all sets are subsets of some set then some author may declare that the empty intersection of these sets be equal to However, the nullary intersection convention is not as commonly accepted as the nullary union convention and this article will not adopt it.
'''Multiple index sets'''

Distributing unions and intersections

Binary ⋂ of arbitrary ⋃'s

and

If all are pairwise disjoint and all are also pairwise disjoint, then so are all .Importantly, if then in general, . The single union on the right hand side be over all pairs The same is usually true for other similar non-trivial set equalities and relations that depend on two indexing sets and . Two exceptions are and, but both of these are among the most trivial of set equalities.: Let and let Let and let Then Furthermore,

Binary ⋃ of arbitrary ⋂'s

andImportantly, if then in general, . The single intersection on the right hand side be over all pairs

Arbitrary ⋂'s and arbitrary ⋃'s

Incorrectly distributing by swapping ⋂ and ⋃

Naively swapping and may produce a different set
The following inclusion always holds:
In general, equality need not hold and moreover, the right hand side depends on how for each fixed the sets are labelled; and analogously, the left hand side depends on how for each fixed the sets are labelled. An example demonstrating this is now given.

Example of dependence on labeling and failure of equality: To see why equality need not hold when and are swapped, let and let and Then
If and are swapped while and are unchanged, which gives rise to the sets and then
In particular, the left hand side is no longer which shows that the left hand side depends on how the sets are labelled.
If instead and are swapped while and are unchanged, which gives rise to the sets and then both the left hand side and right hand side are equal to which shows that the right hand side also depends on how the sets are labeled.

Equality in can hold under certain circumstances, such as in, which is the special case where is, or such as in, which is the special case where is .
For a correct formula that extends the distributive laws, an approach other than just switching and is needed.

Correct distributive laws

Suppose that for each is a non-empty index set and for each let be any set. Let
denote the Cartesian product, which can be interpreted as the set of all functions such that for every Such a function may also be denoted using the tuple notation where for every and conversely, a tuple is just notation for the function with domain whose value at is both notations can be used to denote the elements of
Then
where

Applying the distributive laws

': In the particular case where all are equal, then letting denote this common set, the Cartesian product will be which is the set of all functions of the form The above set equalities and, respectively become:
which when combined with implies:
where
on the left hand side, the indices range over
on the right hand side, the indices range over .
': To apply the general formula to the case of and use and let for all and let for all
Every map can be bijectively identified with the pair . Recall that was

Expanding and simplifying the left hand side gives
and doing the same to the right hand side gives:
Thus the general identity reduces down to the previously given set equality :

Distributing subtraction over ⋃ and ⋂

The next identities are known as De Morgan's laws.
The following four set equalities can be deduced from the equalities - above.
In general, naively swapping and may produce a different set.
The equalities
found in and are thus unusual in that they state exactly that swapping and will change the resulting set.

Commutativity and associativity of ⋃ and ⋂

Commutativity:
Unions of unions and intersections of intersections:
and
and if then also:

Cartesian products of arbitrarily many sets

Intersections of

If is a family of sets then

Moreover, a tuple belongs to the set in above if and only if for all and all

In particular, if and are two families indexed by the same set then
So for instance,
and
Intersections of products indexed by different sets
Let and be two families indexed by different sets.
Technically, implies
However, sometimes these products are somehow identified as the same set through some bijection or one of these products is identified as a subset of the other via some injective map, in which case this intersection may be equal to some other set.

For example, if and with all sets equal to then and where, for example, is identified as a subset of through some injection, such as maybe for instance; however, in this particular case the product actually represents the -indexed product where
For another example, take and with and all equal to Then and which can both be identified as the same set via the bijection that sends to Under this identification,

Binary distributes over arbitrary and

The binary Cartesian product ⨯ distributes over arbitrary intersections and over arbitrary unions:

Distributing arbitrary over arbitrary

Suppose that for each is a non-empty index set and for each let be any set. Let
denote the Cartesian product, which can be interpreted as the set of all functions such that for every.
Then
where

Unions of

For unions, only the following is guaranteed in general:
where is a family of sets. Example where equality fails: Let let let and let Then More generally, if and only if for each at least one of the sets in the -indexed collections of sets is empty, while if and only if for each at least one of the sets in the -indexed collections of sets is not empty.
However,

