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20+Practice+Midterm+Solutions

# 20+Practice+Midterm+Solutions - Handout#20 CS103 Robert...

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Handout #20 CS103 April 18, 2011 Robert Plummer Practice Midterm Solutions 1. Propositional Logic (20 points) 1. (W P) I 2. I (C S) 3. S U 4. ¬C ¬U 5. U Simplification, 4 6. S Modus tollens, 3, 5 7. C Simplification, 4 8. C S Conjunction, 6,7 9. (C S) De Morgan's, 8 10. I Modus tollens, 9, 2 11. (W P) Modus tollens, 10, 1 12. W P De Morgan's, 11 13. W Simplification, 12 2. Proof with Quantifiers (20 points) 1. x ( P(x) y(S(y) L(x, y)) ) 2. x (P(x) y(C(y) ¬L(x, y))) 3. Let c be an object such that ( P(c) y(S(y) L(c, y)) Exist. Inst., 1 4. P(c) Simplification, 3 5. y (S(y) L(c, y)) Simplification, 3 6. Suppose a is an arbitrary object 7. Suppose S(a) 8. S(a) L(c, a) Universal Inst., 5 9. L(c, a) Modus ponens, 6, 7 10. P(c) y(C(y) L(c, y)) Universal Inst., 2 11. y(C(y) L(c, y)) Modus ponens, 4, 10 12. C(a) L(c, a) Universal Inst., 11 13. C(a) Modus tollens, 12, 9 14. x(S(x) ¬C(x)) General Conditional Proof, 6-14

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