The Wolfram Language has a flexible system for specifying arbitrary symbolic assumptions about variables. It uses a wide range of sophisticated algorithms to infer the consequences of assumptions—often in the process automatically proving a sequence of necessary mathematical theorems.

Element (∈ ) — specify membership in a domain (entered as elem)

NotElement (∉ ) — specify exclusion from a domain (!elem)

Less (< ), Greater (> ), ... — define inequalities, implicitly for real numbers

ForAll (∀ ) — universal quantifier (entered as fa)

Exists (∃ ) — existential quantifier (entered as ex)

Domains

Reals   ▪ Integers   ▪ Complexes   ▪ Algebraics   ▪ Primes   ▪ Rationals   ▪ Booleans

PositiveReals   ▪ NegativeReals   ▪ NonNegativeReals   ▪ NonPositiveReals

PositiveRationals   ▪ NegativeRationals   ▪ NonNegativeRationals   ▪ NonPositiveRationals

PositiveIntegers   ▪ NegativeIntegers   ▪ NonNegativeIntegers   ▪ NonPositiveIntegers

Refine — evaluate an expression using assumptions

Simplify , FullSimplify — simplify using assumptions

FunctionExpand — expand in terms of simpler functions, using assumptions

Assuming — set up assumptions to be used by functions inside

$Assumptions — global default for Assumptions option

Functions Allowing Domain Specifications

Reduce   ▪ Resolve   ▪ FindInstance   ▪ Minimize   ▪ NMinimize   ▪ ...

Computing Domain and Range of Functions

FunctionDomain — find the domain of a function

FunctionRange — find the range of a function

Related Tech Notes

▪
• Using Assumptions
▪
• Simplifying with Assumptions
▪
• Quantifiers
▪
• Equations and Inequalities over Domains
▪
• Complex Polynomial Systems

Related Guides

▪
• Polynomial Systems
▪
• Inequalities
▪
• Logic & Boolean Algebra