lecture5 - Lecture 5 Rules of Inference (cont) Proofs...

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Lecture 5 Rules of Inference (cont) Proofs
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Recap: nested quantifiers x y: There exists an x such that for all y … x y: For all x, there exits a y, …
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Recap: nested quantifiers. . and their order x y and x y are not equivalent! x y P(x,y) P(x,y) = (x+y == 0) is true x y P(x,y) P(x,y) = (x+y == 0) is false
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Summary: Rules of Inference Modus ponens p p q q Modus tollens q p q p Hypothetical syllogism p q q r p r Disjunctive syllogism p q p q Addition p p q Simplification p q p Conjunction p q p q Resolution p q p r q r
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Example “If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on” ( r f) (s d) “If the sailing race is held, then the trophy will be awarded” s t “The trophy was not awarded” t Can you conclude: “It rained”?
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Example proof 1. ¬t 3 rd premise 2. s → t 2 nd premise 3. ¬s 4. (¬r ¬f)→(s d) 1 st premise 5. ¬(s d)→¬(¬r ¬f) Contrapositive of step 4 6. (¬s ¬d)→(r f) DeMorgan’s law, double negation law 7. ¬s ¬d Addition from step 3 8. r f 9. r Simplification using step 8
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Example (#35 on p74) If Superman were able and willing to prevent evil, he would do so. If Superman were unable to prevent
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lecture5 - Lecture 5 Rules of Inference (cont) Proofs...

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