This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
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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.
Element () specify membership in a domain (entered as Esc elem Esc)
NotElement () — specify exclusion from a domain (Esc !elem Esc)
Less (<), Greater (>), ... define inequalities, implicitly for real numbers
ForAll () — universal quantifier (entered as Esc fa Esc)
Exists () — existential quantifier (entered as Esc ex Esc)
Domains

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
TUTORIALS