This article lists
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
properties and laws of
sets, involving the set-theoretic
operations of
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
,
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
, and
complementation and the
relations of set
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elite ...
and set
inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society.
** Inclusion (disability rights), promotion of people with disabiliti ...
. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
The
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s of set union (
) and intersection (
) satisfy many identities. Several of these identities or "laws" have well established names.
Notation
Throughout this article, capital letters such as
and
will denote sets and
will denote the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
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)
Typically, the set
will denote the eft most set,
the iddle set, and
the ight most set.
For sets
and
define:
and
where the
is sometimes denoted by
and equals:
If
is a set that is understood (say from context, or because it is clearly stated) to be a subset of some other set
then the complement of a set
may be denoted by:
The definition of
may depend 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
Finitely many sets
One subset involved
Assume
Identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
:
Definition:
is called a
left identity element of a
binary operator
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary ope ...
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
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
.
but
so
Idempotence
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
and
Nilpotence :
Domination/
Zero element
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive identi ...
:
but
so
Double complement or
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
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
should be assumed to be subsets of some universe set
so that
De Morgan's laws
De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...
state that for
Commutativity
Unions, intersection, and symmetric difference are
commutative operation
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
s:
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
(because
must hold by definition of
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elite ...
), 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
In abstract algebra, alternativity is a property of a binary operation. A magma ''G'' is said to be if (xx)y = x(xy) for all x, y \in G and if y(xx) = (yx)x for all x, y \in G. A magma that is both left and right alternative is said to be () ...
nor
right alternative
In abstract algebra, alternativity is a property of a binary operation. A magma ''G'' is said to be if (xx)y = x(xy) for all x, y \in G and if y(xx) = (yx)x for all x, y \in G. A magma that is both left and right alternative is said to be () ...
; instead,
if and only if
if and only if
Set subtraction is
quasi-commutative In mathematics, the quasi-commutative property is an extension or generalization of the general commutative property. This property is used in specific applications with various definitions.
Applied to matrices
Two matrices p and q are said to ha ...
and satisfies the
Jordan identity
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
# xy = yx (commutative law)
# (xy)(xx) = x(y(xx)) ().
The product of two elements ''x'' and ''y'' in a Jordan al ...
.
Other identities involving two sets
Absorption law
In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations.
Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if:
:''a'' ¤ (''a'' ⁂ ''b'') = ''a'' ⁂ (''a'' ¤ ''b ...
s:
Other properties
Intervals: