# Reduce

Usage

Reduce[expr, vars] reduces the statement expr by solving equations or inequalities for vars and eliminating quantifiers.
Reduce[expr, vars, dom] does the reduction over the domain dom. Common choices of dom are Reals, Integers and Complexes.

Notes

• 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 ForAll[x, cond, expr] universal quantifiers Exists[x, cond, expr] existential quantifiers
• The result of Reduce[expr, vars] always describes exactly the same mathematical set as expr.
Reduce[{, , ... }, vars] is equivalent to Reduce[ && && ... , vars].
Reduce[expr, vars] assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.
Reduce[expr, vars, dom] restricts all variables and parameters to belong to the domain dom.
• If dom is Reals, or a subset such as Integers or Rationals, then all constants and function values are also restricted to be real.
Reduce[expr && vars Reals, vars, Complexes] performs reductions with variables assumed real, but function values allowed to be complex.
Reduce[expr, vars, Integers] reduces Diophantine equations over the integers.
Reduce[expr, {, , ... }, ... ] effectively writes expr as a combination of conditions on , , ... , where each condition involves only the earlier .
• Algebraic variables in expr free of the are treated as independent parameters.
• Applying LogicalExpand to the results of Reduce[expr, ... ] yields an expression of the form || || ... , where each of the can be thought of as representing a separate component in the set defined by expr.
• The may not be disjoint, and may have different dimensions. After LogicalExpand, each of the have the form e && e && ... .
• Without LogicalExpand, Reduce by default returns a nested collection of conditions on the , combined alternately by Or and And on successive levels.
• When expr involves only polynomial equations and inequalities over real or complex domains then Reduce can always in principle solve directly for all the .
• When expr involves transcendental conditions or integer domains Reduce will often introduce additional parameters in its results.
• When expr involves only polynomial conditions, Reduce[expr, vars, Reals] gives a cylindrical algebraic decomposition of expr.
Reduce can give explicit representations for solutions to all linear equations and inequalities over the integers, and can solve a large fraction of Diophantine equations described in the literature.
• When expr involves only polynomial conditions over real or complex domains, Reduce[expr, vars] will always eliminate quantifiers, so that quantified variables do not appear in the result.
• The following options can be given:
 Backsubstitution whether to give results unwound by backsubstitution Cubics whether to use explicit radicals to solve all cubics how to name parameters that are generated 0 modulus to assume for integers Quartics whether to use explicit radicals to solve all quartics
Reduce[expr, {, , ... }, Backsubstitution->True] yields a form in which values from equations generated for earlier are backsubstituted so that the conditions for a particular have only minimal dependence on earlier .
• Implementation notes: see Section A.9.5.
• New in Version 1; modified in 5.

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