Conditional Proof Proof by Cases Proof by Contradiction Proof by Induction
Intro
There are 3 common types of proofs
- If \(P\), then \(Q\). (\(P \implies Q\))
- \(P\) iff \(Q\) (\(P \iff Q\)) (
iff
stands forif and only if
) - If \(P\) then (a), (b) and (c) are equivalent
If \(P\), then \(Q\)
There are two ways to prove
- Conditional Proof
- Proof by Contradiction
- Proof by Contrapositive
- Proof by Induction
\(P\) iff \(Q\)
There are two ways to prove
- Prove \(P \implies Q\) and then prove \(Q \implies P\) (First prove forward \(\implies\) and then prove backward \(\impliedby\))
- A chain of
iff
\[ \begin{align*} &P \\ &\iff R \\ &\iff \dots\\ &\iff Q \end{align*} \]
Where to start
- Start from \(P\) (Which part of \(P\)?)
- Start from \(Q\) (Which part of \(Q\)?)
- Start from \(P\) and \(Q\) at the same time (Which part of \(P\), \(Q\)?)
by what?
- by the sense of direction
- by cases
- by examples
- by graph
- by attempts
- by analogy
- by pattern-method
- by experiences
- by intuition
Now we know where to start
The first step and further steps?
- by definition
- by theorem or propositions
- by transformations/operations
- by techniques
- by axioms
Change the focus point
When you
- (have no idea how to carry on) get stuck
- (know how to carry on but) find it super hard/complex to going forward in this direction
- finished this part
- feeling I might ignored some important points/things