({{, (RefLink[Element, paclet:ref/Element])[x,dom]
or x∈dom asserts that x is an element of the domain dom. }, {, (RefLink[Element, paclet:ref/Element])[x,reg]
or x∈reg asserts that x is an element of the region reg.}, {, (RefLink[Element, paclet:ref/Element])[(Subscript[x, 1])|(Subscript[x, 2])|…,dom]
asserts that all the (Subscript[x, i]) are elements of dom. }, {, (RefLink[Element, paclet:ref/Element])[patt,dom]
asserts that any expression matching the pattern patt is an element of dom. }})

OpenersNotesSection(Spacer1)Details

x∈dom can be entered as x (Key0[esc]) el (Key0[esc]) dom or x∖[Element]dom.

(RefLink[Element, paclet:ref/Element]) can be used to set up assumptions in (RefLink[Simplify, paclet:ref/Simplify]) and related functions.

dom may be a numeric domain or a region in (ℝ^n).

Possible domains dom are:

Possible regions reg are defined by (RefLink[RegionQ, paclet:ref/RegionQ]).

x∈dom if possible evaluates immediately when x is numeric.

For a domain dom, {(Subscript[x, 1]),(Subscript[x, 2]),…}∈dom is equivalent to ((Subscript[x, 1])|(Subscript[x, 2])|…)∈dom.

For a region reg, {(Subscript[x, 1]),(Subscript[x, 2]),…}∈reg asserts that the point with coordinates (Subscript[x, 1]),(Subscript[x, 2]),… belongs to reg.

{(Subscript[x, 1]),(Subscript[x, 2]),…}∈dom evaluates to ((Subscript[x, 1])|(Subscript[x, 2])|…)∈dom if its truth or falsity cannot immediately be determined.

OpenersPrimaryExamplesSection(Spacer1)Examples  (17)

OpenersExampleSection0(Spacer1)Basic Examples  (5)

Test whether π is an element of the reals:

Pi ∈ Reals

Test whether the point {(1/2),(1/3)} belongs to the unit disk:

{1/2,1/3}∈Disk[]

Express domain membership for an expression:

Element[x+y,Reals]

Assert that the point {x,y,z} belongs to the unit ball:

Element[{x,y,z},Ball[]]

Use element assertions to integrate over a region:

Integrate[1,{x,y,z}∈Ball[]]

Or to optimize over a region:

MinValue[x+y,{x,y,z}∈Ball[]]

Enter using (Key0[esc]) elem (Key0[esc]):

x∈Reals

OpenersExampleSection1(Spacer1)Scope  (9)

Test domain membership:

Element[Sqrt[2],#]&/@{Complexes,Algebraics,Reals,Rationals,Integers,Primes}

Test region membership:

(Subscript[ℛ, 1])=Disk[{1,1}];
(Subscript[ℛ, 2])=Rectangle[{0,0},{1,1}];
(Subscript[ℛ, 3])=ImplicitRegion[x^2≤y^3+1,{x,y}];
(Subscript[ℛ, 4])=RegionUnion[Disk[{-1,0}],Line[{{0,0},{2,2}}]];

Element[{3/2,3/2},#]&/@{(Subscript[ℛ, 1]),(Subscript[ℛ, 2]),(Subscript[ℛ, 3]),(Subscript[ℛ, 4])}

Plot it:

Show[{DiscretizeRegion[(Subscript[ℛ, 3])],Graphics[{{Opacity[0.5],{Orange,(Subscript[ℛ, 1])},{Yellow,(Subscript[ℛ, 2])},{Green,Disk[{-1,0}],Line[{{0,0},{2,2}}]}},{Red,PointSize[Large],Point[{3/2,3/2}]}}]}]

Make domain membership assumptions:

Refine[Sin[Pi x],Element[x,Integers]]

Integrate[Abs[1-Abs[x+2]],x,Assumptions->Element[x,Reals]]

Express region membership:

Refine[Sqrt[(x+y-2)^2],Element[{x,y},Disk[]]]

Solve[x+y==1/2&&Element[{x,y},Circle[]],{x,y}]

Integrate[1,Element[{x,y},Circle[]]]

Test domain membership using assumptions:

Refine[Element[x^y+Log[x+y],Reals],x>0&&y>0]

Test region membership using assumptions:

