List of set identities and relations

Summary

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

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Throughout this article, capital letters (such as and ) will denote sets. On the left hand side of an identity, typically,

  • will be the Left most set,
  • will be the M iddle set, and
  • will be the R ight most set.

This is to facilitate applying identities to expressions that are complicated or use the same symbols as the identity.[note 1] For example, the identity may be read as:

Elementary set operations

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For sets and define: and where the symmetric difference is sometimes denoted by and equals:[1][2]

One set is said to intersect another set if Sets that do not intersect are said to be disjoint.

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 (say from context, or because it is clearly stated what the superset is). 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 (the universe)

One subset involved

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Assume

Identity:[3]

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[3] and Nilpotence :

Domination[3]/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:

[3]

[3]

Two sets involved

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

Formulas for binary set operations ⋂, ⋃, \, and ∆

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

De Morgan's laws

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De Morgan's laws state that for

Commutativity

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Unions, intersection, and symmetric difference are commutative operations:[3]

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 only 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 every binary operation (because must hold by definition of equality), 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

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Absorption laws:

Other properties

Intervals:

Subsets ⊆ and supersets ⊇

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The following statements are equivalent for any [3]

    • Definition of subset: if then
  1. and are disjoint (that is, )
  2. (that is, )

The following statements are equivalent for any

  1. There exists some

Set equality

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The following statements are equivalent:

  • If then if and only if
  • Uniqueness of complements: If then
Empty set
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A set is empty if the sentence is true, where the notation is shorthand for

If is any set then the following are equivalent:

  1. is not empty, meaning that the sentence is true (literally, the logical negation of " is empty" holds true).
  2. (In classical mathematics) 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 (where by definition, " is empty" means that the statement is true) might not have an inhabitant (which is an such that ).
  3. for some set

If is any set then the following are equivalent:

  1. is empty (), meaning:
  2. for every set
  3. for every set
  4. for some/every set

Given any the following are equivalent:

Moreover,

Meets, Joins, and lattice properties

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Inclusion is a partial order: Explicitly, this means that inclusion which is a binary operation, has the following three properties:[3]

  • Reflexivity:
  • Antisymmetry:
  • Transitivity:

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:[3]

The union is the join/supremum of and with respect to because:

  1. and and
  2. if is a set such that and then

The intersection is the join/supremum of and with respect to

Meets/infimums exist:[3]

The intersection is the meet/infimum of and with respect to because:

  1. if and and
  2. 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 [3]

Three sets involved

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In the left hand sides of the following identities, is the L eft most set, is the M iddle set, and is the R 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 (where has precedence over ) 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 (subtraction) 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 (where was rewritten as ), the formula would refer to the set moreover, since this set is also equal to (other set identities can similarly be deduced from propositional calculus identities in this way). 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) (also sometimes denoted by ), in which case if the order of precedence from highest to lowest is then the order of precedence (from highest to lowest) 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

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Definition: A binary operator is called associative if always holds.

The following set operators are associative:[3]

For set subtraction, instead of associativity, only the following is always guaranteed: where equality holds if and only if (this condition does not depend on ). 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

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Definition: If are binary operators then left distributes over if while right distributes over if The operator distributes over if it both left distributes and right distributes over In the definitions above, to transform one side to the other, the innermost operator (the operator inside the parentheses) becomes the outermost operator and the outermost operator becomes the innermost operator.

Right distributivity:[3]

Left distributivity:[3]

Distributivity and symmetric difference ∆

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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 (where represents ), might not be a subset, nor a superset, of (and the same is true for ).

Distributivity and set subtraction \

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Failure of set subtraction to left distribute:

Set subtraction is right distributive over itself. However, set subtraction is not 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 (both of) De Morgan's laws are all related: always holds (the equalities on the left and right are De Morgan's laws) but equality is not guaranteed in general (that is, the containment might be strict). Equality holds if and only if which happens if and only if

This observation about De Morgan's laws shows that is not left distributive over or because only the following are guaranteed in general: where equality holds for one (or equivalently, for both) of the above two inclusion formulas if and only if

The following statements are equivalent:

  1. that is, left distributes over for these three particular sets
  2. that is, left distributes over for these three particular sets
  3. 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 (relative) complexity of formulas involving set subtraction (compared to those without it) 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

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Set subtraction is not associative in general: since only the following is always guaranteed:

(L\M)\R

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L\(M\R)

