Learn about some of the methods mathematicians use to prove things!

Proof by contradiction: If you want to prove a statement A, make an assumption that A is false. Then, show why this assumption is impossible.

• Example: we want to prove that the Earth is not flat. First, assume that the claim is false (i.e, assume that the Earth is flat). Then, if you keep walking forward in any direction you should eventually reach an "edge" where if you jumped off, you will float into space. However, this is impossible since this "edge" doesn't exist. Therefore, the Earth is not flat

Proof by contraposition: you can prove that that A implies B by also showing that "not B" implies "not A"

• Example: we want to prove that not doing any assignments (claim A) for a class will result in an F for that class (claim B). We can prove this by first assuming that a student did not get an F in a class(not B). This means the student got higher than 0% in the class, which implies that the student did at least 1 assignment (not A). We have proven that (not B) implies (not A), so A implies B

Proof by (strong) induction: If you want prove that a relation holds for all natural numbers, first show that it holds for the first natural number (base case). Then, show that if the relation holds for an arbitrary natural number (or, for strong induction, all smaller natural numbers), it must hold for the next natural number.

• This is like a domino effect, since if the relation keeps holding for the next natural number it will hold for all

Direct proof: Nothing special here. Just prove a statement directly from the given axioms