# Terminology: Difference between Lemma, Theorem, Definition, Hypothesis, Postulate and a Proposition

Based on observation after reading few books and papers, I think that

Lemma : Lemma contains some information that is commonly used to support a theorem. So, a Lemma introduces a Theorem and comes before that Theorem. The information contained in the Lemma is generally used in the proof of the Theorem.

Q1: What is a Proposition ?

Q2: What is the difference between Proposition and Theorem ? A Proposition can also be proved, in the same way as a Theorem is proven.

Hypothesis : A hypothesis is like a statement for a guess, and we need to prove that analytically or experimentally.

Q3: What is the difference between Theorem and Hypothesis, for example Null hypothesis in statistics? In general, if a Theorem is always proven to be true then it no longer becomes a Hypothesis? Am I correct?

Q4: What is the difference between Postulate and Theorem?

Q5 : What is the difference between a proposition and a definition?

Looking for easy to remember answers. Thank you for help.

• A definition is just a 'naming' of an object and not like a theorem at all... a proposition is a theorem... usually an 'easy theorem' setting out some basic facts. – JP McCarthy Apr 22 '15 at 17:38

I'm not the authority on this, but this is how I interpret all of these words in math literature:

Definition - This is an assignment of language and syntax to some property of a set, function, or other object. A definition is not something you prove, it is something someone assigns. Often you will want to prove that something satisfies a definition. Example: We call a mapping $f:X\to Y$ injective if whenever $f(x) = f(y)$ then $x=y$.

Proposition - This is a property that one can derive easily or directly from a given definition of an object. Example: the identity element in a group is unique.

Lemma - This is a property that one can derive or prove which is usually technical in nature and is not of primary importance to the overall body of knowledge one is trying to develop. Usually lemmas are there as precursors to larger results that one wants to obtain, or introduce a new technique or tool that one can use over and over again. Example: In a Hausdorff space, compact subsets can be separated by disjoint open subsets.

Theorem - This is a property of major importance that one can derive which usually has far-sweeping consequences for the area of math one is studying. Theorems don't necessarily need the support of propositions or lemmas, but they often do require other smaller results to support their evidence. Example: Every manifold has a simply connected covering space.

Corollary - This is usually a result that is a direct consequence of a major theorem. Often times a theorem lends itself to other smaller results or special cases which can be shown by simpler methods once a theorem is proven. Example: A consequence to the Hopf-Rinow theorem is that compact manifolds are geodesically complete.

Conjecture - This is an educated prediction that one makes based on their experience. The difference between a conjecture and a lemma/theorem/corollary is that it is usually an open research problem that either has no answer, or some partial answer. Conjectures are usually only considered important if they are authored by someone well-known in their respective area of mathematics. Once it is proven or disproven, it ceases to be a conjecture and either becomes a fact (backed by a theorem) or there is some interesting counterexample to demonstrate how it is wrong. Example: The Poincar$\acute{\text{e}}$ conjecture was a famous statement that remained an open research problem in topology for roughly a century. The claim was that every simply connected, compact 3-manifold was homeomorphic to the 3-sphere $\mathbb{S}^3$. This statement however is no longer a conjecture since it was famously proven by Grigori Perelman in 2003.

Postulate - I would appreciate community input on this, but I haven't seen this word used in any of the texts/papers I read. I would assume that this is synonymous with proposition.

• I know Postulate is a synonym of axiom. Very used word in italian, but more in physics than mathematics. See wikipedia en.wikipedia.org/wiki/Axiom – user3621272 Apr 23 '15 at 9:24
• @Mnifldz and user3621272: Thank you very much for the answers; it is easy to understand and clear. – SKM Apr 23 '15 at 17:01