Wolfram Language & System 11.0 (2016)|Legacy Documentation

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

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

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


  • Exists[x,expr] can be entered as  exists _(x)expr. The character  exists can be entered as EscexEsc 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.

ExamplesExamplesopen allclose all

Basic Examples  (1)Basic Examples  (1)

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

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

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Reduce gives the condition in a solved form:

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Introduced in 2003