Unformatted text preview: Lecture 9 Lecture 9 Counterexamples Counterexamples
Counterexample Equivalence: A truthvalue assignment to the sentential variables is a counterexample to the lefthand side [of a law] iff it is a counterexample to at least one of the implications on the righthand side. (p.128) Elementary Implications Elementary Implications I. II. Definition Claims If an elementary implication is valid, then it is selfevident. If an elementary implication is not self evident then there is a unique assignment to its sentential variables that constitutes a counterexample. FoolProof Method FoolProof Method Two kinds of self-evident implications Laws for conjunction: (∧╞) and (╞,∧ , ) Laws for disjunction: (∨ and (╞,∨ ,╞) ) Laws for conditional: (→,╞) and (╞,→) Laws for bi-conditional: (↔,╞) and (╞,↔) Laws for negated compounds. If you know (∧╞) ,(╞,∧ (∨ and (╞,→), then you , ), ,╞) If ), shouldn’t have much difficulty in recalling the others. Example Problems Example Problems [A → (B∨ ∧ ╞ ¬C → A C)] B A∧ , ¬B → ¬ C╞ C B B, ¬C → A ╞ [A → (B∨ ∧ C)] B Proofs by Contradiction Proofs by Contradiction Take ⊥ to be your favorite contradiction. I’ll assume that you have one and that it is unique. Observe that Γ ╞ A if and only if Γ,¬A ╞ ⊥ This suggests the following proof strategy: Replace initial goal Γ ╞ A with equivalent A with equivalent goal Γ,¬A ╞ ⊥ . Examples Examples ¬A, A → B╞ ¬B A∧ , ¬B → ¬ C╞ C B B, A →B, B → C╞ ¬C → ¬A ...
View Full Document
- Spring '11
- Counterexamples in Topology, Counterexamples Counterexamples, Implications Elementary Implications, sentential variables