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