Summary
Proving implications
- direct proof using modus ponens
- proof by contradiction, very little information
- proof by contrapositive using modus tollens
Counter-example (disprove)
Proving biconditonals
- prove the implication in both directions
- proving logical equivalences
Proof by cases
Concept
Builds upon the rules of inference
Universal instantiation
- if true for all, then true for one instance
Existential generalization
- if true for one, then there exists one or more
Quantified statements in conditional form
Application
Pigeonhole principle, if there are more cards then boxes, then there is a(at least one) box with more than one card
- general
- no ordering