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I read somewhere that mathematics is all about statements describing properties of some abstract objects. These are:

  1. Predicate: Statement which can be either True or False.
  2. PrepositionProposition: Need clarification.
  3. Theorem: A statement which is proved to be True
  4. Lemma: Intermediate proof (I think it as of a subroutine in programming).
  5. Corollary: Need clarification.
  6. Axiom: A statement too trivial to be proved.
  7. Premise: A previous statement from which another statement follows as a conclusion (Seems like lemma or axiom to me).

These are the only ones known to me.

Note:

  • I have collected all the information from different source as a result of Google search.
  • Its not a homework, I have been always wanted a single source of information, right now I feel its scattered.
vivek
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  • You probably meant proposition. –  Oct 28 '15 at 09:03
  • @YvesDaoust done. – vivek Oct 28 '15 at 09:06
  • A common misconception is that an axiom is so trivial that we don't need to prove it. The fact is that we can't! We assume it as true to build math over it but we can't prove it anyway, because all the proof come after it. – AlienRem Oct 28 '15 at 11:01

3 Answers3

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In mathematics, we start from axioms in order to prove statements, which are usually called theorems, lemmas, corollaries. There is not much difference between these types of statements: all need proofs.

Axiom : a statement assumed to be true without proof.

Theorem : a statement proved from axioms or previously proved theorems.

Corollary : a statement that follow easily from other results; usually, a "particular case", or a consequence of a theorem that needs few inference steps to be derived.

Lemma : is a statement used in the proofs of other results; in case of a complex proof of a theorem, can be useful to split the proof in parts : some preliminary results, called lemmata, and a final one : the theorem itself.

Premise : a statement assumed as true in an argument; the consequences of the premises are true, provided that the premises are.

Proposition : a statement "asserting" a fact (questions, e.g. are not usually considered propositions), like "$2$ is even". It must be true or false.

Predicate : an expression relative to a property, like "$x$ is even" (unary predicate) or a relation, like "$x$ is greater than $y$" (binary predicate).

Thus, if you consider the statement "$x$ is even" with the variable $x$, it is correct to say that it is neither true nor false, because it works like a "recipe" to produce true or false statements, according to the value assigned to the variable $x$.

For $3$ as value for $x$, we get the statement "$3$ is even", that is false, while for $2$ as value for $x$, we get the statement "$2$ is even", that is true.

Mauro ALLEGRANZA
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  • Premise and Proposition seems similar to me. Am I missing something? – vivek Oct 28 '15 at 09:09
  • @vivekpoddar - A premise is a proposition, but it is asserted in the context of an argument. Not necessarily so for a proposition : I can assert the prop "Obama is the Pres of U.S." without "building" any argument with it. – Mauro ALLEGRANZA Oct 28 '15 at 10:12
  • Hmm, makes sense and the example you gave me for prop above "2 is even" seems like its a special case of the predicate where predicate gives us a nice function when filled with some value gives us yes or no answer. – vivek Oct 29 '15 at 02:37
  • Suppose there is a statement "x is even when x^2 is even" then this statement is Lemma, axiom or anything else and why? – vivek Oct 29 '15 at 02:39
  • @vivekpoddar - Very useful : Ethan Bloch, Proofs and Fundamentals : A First Course in Abstract Mathematics (2nd ed - 2011) : Ch.2.6 Writing Mathematics, page 80-on. – Mauro ALLEGRANZA Oct 29 '15 at 07:09
  • @vivekpoddar - yes; a predicate is a sort of "propositional function". In the same way as the numerical function $f(x)=x^2$ produces a number as output every time you "input" a number (i.e. you assign a value to the variable $x$), the "propositional function "$x$ is even" produces a prioposition every time you assign a value to $x$. The output is a proposition because it is definitely true or false. An other example of prop function is "$x$ is a philosopher"; for $Socrates$ as value for $x$, we get a true pro, while with $Napoleon$ we get a false prop. – Mauro ALLEGRANZA Oct 29 '15 at 07:13
  • So, basically all these statements take either True or False and nothing else, right? – vivek Oct 30 '15 at 05:09
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As for Lemma/Proposition/Theorem/Corollary, there is no formal difference between them.

The way they are most commonly used are as markers of "importance".

  • A Theorem is a major result, likely to be useful and applicable in other contexts
  • A Proposistion is a minor result, but still interesting enough to formulate on its own
  • A Lemma is usually a technical result, probably not very useful on its own, but useful to simplify the proofs of a theorem or proposition. The analogy with subroutines from programming is pretty good.
  • A Corollary is an almost immediate consequence of a theorem/proposition, whose proof is almost obvious once we have established the main result.
mrf
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There's no formal distinction between "proposition", "theorem", "lemma" and "corollary", but conventionally a lemma is just a minor theorem just used to prove a more important theorem (often formulated and proved as part of the proof of the theorem that uses it) and a corallary is something that follows (immediately) from a theorem (normally stated immediately after). But there are theorems that are called lemmas that are quite central. Some authors consequently call everything a theorem or proposition.

A premise is the left hand side of a implication and is something that doesn't need to be universally true (it even doesn't have to be possibly true). For example in Pythagoras theorem the premise is that the triangle is right - this is not universally true for triangles. Normally the premise is of the form that it states something about some entity (in Pythagoras theorem it states something about the triangle) and that the consequence of the theorem will therefore only apply to entities that satisfy the premise.

An axiom on the other hand is a statement that is taken to be true without proof (this is different from a lemma which is a statement which is taken to true because it has been proven). It's often a statement that is universally true, but it can also be a statement of existence of something. For example Peano's axioms contains both of these - the first axioms states existence, but the third (that if the successor of two numbers are equals then the numbers them selves ar equal) is a statement that's universally true (for numbers).

skyking
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