Lecture 18: summary of proof techniques

Reading: fa15 notes
Systematic review of proof techniques
Suggestions for when you get stuck on a proof

Proof advice:

it's often useful to use the definitions you have, either because you know them or because they're given in the problem
statements to be proven often suggest a proof structure. For example, a proof that $∀ x, P(x)$ often starts with "choose an arbitrary $x$ ." See the fa15 notes for a systematic list of proof structures.

Looking at induction a different way:

One way to summarize induction is by the statement:

Induction principle: $P(0)$ and $∀n, (P(n) \implies P(n+1))$ together imply $∀n, P(n)$ .

Breaking this down by the rules linked above, to use this prove $∀n, P(n)$ , you can first prove ( $P(0)$ and $∀n, (P(n) \implies P(n+1))$ . Proving $P(0)$ is of course the base case, while proving $∀n, (P(n)\implies P(n+1))$ gives the inductive step. How does one prove $∀n, (P(n) \implies P(n+1))$ ? By the table linked above, you start by choosing an arbitrary $n$ . Then, you are trying to prove $P(n) \implies P(n+1)$ for that $n$ ; so you assume $P(n)$ , and your goal becomes showing that $P(n+1)$ holds.