DS_Lecture_3.pdf - Lecture 3 Predicate Logic and Quantifiers Dr Chengjiang Long Computer Vision Researcher at Kitware Inc Adjunct Professor at SUNY at

DS_Lecture_3.pdf - Lecture 3 Predicate Logic and...

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Lecture 3: Predicate Logic and Quantifiers Dr. Chengjiang Long Computer Vision Researcher at Kitware Inc. Adjunct Professor at SUNY at Albany. Email: [email protected]
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C. Long Lecture 3 September 4, 2018 2 ICEN/ICSI210 Discrete Structures Outline Logical Equivalences Introduction to Predicate Logic Quantifiers
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C. Long Lecture 3 September 4, 2018 3 ICEN/ICSI210 Discrete Structures Outline Logical Equivalences Introduction to Predicate Logic Quantifiers
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C. Long Lecture 3 September 4, 2018 4 ICEN/ICSI210 Discrete Structures Logical Equivalences: Example 1 Are the propositions ( p ® q ) and ( ¬ p Ú q ) logically equivalent? To find out, we construct the truth tables for each: p q p ® q ¬ p ¬ p Ú q 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 0 1 The two columns in the truth table are identical, thus we conclude that ( p ® q ) º ( ¬ p Ú q )
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C. Long Lecture 3 September 4, 2018 5 ICEN/ICSI210 Discrete Structures Logical Equivalences: Example 2 Show that ( p ® r ) Ú ( q ® r ) º ( p Ù q ) ® r p q r p ® r q ® r (p ® r) Ú ( q ® r ) p Ù q ( p Ù q ) ® r 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 1 1 1 1 1
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C. Long Lecture 3 September 4, 2018 6 ICEN/ICSI210 Discrete Structures Logical Equivalences Involving Conditional Statements. p q ¬p q p q ¬q ¬p p q ¬p q p q ¬(p ¬q) ¬(p q) p ¬q (p q) (p r) p (q r) (p r) (q r) (p q) r (p q) (p r) p (q r) (p r) (q r) (p q) r Logical Equivalences: Cheat Sheet (1)
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C. Long Lecture 3 September 4, 2018 7 ICEN/ICSI210 Discrete Structures Identities (Equivalences) Name p T p p F p Identity laws p T T p F F Domination laws p p p p p p Idempotent laws ¬(¬p) p Double negation law p q q p p q q p Commutative laws ( p q) r p ( q r ) ( p q) r p ( q r ) Associative laws p ( q r) (p q) ( p r ) p ( q r) (p q) ( p r ) Distributive laws ¬(p q) ¬p ¬q ¬(p q) ¬p ¬q De Morgan’s laws p ( p q) p p ( p q) p Absorption laws p ¬p T p ¬p F Negation laws Logical Equivalences: Cheat Sheet (2)
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C. Long Lecture 3 September 4, 2018 8 ICEN/ICSI210 Discrete Structures Using Logical Equivalences: Example 1 Logical equivalences can be used to construct additional logical equivalences Example: Show that ( p Ù q ) ® q is a tautology 0. ( p Ù q ) ® q 1. º ¬ ( p Ù q ) Ú q Implication Law on 0 2. º ( ¬ p Ú ¬ q ) Ú q De Morgan’s Law (1 st ) on 1 3. º ¬ p Ú ( ¬ q Ú q) Associative Law on 2 4. º ¬ p Ú 1 Negation Law on 3 5. º 1 Domination Law on 4
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C. Long Lecture 3 September 4, 2018 9 ICEN/ICSI210 Discrete Structures Using Logical Equivalences: Example 3 Example (Exercise 17)*: Show that ¬ ( p « q ) º ( p « ¬ q ) Sometimes it helps to start with the second proposition ( p « ¬ q ) 0. ( p « ¬ q ) 1. º ( p ® ¬ q ) Ù ( ¬ q ® p ) Equivalence Law on 0 2. º ( ¬ p Ú ¬ q ) Ù ( q Ú p ) Implication Law on 1 3. º
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