Exists

Exists[x,expr]

represents the statement that there exists a value of x for which expr is True.

Exists[x,cond,expr]

states that there exists an x satisfying the condition cond for which expr is True.

Exists[{x1,x2,},expr]

states that there exist values for all the x_(i) for which expr is True.

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 xIntegers.
  • 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

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Basic Examples  (1)

This states that there exists a positive solution to the equation :

Use Resolve to get a condition on real parameters for which the statement is true:

Reduce gives the condition in a solved form:

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:

Wolfram Research (2003), Exists, Wolfram Language function, https://reference.wolfram.com/language/ref/Exists.html.

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_2023_exists, author="Wolfram Research", title="{Exists}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/Exists.html}", note=[Accessed: 18-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_exists, organization={Wolfram Research}, title={Exists}, year={2003}, url={https://reference.wolfram.com/language/ref/Exists.html}, note=[Accessed: 18-March-2024 ]}