Theorems, Lemmas and Other definitions

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 called one-to-one if 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: A yellow number is 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 x2 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.

Posted by Kristian

4 comments

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!

MOLOYA BHUYAN

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.

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