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?"
And i couldn't answer him.
Can someone with more experience help me with this? how to explain variables and constants in proofs?