# Teaching introduction to mathematical proofs

I recently begin to teach how to write and read mathematical proofs to some close friends. I started showing that, in math, we use logic to show that statements are true, and our first theorem was:

"The sum of two even integers is always a even integer".

The proof goes like this:

Suppose $a$ and $b$ are two even integers. Thus, by definition of even integers, there exists two integers $c$ and $d$ such that

$a = 2c$ and $b = 2d$.

it follows that

$a + b = 2c + 2d = 2(c+d)$ (by distributive property)

Note that $c + d$ is a integer, because the sum of two integers is also a integers. Hence there exits a integer (namely $c+d$) such that $a+b$ is a twice that number. therefore, by definition of even integer, $a + b$ is a even integer.

And one of them, ask me the following question :

"How did you prove that the sum of two even integers is a even integers if you only show that the sum of a and b (which are letters not numbers) is a even integer?"

Can someone with more experience help me with this? how to explain variables and constants in proofs?

Thanks.

• a and b are symbols that represent two arbitrary even integers (so they have all the properties of even integers). By showing it works for a and b, you are in a sense proving this for any two even integers. Another way to think of this is that the only facts about a and b assumed in this proof apply to all even integers, so you can sort of substitute in any of your desired even integers to get a proof that their sum in particular is even. – Mauve Dec 4 '17 at 15:53
• It depends on how deep they want to get into math. I would say you could start with very basic logic, like propositional logic; at least that's how I was taught. I'd make sure they grasp the concept of implication, negation etc, and then have it apply to more general ideas. – Marc Dec 4 '17 at 15:59
• Ask them if they've ever heard of someone discussing some general business notion (sales, marketing, etc.) using the term widget. But, to more fully address your issue, if it were me, then I would simply repeat the argument for several specific even numbers. – Dave L. Renfro Dec 4 '17 at 16:03
• We already studied some propositional calculus (even notions of FOL), what i'm having trouble is in explaning the use of variables in proofs. – Vinicius L. Deloi Dec 4 '17 at 16:04

You state at the beginning of the proof that $a$ and $b$ are even integers. They are not letters they are numbers axiomatically. You have assumed this truth as the starting point of the proof. Math always begins with unjustified assumptions which we choose to accept but there is no limit to how skeptical you can be. Ultimately what axioms you choose to accept as true is rather arbitrary and subject to scrutiny. For example Euclidean and non-Euclidian geometries each have axioms that directly contradict each other however they are self-contained and can each be studied separately by agreeing to different axioms as a starting point. As long as the axioms are internally consistent we can have a meaningful conversation about the consequences of those assumptions even if we don't believe them.