This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Exists

 Existsrepresents the statement that there exists a value of for which expr is True. Existsstates that there exists an satisfying the condition cond for which expr is True. Existsstates that there exist values for all the for which expr is True.
• Exists can be entered as . The character can be entered as Esc ex Esc or \[Exists]. The variable is given as a subscript.
• The condition cond is often used to specify the domain of a variable, as in Integers.
• The value of in Exists is taken to be localized, as in Block.
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:
This states that there exists a positive solution to the equation :
 Out[1]=
Use Resolve to get a condition on real parameters for which the statement is true:
 Out[2]=
Reduce gives the condition in a solved form:
 Out[3]=
 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:
 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 and not :
The statement is false, hence the region defined by is included in the region defined by :
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:
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:
New in 5