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FindInstance—Wolfram Language Documentation

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• See Also
  • Solve
  • Reduce
  • RandomSeeding
  • FindRoot
  • AsymptoticSolve
  • PowersRepresentations
  • IntegerPartitions
  • Minimize
  • RegionPlot
  • RegionPlot3D
  • CylindricalDecomposition
  • SemialgebraicComponentInstances
  • ChineseRemainder
  • Resolve
  • SatisfiabilityInstances
  • PolynomialSumOfSquaresList
• Related Guides
  • Equation Solving
  • Plane Geometry
  • Solid Geometry
  • Inequalities
  • Number Theory
  • Diophantine Equations
  • Prime Numbers
  • Theorem Proving
  • Finite Fields
  • Solvers over Regions
  • Logic & Boolean Algebra
  • Polynomial Systems
  • Symbolic Vectors, Matrices and Arrays
• Tech Notes
  • Inequalities
  • Equations and Inequalities over Domains
  • Complex Polynomial Systems

FindInstance[expr,vars]

finds an instance of vars that makes the statement expr be True.

• FindInstance[expr,{x₁,x₂,…}] gives results in the same form as Solve: {{x₁->val₁,x₂->val₂,…}} if an instance exists, and {} if it does not.
• expr can contain equations, inequalities, domain specifications and quantifiers, in the same form as in Reduce.
• The statement expr can be any logical combination of:
• lhs==rhs equations lhs!=rhs inequations lhs>rhs or lhs>=rhs inequalities expr∈dom domain specifications {x,y,…}∈reg region specification ForAll[x,cond,expr] universal quantifiers Exists[x,cond,expr] existential quantifiers
• With exact symbolic input, FindInstance gives exact results.
• Even if two inputs define the same mathematical set, FindInstance may still pick different instances to return.
• The instances returned by FindInstance typically correspond to special or interesting points in the set.
• FindInstance[expr,vars] assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.
• FindInstance[expr,vars,Integers] finds solutions to Diophantine equations.
• FindInstance[expr,vars,Booleans] solves Boolean satisfiability for expr.
• FindInstance[expr,vars,Reals] assumes that not only vars but also all function values in expr are real. FindInstance[expr&&vars∈Reals,vars] assumes only that the vars are real.
• FindInstance[…,x∈reg,Reals] constrains x to be in the region reg. The different coordinates for x can be referred to using Indexed[x,i].
• FindInstance may be able to find instances even if Reduce cannot give a complete reduction.
• By default, every time you run FindInstance with a given input, it will return the same output.
• FindInstance[expr,vars,dom,n] will return a shorter list if the total number of instances is less than n.
• The following options can be given:

Find a solution instance of a system of equations:

Find a real solution instance of a system of equations and inequalities:

Find an integer solution instance:

Find Boolean values of variables that satisfy a formula:

Find several instances:

Find a point in a geometric region:

A linear system:

A univariate polynomial equation:

Five roots of a polynomial of a high degree:

A multivariate polynomial equation:

Systems of polynomial equations and inequations:

This gives three solution instances:

If there are no solutions FindInstance returns an empty list:

If there are fewer solutions than the requested number, FindInstance returns all solutions:

Five out of a trillion roots of a polynomial system:

Quantified polynomial system:

An algebraic system:

Transcendental equations:

In this case there is no solution:

A solution in terms of transcendental Root objects:

Five roots of an unrestricted equation:

Systems of transcendental equations:

Three roots of a transcendental system:

A linear system:

A univariate polynomial equation:

A univariate polynomial inequality:

A multivariate polynomial equation:

A multivariate polynomial inequality:

Systems of polynomial equations and inequalities:

Get four solution instances:

If there are no solutions FindInstance returns an empty list:

If there are fewer solutions than the requested number, FindInstance returns all solutions:

A quantified polynomial system:

An algebraic system:

Piecewise equations:

Piecewise inequalities:

Transcendental equations:

A solution in terms of transcendental Root objects:

Transcendental inequalities:

Transcendental systems:

A linear system of equations:

A linear system of equations and inequalities:

Find more than one solution:

A univariate polynomial equation:

A univariate polynomial inequality:

Binary quadratic equations:

A Thue equation:

If there are fewer solutions than the requested number, FindInstance returns all solutions:

A sum of squares equation:

The Pythagorean equation:

A bounded system of equations and inequalities:

A high-degree system with no solution:

Transcendental Diophantine systems:

A polynomial system of congruences:

A linear system:

A univariate polynomial equation:

A multivariate polynomial equation:

Find seven instances:

A system of polynomial equations and inequations:

A quantified polynomial system:

Univariate equations:

Systems of linear equations:

Systems of polynomial equations:

Find three instances:

Systems involving quantifiers:

Mixed real and complex variables:

Find a real value of and a complex value of for which is real and less than :

An inequality involving Abs[z]:

Find instances in basic geometric regions in 2D:

Plot it:

Find instances in basic geometric regions in 3D:

Plot it:

Find a point in the projection of a region:

Plot it:

An implicitly defined region:

A parametrically defined region:

Derived regions:

Plot it:

Regions dependent on parameters:

Find values of parameters , , and for which the circles contain the given points:

Plot it:

Use to specify that is a vector in :

In this case is a vector in :

Find a solution over the integers modulo 9:

Find three solutions:

Finding instances often involves random choice from large solution sets:

By default, FindInstance chooses the same solutions each time:

Use RandomSeedingAutomatic to generate potentially new instances each time:

The region ℛ is a subset of  if is empty. Show that Disk[{0,0},{2,1}] is a subset of Rectangle[{-2,-1},{2,1}]:

Plot it:

Show that Rectangle[] is a not a subset of Disk[{0,0},7/5]:

Plot it:

Show that Cylinder[]⊆Ball[{0,0,0},2]:

Plot it:

Show that Cylinder[]⊈Ball[{0,0,0},7/5]:

Plot it:

Find a point in the intersection of two regions:

Find a counterexample to a geometric conjecture:

Prove the conjecture using stronger assumptions:

Prove that a statement is a tautology:

This can be proven with TautologyQ as well:

Show that a statement is not a tautology; get a counterexample:

This can be done with SatisfiabilityInstances as well:

Find a Pythagorean triple:

Find Pythagorean triples when they exist:

Two instances are now found when :

Find Pythagorean quadruples and visualize the result:

Generate all solutions for and visualize the result:

Show that there are no 2×2 magic squares with all numbers unequal:

Solution instances satisfy the input system:

Use RootReduce to prove that algebraic numbers satisfy equations:

When there are no solutions, FindInstance returns an empty list:

If there are fewer solutions than the requested number, FindInstance returns all solutions:

To get a complete description of the solution set use Reduce:

To get a generic solution of a system of complex equations use Solve:

Solving a sum of squares representation problem:

Use SquaresR to find the number of solutions to sum of squares problems:

Solving a sum of powers representation problem:

Use PowersRepresentations to enumerate all solutions:

Find instances satisfying a Boolean statement:

Use SatisfiabilityInstances to obtain solutions represented as Boolean vectors:

FindInstance shows that the polynomial is non-negative:

Use PolynomialSumOfSquaresList to represent as a sum of squares:

The Motzkin polynomial is non-negative, but is not a sum of squares:

Integer solutions for a Thue equation:

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