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CS2013 Mathematics for Computing Science Adam Wyner University of Aberdeen Computing Science

CS2013 Mathematics for Computing Science Adam Wyner University of Aberdeen Computing Science. Slides adapted from Michael P. Frank ' s course based on the text Discrete Mathematics & Its Applications (5 th Edition) by Kenneth H. Rosen. Proof Replacement & Quantifiers. Topics.

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CS2013 Mathematics for Computing Science Adam Wyner University of Aberdeen Computing Science

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  1. CS2013Mathematics for Computing ScienceAdam WynerUniversity of AberdeenComputing Science Slides adapted from Michael P. Frank's course based on the textDiscrete Mathematics & Its Applications(5th Edition)by Kenneth H. Rosen

  2. ProofReplacement & Quantifiers

  3. Topics • Equivalences that can be used to replace formulae in proofs • Further examples of Propositional Logic proofs. • Proof rules with quantifiers. Frank / van Deemter / Wyner

  4. Equivalences Equivalence expressions can be substitutedsince they do not change truth. Frank / van Deemter / Wyner

  5. Equivalences Frank / van Deemter / Wyner

  6. A Direct Proof 1. ((A ∨ ¬ B) ∨ C)(D  (E F)) 2. (A ∨ ¬ B)((F G)H) 3. A  ((EF)(FG)) 4. A 5. Show: D  H 6. A ∨ ¬ B 7. (A ∨ ¬ B) ∨ C 8. (D  (E F)) 9. (E F) (F G) 10. D (F G) 11. (F G) H 12. DH Frank / van Deemter / Wyner

  7. A Direct Proof 1. ((A ∨ ¬ B) ∨ C)(D  (E F)) 2. (A ∨ ¬ B)((F G)H) 3. A  ((EF)(FG)) 4. A 5. Show: D  H DD 12 6. A ∨ ¬ B DI 4 7. (A ∨ ¬ B) ∨ C DI 6 8. (D  (E F)) IE 1,7 9. (E F) (F G) IE 3,4 10. D (F G) HS 8,9 11. (F G) H IE 2,6 12. DH HS 10,11 Frank / van Deemter / Wyner

  8. A Conditional Proof 1. (A ∨ B)(C ∧ D) 2. (D ∨ E)F 3. Show: A  F 4. A 5. Show: F 6. A ∨ B 7. C ∧ D 8. D 9. (D ∨ E) 10. F Frank / van Deemter / Wyner

  9. A Conditional Proof 1. (A ∨ B)(C ∧ D) 2. (D ∨ E)F 3. Show: A  F CD 4, 5 4. A Assumption 5. Show: F DD 10 6. A ∨ B DI 4 7. C ∧ D IE 1,6 8. D CE 7 9. (D ∨ E) DI 8 10. F IE 2,9 Frank / van Deemter / Wyner

  10. An Indirect Proof 1. A (B ∧ C) 2. (B ∨ D)E 3. (D ∨ A) 4. Show: E ID 13 5. Assumption 6. IE 2,5 7. Second De Morgan 6 8. CE 7 9. DE 4,8 10. IE 1,9 11. CE 10 12. CE 7 13.ContraI 11,12 Frank / van Deemter / Wyner

  11. An Indirect Proof 1. A (B ∧ C) 2. (B ∨ D)E 3. (D ∨ A) 3. Show: E 4. ¬ E 5. ¬ (B ∨ D) 6. ¬ B ∧¬ D 7. ¬ D 8. A 9. B ∧C 10. B 11. ¬ B 12. B ∧¬ B Frank / van Deemter / Wyner

  12. An Indirect Proof 1. A (B ∧ C) 2. (B ∨ D)E 3. (D ∨ A) 4. Show: E ID 13 5. ¬ E Assumption 6. ¬ (B ∨ D) IE 2,5 7. ¬ B ∧¬ D Second De Morgan 6 8. ¬ D CE 7 9. A DE 4,8 10. B ∧C IE 1,9 11. B CE 10 12. ¬ B CE 7 13. B ∧¬ B ContraI 11,12 Frank / van Deemter / Wyner

  13. A Logical Equivalence Prove: ¬ (p ∨ (¬ p ∧ q)) is equivalent to ¬ p ∧¬ q 1. ¬ (p ∨ (¬ p ∧ q))  .... Frank / van Deemter / Wyner

