Built-in Mathematica Symbol

Exists ()

Exists[x, expr]
represents the statement that there exists a value of

for which expr is True.

Exists[x, cond, expr]
states that there exists an

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.

MORE INFORMATION

Exists[x, expr] can be entered as . The character can be entered as Esc ex Esc 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.

EXAMPLES

CLOSE ALL

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:

This states that there exists a positive solution to the equation

:

In[1]:=

Out[1]=

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

In[2]:=

Out[2]=

Reduce gives the condition in a solved form:

In[3]:=

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:

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: