6: Definitions and proof methods
- Page ID
- 83425
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- 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.