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Unformatted text preview: Logic & Proof Discrete Math MATH 135 David Oury Prop. logic Propositions Truth tables Equivalences Predicate logic Predicates Quantifiers Equivalences Logic exercises Proof Direct Cases Contrapositive Contradiction Equivalence Induction Exercises Logic & Proof Discrete Math MATH 135 David Oury January 20, 2012 Chapter 1, Sections 1, 2, 3, 4, 5, 7 Chapter 5, Sections 1, 2 of Discrete Mathematics and its Applications, by Rosen Logic & Proof Discrete Math MATH 135 David Oury Prop. logic Propositions Truth tables Equivalences Predicate logic Predicates Quantifiers Equivalences Logic exercises Proof Direct Cases Contrapositive Contradiction Equivalence Induction Exercises Propositions/statements Proposition A declarative sentence that is either true or false. Examples Today is Friday. 2 = 1. Dogs are mammals. Not an example x + 1 = 2. Since x is a variable the sentence is neither true or false. We will see that this sentence is a predicate. Logic & Proof Discrete Math MATH 135 David Oury Prop. logic Propositions Truth tables Equivalences Predicate logic Predicates Quantifiers Equivalences Logic exercises Proof Direct Cases Contrapositive Contradiction Equivalence Induction Exercises Negated propositions Negation The negation p of a proposition p is the proposition It is not the case that p . Examples Today is not Friday. 2 6 = 1 . Logic & Proof Discrete Math MATH 135 David Oury Prop. logic Propositions Truth tables Equivalences Predicate logic Predicates Quantifiers Equivalences Logic exercises Proof Direct Cases Contrapositive Contradiction Equivalence Induction Exercises Conjunction & disjunction Conjunction The conjunction p q of propositions p and q is p and q . Disjunction The disjunction p q of propositions p and q is p or q . Examples Suppose p is the proposition Dogs are mammals. q is the proposition 2 = 1 . Then p q is the proposition Dogs are mammals and 2 = 1. p q is the proposition Dogs are mammals or 2 = 1 . Logic & Proof Discrete Math MATH 135 David Oury Prop. logic Propositions Truth tables Equivalences Predicate logic Predicates Quantifiers Equivalences Logic exercises Proof Direct Cases Contrapositive Contradiction Equivalence Induction Exercises Compound propositions A compound proposition is An (atomic) proposition. The negation of a compound proposition. The conjunction of two compound propositions. The disjunction of two compound propositions. Examples Suppose p , q and r are (compound) propositions. Then p p q ( p q ) r are all compound propositions. Logic & Proof Discrete Math MATH 135 David Oury Prop. logic Propositions Truth tables Equivalences Predicate logic Predicates Quantifiers Equivalences Logic exercises Proof Direct Cases Contrapositive Contradiction Equivalence Induction Exercises Truth tables Truth table A table listing all truth values for compound propositions....
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This note was uploaded on 02/11/2012 for the course MATH 135 taught by Professor Dr.oury during the Spring '12 term at MO St. Louis.
 Spring '12
 Dr.Oury
 Math, Logic

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