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Assumptions and Domains
Mathematica 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.
Reference
Element (
) — specify membership in a domain (entered as EscelemEsc)
NotElement (
) — specify exclusion from a domain (Esc!elemEsc)
Less (<), Greater (>), ... — define inequalities, implicitly for real numbers
ForAll (
) — universal quantifier (entered as EscfaEsc)
Exists (
) — existential quantifier (entered as EscexEsc)
Domains
Reals ▪ Integers ▪ Complexes ▪ Algebraics ▪ Primes ▪ Rationals ▪ Booleans
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 ▪ 
