Direct proof
You are given a hypothesis and asked for a conclusion. How do you get from one to the other with every step justified?
Where you are 2 inside
What this area is about
A direct proof of “if p, then q” assumes p and reasons forward, one justified step at a time, until it reaches q. Each step uses a definition, an earlier result or a rule of algebra or logic, and nothing else. The two lessons here practice it on integers, where the key move is the same both times: replace a word such as “even” or “divides” by the equation that defines it, do algebra on the equation, then read the word back off the result.
How these fit together
Both children are procedures built on a definition.
Proving statements about even and odd numbers uses n = 2k for even and n = 2k + 1 for odd, with k an integer, to prove rules for sums, products and powers, and to show some equations have no integer solution.
Proving statements about divisibility uses the definition that, for integers a and b, a \mid b means b = ak for some integer k. It proves the divisibility rules that later proofs cite and applies them to expressions.
A path through it
Even and odd first. The divisibility lesson reuses its shape, with the divisor 2 replaced by any integer, so it reads best second.
Direct proof is the base the other methods start from: proof by contradiction, proof by contraposition, proof by cases and proving a set identity all assume it.