or xdom asserts that x is an element of the domain dom.


or xreg asserts that x is an element of the region reg.


asserts that all the xi are elements of dom.


asserts that any expression matching the pattern patt is an element of dom.


  • xdom can be entered as x el dom or x \[Element] dom.
  • Element can be used to set up assumptions in Simplify and related functions.
  • dom may be a numeric domain or a region in .
  • Possible domains dom are:
  • Algebraicsalgebraic numbers
    BooleansTrue or False
    Complexescomplex numbers
    Primesprime numbers
    Rationalsrational numbers
    Realsreal numbers
  • Possible regions reg are defined by RegionQ.
  • xdom if possible evaluates immediately when x is numeric.
  • For a domain dom, {x1,x2,}dom is equivalent to (x1|x2|)dom.
  • For a region reg, {x1,x2,}reg asserts that the point with coordinates x1,x2, belongs to reg.
  • {x1,x2,}dom evaluates to (x1|x2|)dom if its truth or falsity cannot immediately be determined.


open allclose all

Basic Examples  (5)

Test whether is an element of the reals:

Test whether the point belongs to the unit disk:

Express domain membership for an expression:

Assert that the point belongs to the unit ball:

Use element assertions to integrate over a region:

Or to optimize over a region:

Enter using elem:

Scope  (9)

Test domain membership:

Test region membership:

Plot it:

Make domain membership assumptions:

Express region membership:

Test domain membership using assumptions:

Test region membership using assumptions:

Specify variable domains:

Specify assumptions on objects matching a pattern:

TraditionalForm formatting:

Properties & Relations  (2)

For a single variable, the negation of Element is automatically converted to NotElement:

For multiple variables, the negation of Element is not automatically simplified:

Use LogicalExpand to find the representation in terms of NotElement:

Element asserts region membership:

RegionMember gives explicit region membership conditions:

Possible Issues  (1)

When domain membership cannot be decided the Element statement remains unevaluated:

Introduced in 1999
Updated in 2003