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Tuesday, October 23, 2012

Some basic definitions in Geometry

Definition — a precise and unambiguous description of the meaning of a mathematical term.  It characterizes the meaning of a word by giving all the properties and only those properties that must be true.
Theorem — a mathematical statement that is proved using rigorous mathematical reasoning.  In a mathematical paper, the term theorem is often reserved for the most important results.
Lemma — a minor result whose sole purpose is to help in proving a theorem.  It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own (Zorn’s lemmaUrysohn’s lemma, Burnside’s lemma, Sperner’s lemma).
Corollary — a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”).
Proposition — a proved and often interesting result, but generally less important than a theorem.
Conjecture — a statement that is unproved, but is believed to be true (Collatz conjecture, Goldbach conjecture, twin prime conjecture).
Claim — an assertion that is then proved.  It is often used like an informal lemma.
Axiom/Postulate — a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved (Euclid’s five postulates, Zermelo-Fraenkel axioms, Peano axioms).
Identity — a mathematical expression giving the equality of two (often variable) quantities (trigonometric identitiesEuler’s identity).
Paradox — a statement that can be shown, using a given set of axioms and definitions, to be both true and false. Paradoxes are often used to show the inconsistencies in a flawed theory (Russell’s paradox).  The term paradox is often used informally to describe a surprising or counterintuitive result that follows from a given set of rules (Banach-Tarski paradox, Alabama paradox, Gabriel’s horn).

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