Exists
Details
- Exists [x,expr] can be entered as exists _(x)expr. The character exists can be entered as ex or \[Exists]. The variable x is given as a subscript.
- Exists [x,cond,expr] can be entered as exists _(x,cond)expr.
- In StandardForm , Exists [x,expr] is output as exists _(x)expr.
- Exists [x,cond,expr] is output as exists _(x,cond)expr.
- Exists can be used in such functions as Reduce , Resolve , and FullSimplify .
- The condition cond is often used to specify the domain of a variable, as in x∈Integers .
- Exists [x,cond,expr] is equivalent to Exists [x,cond&&expr].
- Exists [{x1,x2,…},…] is equivalent to exists _(x_(1)) exists _(x_(2))....
- The value of x in Exists [x,expr] is taken to be localized, as in Block .
Examples
open all close allBasic Examples (1)
Scope (6)
This states that there exists for which the equation is true:
Use Resolve to prove that the statement is true:
This states that there exists a real for which the equation is true:
Use Resolve to prove that the statement is false:
This states that there exists a pair for which the inequality is true:
With domain not specified, Resolve considers algebraic variables in inequalities to be real:
With domain Complexes , complex values that make the inequality True are allowed:
This states that the negation of a tautology is satisfiable:
Use Resolve to prove it False :
If the expression does not explicitly contain the variable, Exists simplifies automatically:
TraditionalForm formatting:
Applications (4)
This states that a quadratic attains negative values:
This gives explicit conditions on real parameters:
Test whether one region is included in another:
This states that there are points satisfying R1 and not R2:
The statement is false, hence the region defined by R1 is included in the region defined by R2:
Plot the relationship:
Test geometric conjectures:
This states that there is a triangle for which the conjecture is not true:
The statement is true, hence the conjecture is not true for arbitrary triangles:
This states that there is an acute triangle for which the conjecture is not true:
The statement is false, hence the conjecture is true for all acute triangles:
Prove that a statement is a tautology:
This proves that there are no values of for which the statement is not true:
This can be proven with TautologyQ as well:
Properties & Relations (5)
Negation of Exists gives ForAll :
Quantifiers can be eliminated using Resolve or Reduce :
This eliminates the quantifier:
This eliminates the quantifier and solves the resulting equations and inequalities:
This shows that a system of inequalities has solutions:
Use FindInstance to find an explicit solution instance:
This states that there exists a complex for which the equations are satisfied:
Use Resolve to find conditions on and for which the statement is true:
This solves the same problem using Eliminate :
This finds the projection of the complex algebraic set along the axis:
This finds the projection of the real unit disc along the axis:
See Also
ForAll FindInstance Resolve Disjunction Reduce Element Eliminate SatisfiableQ AnyTrue
Characters: \[Exists]
Tech Notes
Related Guides
Related Links
History
Introduced in 2003 (5.0)
Text
Wolfram Research (2003), Exists, Wolfram Language function, https://reference.wolfram.com/language/ref/Exists.html.
CMS
Wolfram Language. 2003. "Exists." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Exists.html.
APA
Wolfram Language. (2003). Exists. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Exists.html
BibTeX
@misc{reference.wolfram_2025_exists, author="Wolfram Research", title="{Exists}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/Exists.html}", note=[Accessed: 06-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_exists, organization={Wolfram Research}, title={Exists}, year={2003}, url={https://reference.wolfram.com/language/ref/Exists.html}, note=[Accessed: 06-January-2026]}