How to write a proof

Conditional Proof Proof by Cases Proof by Contradiction Proof by Induction

Intro

There are 3 common types of proofs

1. If \(P\), then \(Q\). (\(P \implies Q\))
2. \(P\) iff \(Q\) (\(P \iff Q\)) (iff stands for if and only if)
3. If \(P\) then (a), (b) and (c) are equivalent

If \(P\), then \(Q\)

There are two ways to prove

1. Conditional Proof
2. Proof by Contradiction
3. Proof by Contrapositive
4. Proof by Induction

\(P\) iff \(Q\)

There are two ways to prove

1. Prove \(P \implies Q\) and then prove \(Q \implies P\) (First prove forward \(\implies\) and then prove backward \(\impliedby\))
2. A chain of iff \[ \begin{align*} &P \\ &\iff R \\ &\iff \dots\\ &\iff Q \end{align*} \]

Where to start

1. Start from \(P\) (Which part of \(P\)?)
2. Start from \(Q\) (Which part of \(Q\)?)
3. Start from \(P\) and \(Q\) at the same time (Which part of \(P\), \(Q\)?)

by what?

1. by the sense of direction
2. by cases
3. by examples
4. by graph
5. by attempts
6. by analogy
7. by pattern-method
8. by experiences
9. by intuition

Now we know where to start

The first step and further steps?

1. by definition
2. by theorem or propositions
3. by transformations/operations
4. by techniques
5. by axioms

Change the focus point

When you

1. (have no idea how to carry on) get stuck
2. (know how to carry on but) find it super hard/complex to going forward in this direction
3. finished this part
4. feeling I might ignored some important points/things