Editing Set (mathematics) (section)

== Basic operations ==

There are several standard [[operation (mathematics)|operations]] that produce new sets from  given sets, in the same way as [[addition]] and [[multiplication]] produce new numbers from given numbers. The operations that are considered in this section are those such that all elements of the produced sets belong to a previously defined set. These operations are commonly illustrated with [[Euler diagram]]s and [[Venn diagram]]s.<ref>{{Cite book |last=Tanton |first=James |title=Encyclopedia of Mathematics |publisher=Facts On File |year=2005 |isbn=0-8160-5124-0 |location=New York |pages=460–61 |language=en |chapter=Set theory}}</ref>

The main basic operations on sets are the following ones.

===Intersection===
[[File:Venn0001.svg|thumb|<div class="center">The ''intersection'' of {{mvar|A}} and {{mvar|B}}, denoted {{math|''A'' ∩ ''B''}}</div>]]
The ''[[set intersection|intersection]]'' of two sets {{tmath|A}} and {{tmath|B}} is a set denoted {{tmath|A \cap B}} whose elements are those elements that belong to both {{tmath|A}} and {{tmath|B}}. That is,
<math display=block>A \cap B=\{x\mid x\in A \land x\in B\},</math>
where {{tmath|\land}} denotes the [[logical and]]. 

Intersection is [[associative]] and [[commutative]]; this means that for proceeding a sequence of intersections, one may proceed in any order, without the need of parentheses for specifying the [[order of operations]]. Intersection has no general [[identity element]]. However, if one restricts intersection to the subsets of a given set {{tmath|U}}, intersection has {{tmath|U}} as identity element.

If {{tmath|\mathcal S}} is a nonempty set of sets, its intersection, denoted 
<math display=inline>\bigcap_{A\in \mathcal S} A,</math> 
is the set whose elements are those elements that belong to all sets in {{tmath|\mathcal S}}. That is,
<math display=block>\bigcap_{A\in \mathcal S} A =\{x\mid (\forall A\in \mathcal S)\; x\in A\}.</math>

These two definitions of the intersection coincide when {{tmath|\mathcal S}} has two elements. 

===Union===
[[File:Venn0111.svg|thumb|<div class="center">The ''union'' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
The ''[[set union|union]]'' of two sets {{tmath|A}} and {{tmath|B}} is a set denoted {{tmath|A \cup B}} whose elements are those elements that belong to {{tmath|A}} or {{tmath|B}} or both. That is,
<math display=block>A \cup B=\{x\mid x\in A \lor x\in B\},</math>
where {{tmath|\lor}} denotes the [[logical or]]. 

Union is [[associative]] and [[commutative]]; this means that for proceeding a sequence of intersections, one may proceed in any order, without the need of parentheses for specifying the [[order of operations]]. The empty set is an [[identity element]] for the union operation.

If {{tmath|\mathcal S}} is a  set of sets, its union, denoted 
<math display=inline>\bigcup_{A\in \mathcal S} A,</math> 
is the set whose elements are those elements that belong to at least one set in {{tmath|\mathcal S}}. That is,
<math display=block>\bigcup_{A\in \mathcal S} A =\{x\mid (\exists A\in \mathcal S)\; x\in A\}.</math>

These two definitions of the union coincide when {{tmath|\mathcal S}} has two elements. 

===Set difference===
[[File:Venn0100.svg|thumb|<div class="center">The ''set difference'' {{math|''A'' \ ''B''}}</div>]]
The ''set difference'' of two sets {{tmath|A}} and {{tmath|B}}, is a set, denoted {{tmath|A \setminus B}} or {{tmath|A - B}}, whose elements are those elements that belong to {{tmath|A}}, but not to {{tmath|B}}. That is,
<math display=block>A \setminus B=\{x\mid x\in A \land x\not\in B\},</math>
where {{tmath|\land}} denotes the [[logical and]]. 

[[File:Venn1010.svg|thumb|<div class="center">The ''complement'' of ''A'' in ''U''</div>]]
When {{tmath|B\subseteq A}} the difference {{tmath|A \setminus B}} is also called the ''[[set complement|complement]]'' of {{tmath|B}} in {{tmath|A}}. When all sets that are considered are subsets of a fixed ''universal set'' {{tmath|U}}, the complement {{tmath|U \setminus A}} is often called the ''absolute complement'' of {{tmath|A}}.

[[File:Venn0110.svg|thumb|<div class="center">The ''symmetric difference'' of ''A'' and ''B''</div>]]
The ''[[symmetric difference]]'' of two sets {{tmath|A}} and {{tmath|B}}, denoted {{tmath|A\,\Delta\,B}}, is the set of those elements that belong to {{mvar|A}} or {{mvar|B}} but not to both:
<math display =block>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>

===Algebra of subsets===
{{main|Algebra of sets}}
The set of all subsets of a set {{tmath|U}} is called the [[powerset]] of {{tmath|U}}, often denoted {{tmath|\mathcal P(U)}}. The powerset is an algebraic structure whose main operations are union, intersection, set difference, symmetric difference and absolute complement (complement in {{tmath|U}}).

The powerset is a [[Boolean ring]] that has the symmetric difference as addition, the intersection as multiplication, the empty set as [[additive identity]], {{tmath|U}} as [[multiplicative identity]], and complement as additive inverse.

The powerset is also a [[Boolean algebra (structure)|Boolean algebra]] for which the ''join'' {{tmath|\lor}} is the union {{tmath|\cup}}, the ''meet'' {{tmath|\land}} is the intersection {{tmath|\cap}}, and the negation is the set complement. 

As every Boolean algebra, the power set is also a [[partially ordered set]] for set inclusion. It is also a [[complete lattice]].

The axioms of these structures induce many [[identities (mathematics)|identities]] relating subsets, which are detailed in the linked articles.