Difference of

If and are two families of sets then:
so for instance,
and

Functions and sets

Let be any function.
Let be completely arbitrary sets. Assume

Definitions

Let be any function, where we denote its by and denote its by
Many of the identities below do not actually require that the sets be somehow related to 's domain or codomain so when some kind of relationship is necessary then it will be clearly indicated.
Because of this, in this article, if is declared to be "," and it is not indicated that must be somehow related to or then it is meant that is truly arbitrary.
This generality is useful in situations where is a map between two subsets and of some larger sets and and where the set might not be entirely contained in and/or ; in such a situation it may be useful to know what can and cannot be said about and/or without having to introduce a intersection such as: and/or
Images and preimages of sets
If is set then the is defined to be the set:
while the is:
where if is a singleton set then the or is
Denote by or the or of which is the set:
Saturated sets
A set is said to be or a if any of the following equivalent conditions are satisfied:

There exists a set such that
- Any such set necessarily contains as a subset.
- Any set not entirely contained in the domain of cannot be -saturated.
and
- The inclusion always holds, where if then this becomes
and if and satisfy then
Whenever a fiber of intersects then contains the entire fiber. In other words, contains every -fiber that intersects it.

Explicitly: whenever is such that then
In both this statement and the next, the set may be replaced with any superset of and the resulting statement will still be equivalent to the rest.

The intersection of with a fiber of is equal to the empty set or to the fiber itself.
- Explicitly: for every the intersection is equal to the empty set or to .

For a set to be -saturated, it is necessary that
Compositions and restrictions of functions
If and are maps then denotes the map
with domain and codomain
defined by
The denoted by is the map
with defined by sending to that is,
Alternatively, where denotes the inclusion map, which is defined by

(Pre)Images of arbitrary unions ⋃'s and intersections ⋂'s

If is a family of arbitrary sets indexed by then:
So of these four identities, it is that are not always preserved. Preimages preserve all basic set operations. Unions are preserved by both images and preimages.
If all are -saturated then be will be -saturated and equality will hold in the first relation above; explicitly, this means:
If is a family of arbitrary subsets of which means that for all then becomes:

(Pre)Images of binary set operations

Throughout, let and be any sets and let be any function.
Summary
As the table below shows, set equality is guaranteed for of: intersections, set subtractions, and symmetric differences.

Image	Preimage
		-
		-
		-
		-
		-

Preimages preserve set operations
Preimages of sets are well-behaved with respect to all basic set operations:
In words, preimages distribute over unions, intersections, set subtraction, and symmetric difference.
Images preserve unions
Images of unions are well-behaved:
but images of the other basic set operations are since only the following are guaranteed in general:
In words, images distribute over unions but not necessarily over intersections, set subtraction, or symmetric difference. What these latter three operations have in common is set subtraction: they either set subtraction or else they can naturally as the set subtraction of two sets:
If then where as in the more general case, equality is not guaranteed. If is surjective then which can be rewritten as: if and

Counter-examples: images of operations not distributing

If is constant, and then all four of the set containments
are strict/proper since one side is the empty set while the other is non-empty. Thus equality is not guaranteed for even the simplest of functions.
The example above is now generalized to show that these four set equalities can fail for any constant function whose domain contains at least two points.
: Let be any constant function with image and suppose that are non-empty disjoint subsets; that is, and which implies that all of the sets and are not empty and so consequently, their images under are all equal to

The containment is strict:
In words: functions might not distribute over set subtraction
The containment is strict:
The containment is strict:
In words: functions might not distribute over symmetric difference .
The containment is strict:
In words: functions might not distribute over set intersection .

What the set operations in these four examples have in common is that they either set subtraction and ) or else they can naturally as the set subtraction of two sets and ).
Mnemonic: In fact, for each of the above four set formulas for which equality is not guaranteed, the direction of the containment can always be deduced by imagining the function as being and the two sets as being non-empty disjoint subsets of its domain. This is because equality fails for such a function and sets: one side will be always be and the other non-empty − from this fact, the correct choice of can be deduced by answering: "which side is empty?" For example, to decide if the in
should be pretend
that is constant and that and are non-empty disjoint subsets of 's domain; then the hand side would be empty, which indicates that should be because this is the choice that will make
true.
Alternatively, the correct direction of containment can also be deduced by consideration of any constant with and
Furthermore, this mnemonic can also be used to correctly deduce whether or not a set operation always distribute over images or preimages; for example, to determine whether or not always equals or alternatively, whether or not always equals . The answer to such a question can, as before, be deduced by consideration of this constant function: the answer for the general case is always the same as the answer for this choice of function and disjoint non-empty sets.

