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51 Midterm Solutions

# 51 Midterm Solutions - Handout#51 CS103 Robert Plummer...

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Handout #51 CS103 May 18, 2010 Robert Plummer Midterm Solutions 1. Logic Proof (20 points) (a) In this part, you are proving one of the De Morgan theorems for quantifiers. You are not allowed to use (nor will you need) the other De Morgan theorems for quantifiers. 1. ¬ x P(x) 2. Let c be an arbitrary member of the domain 3. P(c) 4. x P(x) Existential generalization, 3 5. Contradiction, 1, 4 6. P(c) Negation elimination, 3-5 7. x ¬P(x) Universal generalization, 2-6 That is all there is to it! To prove that something is universally true [in this case ¬P(x)], we consider an arbitrary element of the domain and show that the thing we want to prove is true for that element. So the form is the subproof from line 2 to line 6. That is the only way we can do a universal generalization, and the universally quantified result is outside thesubproof. Here is a common erroneous proof: 1. ¬ x P(x) 2. ¬ x ¬P(x) 3. Let c be an arbitrary member of the domain 4. ¬P(c) 5. x ¬P(x) Universal generalization, 4 ERROR! 6. Contradiction, 2, 5 7. P(c) Negation introduction, 4-6 8. x P(x) Existential introduction, 7 ERROR! 9. Contradiction, 1, 8 10. x ¬P(x) Negation introduction, 2-9 You cannot do the universal generalization at step 5. The pattern is not the one shown in the first proof. Step 8 is also incorrect. The options here would have been the existential introduction inside the subproof, or universal introduction outside the subproof (because at this point you would have the right pattern). Neither helps.

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2 Here is another incorrect proof: 1. ¬ x P(x) 2. ¬ P(c) Existential instantiation, 1 ERROR! 3. x ¬P(x) Universal generalization, 2 It is not possible to instantiate the existential quantifier, since it has a negation in front. To do the instantiation, the first symbol has to be . You could get line 2 by first applying De Morgan's Law, but that is what you are tying to prove! (b) In this part, you are allowed to use De Morgan's laws freely, both the quantified and unquantified versions , but cite them as justification if you do. Note that 'b' is a constant in this proof. 1. x ( (A(x) ¬B(x)) C(b) ) 2. ¬ x C(x) 3. y B(y) 4. B(d) For some d, existential instantiation, 3 5. (A(d) ¬B(d)) C(b) Universal instantiation, 1 6. ¬A(d) ¬B(d) C(b) Table 7, alternate form of 7. ¬A(d) C(b) Disjunctive syllogism, 4, 6 8. x ¬C(x) De Morgan's Law, 2 9. ¬C(b) Universal instantiation, 8 10. ¬A(d) Disjunctive syllogism, 7, 9 11. x ¬A(x) Existential instantiation, 10 12.  x A(x) De Morgan's Law, 11 Most of you did well on this part.
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51 Midterm Solutions - Handout#51 CS103 Robert Plummer...

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