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 for which expr is True.

DetailsDetails

  • Exists[x,expr] can be entered as . The character can be entered as EscexEsc or \[Exists]. The variable is given as a subscript.
  • Exists[x,cond,expr] can be entered as .
  • In StandardForm, Exists[x,expr] is output as .
  • Exists[x,cond,expr] is output as .
  • 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 .
  • The value of 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 :

In[1]:=
Click for copyable input
Out[1]=

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

In[2]:=
Click for copyable input
Out[2]=

Reduce gives the condition in a solved form:

In[3]:=
Click for copyable input
Out[3]=
Introduced in 2003
(5.0)