Conditions guaranteeing that images distribute over set operations

Characterizations of when equality holds for sets:
For any function the following statements are equivalent:

is injective.
- This means: for all distinct
.
.
.
.
Any one of the four statements - but with the words "for all" replaced with any one of the following:
1. "for all singleton subsets"
  - In particular, the statement that results from gives a characterization of injectivity that explicitly involves only one point : is injective if and only if
2. "for all disjoint singleton subsets"
  - For statement, this is the same as: "for all singleton subsets".
3. "for all disjoint subsets"

In particular, if a map is not known to be injective then barring additional information, there is no guarantee that any of the equalities in statements - hold.
[|An example above] can be used to help prove this characterization. Indeed, comparison of that example with such a proof suggests that the example is representative of the fundamental reason why one of these four equalities in statements - might not hold.

Conditions for f(L⋂R) = f(L)⋂f(R)

Characterizations of equality: The following statements are equivalent:

- The left hand side is always equal to .
If satisfies then
If but then
Any of the above three conditions - but with the subset symbol replaced with an equals sign

Sufficient conditions for equality: Equality holds if any of the following are true:

is injective.
The restriction is injective.
is -saturated; that is,
is -saturated; that is,
or equivalently,
or equivalently,
or equivalently,

In addition, the following always hold:

Conditions for f(L\R) = f(L)\f(R)

Characterizations of equality: The following statements are equivalent:

Whenever then
- The set on the right hand side is always equal to
- This is the above condition but with the subset symbol replaced with an equals sign

Necessary conditions for equality : If equality holds then the following are necessarily true:

or equivalently
or equivalently,

Sufficient conditions for equality: Equality holds if any of the following are true:

is injective.
The restriction is injective.
or equivalently,
is -saturated; that is,
or equivalently,

Conditions for f(X\R) = f(X)\f(R)

Characterizations of equality: The following statements are equivalent:

is -saturated.
Whenever then

where if then this list can be extended to include:

is -saturated; that is,

Sufficient conditions for equality: Equality holds if any of the following are true:

is injective.
is -saturated; that is,

Conditions for f(L∆R) = f(L)∆f(R)

Characterizations of equality: The following statements are equivalent:

and
and
and
- The inclusions and always hold.
- If this above set equality holds, then this set will also be equal to both and
and

Necessary conditions for equality : If equality holds then the following are necessarily true:

or equivalently

Sufficient conditions for equality: Equality holds if any of the following are true:

is injective.
The restriction is injective.

Exact formulas/equalities for images of set operations

For any function and any sets and

Taking in the above formulas gives:
where the set is equal to the image under of the largest -saturated subset of

In general, only always holds and equality is not guaranteed; but replacing "" with its subset "" results in a formula in which equality is guaranteed:
From this it follows that:
If then which can be written more symmetrically as .

It follows from and the above formulas for the image of a set subtraction that for any function and any sets and

It follows from the above formulas for the image of a set subtraction that for any function and any set
This is more easily seen as being a consequence of the fact that for any if and only if

It follows from the above formulas for the image of a set that for any function and any sets and
where moreover, for any
The sets and [|mentioned above] could, in particular, be any of the sets or for example.

(Pre)Images of set operations on (pre)images

Let and be arbitrary sets, be any map, and let and

Image of preimage	Preimage of image	Additional assumptions on sets
		-
	Equality holds if any of the following are true:

Images of operations on images
Since
Since
Using this becomes and
and so

(Pre)Images and Cartesian products Π

Let and for every let
denote the canonical projection onto
Definitions
Given a collection of maps indexed by define the map
which is also denoted by This is the unique map satisfying
Conversely, if given a map then
Explicitly, what this means is that if
is defined for every then the unique map satisfying: for all or said more briefly,
The map should not be confused with the Cartesian product of these maps, which is by definition is the map
with domain rather than
Preimage and images of a Cartesian product
Suppose
If then
If then
where equality will hold if in which case and
For equality to hold, it suffices for there to exist a family of subsets such that in which case:
and for all

