Proofs

Proof method: Proof by contradiction

I was first presented with a proof by contradiction while I was studying Discrete event systems in Canada. And I was puzzled about it most day. I came to really like it though.

When we want to prove something by contradiction we assume that the statement we want to prove is false and then show that it leads to a logic contradiction at some point, therefore the statement must be true. Don’t be confused just yet. I will come to the examples.

Proof by contradiction is not limited to conditional statements like the the direct proof is. So we don’t need to have a proposition on the form if Q then P. Continue reading →

Posted by Kristian in Math, 11 comments

News on proving the Collatz Conjecture

A few weeks ago Gerhard Opfer posted a preprint of a paper titled An analytic approach to the Collatz 3n+1 problem. The paper claims the proof of the Collatz conjecture. In it self that isn’t something really interesting as there are probably several hundred people every year who think they have proven the Collatz conjecture. However, there is one difference here as the paper comes from a research institute. It might sound a little arrogant and imply that non-mathematicians don’t understand math. That is not what I imply, just that the sender packs a bit more punch by being from a research institute. Continue reading →

Posted by Kristian in Math, 9 comments

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. Continue reading →

Posted by Kristian in Math, 4 comments

Proof method: Direct Proof

In my first post on my journey for improving my mathematical rigour I said that I would go through a few different techniques for conducting proofs.

The first one I want to dabble into is direct proofs.  This is the “simplest” method and sometimes it can seem that the proof isn’t there at all.

It will often go something like “if a then b”. So using some definition of a, we can show that b follows as a direct consequence through an unbroken line of logical arguments such that

a -> … -> b

Lets try it out on some sample problems Continue reading →

Posted by Kristian in Math, 3 comments

Improving my mathematical rigour

I have reached a point in my mathematical journey where I feel the need to learn how to make sound arguments for the validity of a mathematical claim. Or in other words, I want to learn more on how to prove things.

The path I took through the Danish educational system has never dealt much with mathematical proofs, but rather on how to apply the mathematics we have learned. I have developed an intuition for mathematics in some areas. But I lack mathematical rigour, so I often time have to resolve to hand waving instead.

The usual approach to learning proving techniques is through a taught topic where you are presented with some proofs. Through that you will expand your toolbox and learn how to do proofs. However, I would through a series of blog posts dabble into how to prove mathematical things and study different techniques.

Ben Tilly pointed me through his blog – random observations – to a document he wrote on how to do proofs. It has a flow chart which you can also see to the below, which I think is a very thorough way to ensure that you get through the proof. It doesn’t say anything about how to actually make your arguments, but it helps you break down the problem.

Let me spend the rest of this blog post to go through the flow chart and interpret it. Continue reading →

Posted by Kristian in Math, 0 comments