What I learnt from proof techniques

May 12, 2020

Mathematics - a field of study of topics such as number theory (quantity), algebra (structure) and geometry (space). “Mathematicians use patterns to formulate a new conclusion; resolving the truth of such conclusion with a mathematical proof.”

What is a mathematical proof? It is basically arguments along with stated assumptions that logically guarantee that the conclusion (or conjecture) is true. I was first introduced to mathematical proofs via a discrete math course back in my college. Proof by induction was the only proof technique that I picked up in that course. The course was brief in knowledge, touching the surface of every discrete part of math, barely dive deep into the “whys” of math. My hunger for the whys didn’t get satisified. I decided to look further.

In 2020, I embarked on a study abroad journey to a widely known to be the best mathematics school in Canada - University of Waterloo and enrolled myself into the one of the most rigorous math proving courses - Algebra for Hons Mathematics.

The course was monotonous at the start, but things start to get interesting as it dives deep into proofs. I learned at least 4 proof technique tools:

Proving Universally Quantified Statement

To prove a statement that is valid for all cases, you abstract a variable, and use it to define for every case within the domain. If it is true for such an abstract variable, then it will be true for all cases. To disprove this, all we need is a counterexample (a case that yields false).

Proving Existentially Quantified Statement

To prove a statement that is valid for for more than one case (p.s., this is what “existenally” means - true for more than one case), all we need is to find such a variable case, then prove that it is true.

To disprove such a statement, is to say that you want to prove the opposite. In general terms, say we have the statement: there is a dog that is green in colour. A contrary statement would be: there are no dogs in green colour. In other words, all we need is to show that every dog is not in green colour.

Proving Implications

To prove this, it is best illustrated with an example. Say we have a statement: if the fan is switched on, I will be cooled down. This is an implication because an event yields another event to happen. To prove this, we first assume that the fan is switched on, then we reason from this point and try to conclude that I will be cooled down. We first look into what happens when a fan is switched on. The blades got spinned, creating a flow of wind. Wind flows, evaporating the moisture in the air. When moisture in the air have been evaporated, the surrounding gets cooler, then subsequently my body gets cooled down. This proves the implied statement to be true.

Proving Contradiction

The basis of this is that no two events of the opposite of each other can happen simultaneously. For instance: luxury goods are expensive. Is that true? Let’s evaluate it. To do that, we create a self-contraditory statement: luxury goods are cheap. Starting from this point, we try to reach a contradiction (two events seemingly can’t happen at the same time). We know that luxury goods are usually made up of high-quality materials or ingredients. Since they are comprised of high-quality materials, they must be expensive (most businesses use this to yield profits!). But we have luxury goods that are cheap. It is certainly impossible to be cheap and expensive at the same time.

This is what it means to arrive to a contradiction. As we know that the contradictory statement can’t occur, that means the opposite of this is true. “We narrow our perceptions down into two possibilities: yes or no. Bob says the sky is raining. But Bob is wrong. So that means the sky must be raining.”

Four simple and logical proof techniques. They are unpractical on a theoretical sense, but extremely practical in the real world, it has close association and mimicks the logic section of the human brain. They make you to be a better thinker and a better problem solver.