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  • If
  • with equality if and only if

One set subtraction

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(L\M) ⁎ R

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Set subtraction on the left, and parentheses on the left

[4]

L\(M ⁎ R)

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Set subtraction on the left, and parentheses on the right

where the above two sets that are the subjects of De Morgan's laws always satisfy

(L ⁎ M)\R

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Set subtraction on the right, and parentheses on the left

L ⁎ (M\R)

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Set subtraction on the right, and parentheses on the right

[4]

Three operations on three sets

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(L • M) ⁎ (M • R)

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Operations of the form :

(L • M) ⁎ (R\M)

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Operations of the form :

(L\M) ⁎ (L\R)

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Operations of the form :

Other simplifications

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Other properties:

  • If then [4]
  • If then
  • if and only if for any belongs to at most two of the sets

Symmetric difference ∆ of finitely many sets

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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. (Recall that symmetric difference is associative so parentheses are not needed for the set ).

Consequently, the symmetric difference of three sets satisfies:

Cartesian products ⨯ of finitely many sets

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Binary ⨯ distributes over ⋃ and ⋂ and \ and ∆

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The binary Cartesian productdistributes over unions, intersections, set subtraction, and symmetric difference:

But in general, ⨯ does not distribute over itself:

Binary ⋂ of finite ⨯

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Binary ⋃ of finite ⨯

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Difference \ of finite ⨯

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and

Finite ⨯ of differences \

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Symmetric difference ∆ and finite ⨯

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

Arbitrary families of sets

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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

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A family of sets or (more briefly) a family refers to a set whose elements are sets.

An indexed family of sets is a function from some set, called its indexing set, 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 (which is a function) can be canonically associated with its image/range (which is a family of sets). Conversely, any given family of sets may be associated with the -indexed family of sets which is technically the identity map However, this is not a bijective correspondence because an indexed family of sets is not required to be injective (that is, there may exist distinct indices such as ), which in particular means that it is possible for distinct indexed families of sets (which are functions) to be associated with the same family of sets (by having the same image/range).

Arbitrary unions defined[3]

(Def. 1)

If then which is somethings called the nullary union convention (despite being called a convention, this equality follows from the definition).

If is a family of sets then denotes the set:

Arbitrary intersections defined

If then[3]

(Def. 2)

If is a non-empty 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 everything in the universe.

So if and:

  1. if you are working in a model in which there exists some universe set then
  2. otherwise, if you are working in a model in which "the class of all things " is not a set (by far the most common situation) then is undefined because consists of everything, which makes a proper class and not a set.
Assumption: Henceforth, whenever a formula requires some indexing set to be non-empty in order for an arbitrary intersection to be well-defined, then this will automatically be assumed without mention.

A consequence of this is the following assumption/definition:

A finite intersection of sets or an intersection of finitely many sets refers to the intersection of a finite collection of one or more sets.

Some authors adopt the so called nullary intersection convention, 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 (this is due to the fact that unlike the empty union, the value of the empty intersection depends on so if there are multiple sets under consideration, which is commonly the case, then the value of the empty intersection risks becoming ambiguous).

Multiple index sets

Distributing unions and intersections

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Binary ⋂ of arbitrary ⋃'s

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(Eq. 3a)

and[4]

(Eq. 3b)
  • If all are pairwise disjoint and all are also pairwise disjoint, then so are all (that is, if then ).

  • Importantly, if then in general, (an example of this is given below). The single union on the right hand side must be over all pairs The same is usually true for other similar non-trivial set equalities and relations that depend on two (potentially unrelated) indexing sets and (such as Eq. 4b or Eq. 7g[4]). Two exceptions are Eq. 2c (unions of unions) and Eq. 2d (intersections of intersections), but both of these are among the most trivial of set equalities (although even for these equalities there is still something that must be proven[note 2]).
  • Example where equality fails: Let and let Let and let Then Furthermore,

Binary ⋃ of arbitrary ⋂'s

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(Eq. 4a)

and[4]

(Eq. 4b)
  • Importantly, if then in general, (an example of this is given above). The single intersection on the right hand side must be over all pairs

Arbitrary ⋂'s and arbitrary ⋃'s

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Incorrectly distributing by swapping ⋂ and ⋃
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Naively swapping and may produce a different set

The following inclusion always holds:

(Inclusion 1 ∪∩ is a subset of ∩∪)

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