(Pre)Image of a single set

Image	Preimage	Additional assumptions
		-
		-
		-
		-
		-
		None.
		-
		-
		-

Containments ⊆ and intersections ⋂ of images and preimages

Equivalences and implications of images and preimages

Image	Preimage	Additional assumptions on sets
	if and only if	-
if and only if	if and only if	-
if and only if	if and only if	and
implies	implies	-
The following are equivalent:	The following are equivalent: If then if and only if
The following are equivalent when for some	The following are equivalent: and The following are equivalent when	and
The following are equivalent:	The following are equivalent:	and
Equality holds if the following is true: Equality holds if any of the following are true: and is surjective.	Equality holds if the following is true: is -saturated. Equality holds if any of the following are true: is injective.

Intersection of a set and a image
The following statements are equivalent:

Thus for any

Sequences and collections of families of sets

Definitions

A or simply a is a set whose elements are sets.
A is a family of subsets of
The of a set is the set of all subsets of :
Notation for sequences of sets
Throughout, will be arbitrary sets and and will denote a net or a sequence of sets where if it is a sequence then this will be indicated by either of the notations
where denotes the natural numbers.
A notation indicates that is a net directed by which is a sequence if the set which is called the net's indexing set, is the natural numbers and is the natural order on
Disjoint and monotone sequences of sets
If for all distinct indices then is called a or simply a.
A sequence or net of set is called or if if for all indices .
A sequence or net of set is called if it is non-decreasing and also for all indices
It is called if it is non-decreasing or non-increasing and it is called if it is strictly increasing or strictly decreasing.
A sequences or net is said to denoted by or if is increasing and the union of all is that is, if
It is said to denoted by or if is increasing and the intersection of all is that is, if
Definitions of elementwise operations on families
If are families of sets and if is any set then define:
which are respectively called , , ' ', , and the '/' The regular union, intersection, and set difference are all defined as usual and are denoted with their usual notation: and respectively.
These elementwise operations on families of sets play an important role in, among other subjects, the theory of filters and prefilters on sets.
The of a family is the family:
and the is the family:

Definitions of categories of families of sets

The following table lists some well-known categories of families of sets having applications in general topology and measure theory.
A family is called,, or in if and
A family is called if
A family is said to be:

if whenever then .
if whenever are elements of then so is their intersections .
if whenever then

A family of sets is called a/an:

if and is closed under finite-intersections.
* Every non-empty family is contained in a unique smallest −system that is denoted by and called
and is said to have if and
if is a family of subsets of that is a −system, is upward closed in and is also, which by definition means that it does not contain the empty set as an element.
or if it is a non-empty family of subsets of some set whose upward closure in is a filter on
is a non-empty family of subsets of that contains the empty set, forms a −system, and is also closed under complementation with respect to
is an algebra on that is closed under countable unions.

Sequences of sets often arise in measure theory.
Algebra of sets
A family of subsets of a set is said to be ' if and for all all three of the sets and are elements of
The article on this topic lists set identities and other relationships these three operations.
Every algebra of sets is also a ring of sets and a π-system.
Algebra generated by a family of sets
Given any family of subsets of there is a unique smallest algebra of sets in containing
It is called ' and it will be denote it by
This algebra can be constructed as follows:

If then and we are done. Alternatively, if is empty then may be replaced with and continue with the construction.
Let be the family of all sets in together with their complements.
Let be the family of all possible finite intersections of sets in
Then the algebra generated by is the set consisting of all possible finite unions of sets in

Elementwise operations on families

Let and be families of sets over
On the left hand sides of the following identities, is the eft most family, is in the iddle, and is the ight most set.
Commutativity:
Associativity:
Identity:
Domination:

Power set

If and are subsets of a vector space and if is a scalar then

Sequences of sets

Suppose that is any set such that for every index
If decreases to then increases to
whereas if instead increases to then decreases to
If are arbitrary sets and if increases to then increase to

Partitions

Suppose that is any sequence of sets, that is any subset, and for every index let
Then and is a sequence of pairwise disjoint sets.
Suppose that is non-decreasing, let and let for every Then and is a sequence of pairwise disjoint sets.