6: Definitions and proof methods Last updated Save as PDF Page ID 83425 Jeremy Sylvestre University of Alberta Augustana 6.1: Definitions Definitions are used in mathematics to label objects that have special properties, and to group all such objects together. 6.2: Common mathematical statements In mathematics, we often want to prove that some statement P logically implies some other statement Q; i.e. we want to prove that P⇒Q or (∀x)(P(x)⇒Q(x)). 6.3: Direct Proof The argument A→C1,C1→C2,…,Cm−1→Cm,Cm→B∴A→B is valid (Extended Law of Syllogism). 6.4: Reduction to Cases The following logical equivalence holds: (s1∨s2∨⋯∨sm)→t⇔(s1→t)∧(s2→t)∧⋯∧(sm→t). 6.5: Statements Involving Disjunction First, let's consider a conditional statement with a disjunction on the hypothesis side. 6.6: Proving the contrapositive Modus tollens: P→Q⇔¬Q→¬P. 6.7: Proof by counterexample Sometimes we want to prove that P⇏Q; i.e. that P→Q is not a tautology. 6.8: Proving biconditionals We also often want to prove that two statements P,Q are equivalent; i.e. that P⇔Q. 6.9: Proof by Contradiction