Refine[Element[{x,y},Rectangle[{-1,-1},{1,1}]],x^2+y^2≤1]

Specify variable domains:

Reduce[E^x-2x==3 && Element[x,Reals],x]

FindInstance[x^2-2y^2==1&&x>10^10&&Element[x|y,Integers],{x,y}]

Minimize[{x-y,x^2+y^2≤100&&Element[x|y,Integers]},{x,y}]

Specify assumptions on objects matching a pattern:

Simplify[Sin[2Pi x[1]+4x[2]x[3]^2Pi+a Pi],Element[x[_],Integers]]

(RefLink[TraditionalForm, paclet:ref/TraditionalForm]) formatting:

Element[x,Reals]//TraditionalForm

OpenersExampleSection3(Spacer1)Properties & Relations  (2)

For a single variable, the negation of (RefLink[Element, paclet:ref/Element]) is automatically converted to (RefLink[NotElement, paclet:ref/NotElement]):

!Element[a,Reals]

For multiple variables, the negation of (RefLink[Element, paclet:ref/Element]) is not automatically simplified:

!Element[a|b|c,Reals]

Use (RefLink[LogicalExpand, paclet:ref/LogicalExpand]) to find the representation in terms of (RefLink[NotElement, paclet:ref/NotElement]):

LogicalExpand[%]

(RefLink[Element, paclet:ref/Element]) asserts region membership:

Element[{x,y},Disk[]]

(RefLink[RegionMember, paclet:ref/RegionMember]) gives explicit region membership conditions:

RegionMember[Disk[],{x,y}]

OpenersExampleSection4(Spacer1)Possible Issues  (1)

When domain membership cannot be decided the (RefLink[Element, paclet:ref/Element]) statement remains unevaluated:

Element[I^Pi,Algebraics]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Simplify, paclet:ref/Simplify])  ▪  (RefLink[MemberQ, paclet:ref/MemberQ])  ▪  (RefLink[IntegerQ, paclet:ref/IntegerQ])  ▪  (RefLink[Assumptions, paclet:ref/Assumptions])  ▪  (RefLink[Condition, paclet:ref/Condition])  ▪  (RefLink[PatternTest, paclet:ref/PatternTest])  ▪  (RefLink[Equal, paclet:ref/Equal])  ▪  (RefLink[Less, paclet:ref/Less])  ▪  (RefLink[Divisible, paclet:ref/Divisible])  ▪  (RefLink[CoprimeQ, paclet:ref/CoprimeQ])  ▪  (RefLink[Booleans, paclet:ref/Booleans])  ▪  (RefLink[Primes, paclet:ref/Primes])  ▪  (RefLink[Exists, paclet:ref/Exists])  ▪  (RefLink[ForAll, paclet:ref/ForAll])  ▪  (RefLink[Distributed, paclet:ref/Distributed])  ▪  (RefLink[GeometricScene, paclet:ref/GeometricScene])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Simplifying with Assumptions, paclet:tutorial/SimplifyingWithAssumptions])}, {▪(RefLinkPlain[Using Assumptions, paclet:tutorial/UsingAssumptions])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Assumptions and Domains, paclet:guide/AssumptionsAndDomains])}, {▪(RefLinkPlain[Region Properties and Measures, paclet:guide/GeometricPropertiesAndMeasures])}, {▪(RefLinkPlain[Number Theory, paclet:guide/NumberTheory])}, {▪(RefLinkPlain[Number Recognition, paclet:guide/NumberRecognition])}, {▪(RefLinkPlain[Operations on Sets, paclet:guide/OperationsOnSets])}, {▪(RefLinkPlain[Synthetic Geometry, paclet:guide/SyntheticGeometry])}, {▪(RefLinkPlain[Formula Manipulation, paclet:guide/FormulaManipulation])}, {▪(RefLinkPlain[Wolfram Language Syntax, paclet:guide/Syntax])}, {▪(RefLinkPlain[Theorem Proving, paclet:guide/TheoremProving])}})}})

({{(Spacer1), ({{(Spacer1)Related Links}, {(WebLink[DocumentationFooterIconsRelatedLinks-NKS.png(Spacer1)NKS|Online, http://www.wolframscience.com/nksonline/index/search.cgi?SearchIndex=Element]) ((WebLink[A New Kind of Science, http://www.wolframscience.com/nksonline/]))}})}})