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Reduce

The function Reduce has been extended to solve any combination of equalities, inequalities, existential quantifiers, universal quantifiers, and domain specifications, making it the most comprehensive symbolic solving function available today. Automatic switching of algorithms is utilized extensively to achieve the range of capabilities available with Reduce.

Example: Solving x² - 2y² = 1 over Different Domains
The equation

is solved over different domains. In all cases there are infinitely many solutions.

Example: Equations with Infinitely Many Roots

Reduce will generate complete solutions for equations with infinitely many roots. such as the following example.

Example: Equations Including Quantifiers

Equations that include quantifiers, such as the existential and universal quantifiers represented in Mathematica by Exists () and ForAll (), can be solved.

This gives the a and b values that make the quadratic polynomial positive for all real x.

$Reduce[ [ForAll]_(x, x [ElementOf] R) x^2 + a x + b [GreaterThanOrEqualTo] 0, {a, b}, R]$

		Documentation from The Mathematica Book Inequalities Equations and Inequalities over Domains Definite Integrals

		Documentation from the Reference Guide `Reduce` `Assumptions` `ForAll`

		Other Links Automatic Algorithm Selection (Key Technology)