  14. A Logical Equivalence Prove: ¬ (p ∨ (¬ p ∧ q)) is equivalent to ¬ p ∧¬ q 1. ¬ (p ∨ (¬ p ∧ q))  second De Morgan 2.  first De Morgan 3. double negation 4. second distributive 5. negation 6. commutativity 7. identity law for F Frank / van Deemter / Wyner

  15. A Logical Equivalence Prove: ¬ (p ∨ (¬ p ∧ q)) is equivalent to ¬ p ∧¬ q 1. ¬ (p ∨ (¬ p ∧ q))  ¬ p ∧¬ (¬ p ∧ q) 2.  ¬ p ∧(¬ (¬ p) ∨¬ q) 3. ¬ p ∧(p ∨¬ q) 4. (¬ p ∧ p) ∨ ( ¬ p ∧ ¬ q) 5. F ∨ ( ¬ p ∧ ¬ q) 6. ( ¬ p ∧ ¬ q) ∨ F 7. ( ¬ p ∧ ¬ q) Frank / van Deemter / Wyner

  16. A Logical Equivalence Prove: ¬ (p ∨ (¬ p ∧ q)) is equivalent to ¬ p ∧¬ q 1. ¬ (p ∨ (¬ p ∧ q))  ¬ p ∧¬ (¬ p ∧ q) second De Morgan 2.  ¬ p ∧(¬ (¬ p) ∨¬ q) first De Morgan 3. ¬ p ∧(p ∨¬ q) double negation 4. (¬ p ∧ p) ∨ ( ¬ p ∧ ¬ q)second distributive 5. F ∨ ( ¬ p ∧ ¬ q) negation 6. ( ¬ p ∧ ¬ q) ∨ F commutativity 7. ( ¬ p ∧ ¬ q)identity law for F Frank / van Deemter / Wyner

  17. Universal Instantiation • xP(x)P(o) (substitute any constant o) The same for any other variable than x. Frank / van Deemter / Wyner

  18. Existential Generalization • P(o) xP(x) The same for any other variable than x. Frank / van Deemter / Wyner

  19. Universal Generalisation • P(g) xP(x) • This is not a valid inference of course. But suppose you can prove P(g) without using any information about g ... • ... then the inference to xP(x) is valid! • In other words ... Frank / van Deemter / Wyner

  20. Universal Generalisation • P(g) (for g an arbitrary or general constant)xP(x) • Concretely, your strategy should be to choose a new constant g (i.e., that did not occur in your proof so far) and to prove P(g). Frank / van Deemter / Wyner

  21. Existential Instantiation • xP(x)P(c) (substitute a newconstantc) Once again, the inference is not generally valid, but we can regard it as valid if c is a new constant. Frank / van Deemter / Wyner

  22. Simple Formal Proof in Predicate :ogic • Argument: • “All TAs compose quizzes. Ramesh is a TA. Therefore, Ramesh composes quizzes.” • First, separate the premises from conclusions: • Premise #1: All TAs compose quizzes. • Premise #2: Ramesh is a TA. • Conclusion: Ramesh composes quizzes. Frank / van Deemter / Wyner

  23. Rendering in Logic Render the example in logic notation. • Premise #1: All TAs compose easy quizzes. • Let U.D. = all people • Let T(x) :≡ “x is a TA” • Let E(x) :≡ “x composes quizzes” • Then Premise #1 says: x(T(x)→E(x)) Frank / van Deemter / Wyner

  24. Rendering cont… • Premise #2: Ramesh is a TA. • Let r :≡ Ramesh • Then Premise #2 says: T(r) • And the Conclusion says: E(r) • The argument is correct, because it can be reduced to a sequence of applications of valid inference rules, as follows: Frank / van Deemter / Wyner

  25. Formal Proof UsingNatural Deduction StatementHow obtained • x(T(x) → E(x)) (Premise #1) • T(r) → E(r) (Universal instantiation) • T(r) (Premise #2) • E(r) (MP 2, 3) Frank / van Deemter / Wyner

  26. A very similar proof Can you prove: • x(T(x) → E(x)) and E(r) •  T(r). Frank / van Deemter / Wyner

  27.  in Natural Deduction A very simple example: Theorem: From xF(x) it follows that yF(y) • xF(x) (Premiss) • F(a) (Arbitrary a, Exist. Inst.) 3. yF(y) (Exist. Generalisation) Frank / van Deemter / Wyner

  28. Quantifier Rules ß Frank / van Deemter / Wyner

  29. Longer Quantifier Proof Frank / van Deemter / Wyner

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