I was asked by an avid reader (I always wanted to write that), to cover the different terms in mathematics regarding proofs, so here is a post which covers some of the terms which I think we will see a lot more of.

## Definitions

Let us start with one of the fundamental things within the world of mathematics – the definitions**. **In example 1 in the blog post on direct proofs we defined the term divisibility. A definition is an exact and unambiguous definition of a term or phrase. Another example is Also used in the aforementioned blog post is

Definition:A function is calledone-to-oneif no two different elements in D have the same element in R. This is the same as for any pair a,b in X such that f(a) = f(b) then a = b

You can make silly definitions, and definitions which are counter intuitive or different from the normal definition of a whatever you are defining.

I could define that

Definition:Ayellow numberis a natural number that is only divisible by 1 and the number it self.

I could then go on and prove that all prime numbers are yellow numbers. It is a silly definition but based on the definitions I can prove things.

## Axioms

In mathematics an axiom is something which is the starting point for the logical deduction of other theorems. They cannot be proven with a logic derivation unless they are redundant. That means every field in mathematics can be boiled down to a set of axioms. One of the axioms of arithmetic is that a + b = b + a. You can’t prove that, but it is the basis of arithmetic and something we use rather often.

## Propositions

A proposition is a statement that something is in a certain way. I could make the proposition that

Proposition:if x is odd then x^{2}is odd.

Which is something which should be rather easy to prove if we have the right definitions, but until I provide a proof for the statement it is nothing more than a proposition.

## Conjectures

There is very little difference between propositions and conjectures. A conjecture is an unproven statement which is believed to be true but has not yet been proven. Usually a proposition does not turn into a conjecture before it has been accepted public as something which is true but unproven.

There are lot of conjectures out there. The Goldbach conjecture stating that

Conjecture:Every even number larger than 2 can be written as a sum of two primes.

Other examples of conjectures are the Riemann hypothesis and the P vs. NP problem. Both are part of the Millennium prize problems which consists of 7 problems which by the beginning of this millennium was unsolved. Since then one of the problems have been solved, but I think that is the topic of another post.

Wikipedia states that there is a difference between a conjecture and a hypothesis, but I think that they are used interchangeable as we can see with the Riemaan hypothesis.

## Theorem

Now we are getting to the meat of the post. A theorem is a proposition which is true and has proven to be so. That is the short description of it. When you have a theorem with a supplied proof you will usually see a structure like this

Theorem:If P then Q

Proof:Assume P[logical deduction]

Then Q.

-QED

Depending on the style of writing a proof can end with a square either filled or not or QED which is short for the Latin phrase “*quod erat demonstrandum*” which means “what was to be demonstrated”.

One interesting not is that The Poincaré conjecture has changed status to become a theorem since it was proven by Perelman. However, I think it will keep it’s name.

## Lemma

There is not formal difference between a theorem and a lemma. A lemma is a proven proposition just like a theorem. Usually a lemma is used as a stepping stone for proving something larger. That means the convention is to call the main statement for a theorem and then split the problem into several smaller problems which are stated as lemmas. Wolfram suggest that a lemma is a short theorem used to prove something larger.

Breaking part of the main proof out into lemmas is a good way to create a structure in a proof and sometimes their importance will prove more valuable than the main theorem.

Some lemmas has significant impact on mathematics such as Bezout’s lemma and Gauss’s lemma to mention a few within number theory.

## Corollary

A corollary is a proposition which follows as a logical consequence of a theorem. If proposition B is a consequence of A then B is often called a corollary unless the impact of B is as important as A, in that case B is often referred to as a theorem instead.

On curious note on corollaries: Andrew Wiles announced the proof of Fermat’s last theorem as a corollary of his main claim. I think he also partly did this to downplay a pretty big achievement – that didn’t work in the long run.

Yay. An avid reader. That’s a very nice title. 🙂

But thanks for that. I only had a small idea about what some of these things meant.

I hope it covered what you needed for now, otherwise just ask.

Wow! At last a lucid account of something every math lover would like clarified!

Thanks a lot!

We can say that “Lemma” is also a theorem which is proposed and proved for performing the proof of a theorem , more often we can say individually lemma has no relevancy,just only the stair case of a theorem.