Types of Proofs - Discrete Mathematical Structures | CS 2233, Study notes of Discrete Mathematics

Download Types of Proofs - Discrete Mathematical Structures | CS 2233 and more Study notes Discrete Mathematics in PDF only on Docsity! 9/10/08 CS 2233 Discrete Mathematical Structures -- Carola Wenk 1 Types of proofs Direct proof Indirect proof (Proof by contrapositive) Proof by contradiction Proof by cases Proof of equivalence For-all proof Counterexample Existence proof Want to show p→q p→q p p→q (p1 p2 … pn)→q p↔q p↔q↔r x: P(x) ¬ x: P(x) x: P(x) How to show it Assume p is true. Derive a chain of implications which in the end proves that q is true. Prove ¬q→ ¬ p with direct proof Show ¬p→ F Show p ¬q→ F Show (p1→q) (p2→q) … (pn→q) Show (p→q) (q→p) Show (p→q) (q→r) (r→p) Prove P(x) for an arbitrary x Induction Find x for which P(x) is false Constructive: Find x such that P(x) is true. Non-constructive: Show that P(x) is true for some x without finding it. case 1 ∀ ∀ ∀ ∃