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lecture notes9 - Chapter1,PartI:PropositionalLogic...

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Chapter 1 , Part I: Propositional Logic With Question/Answer Animations
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Chapter Summary Propositional Logic The Language of Propositions Applications Logical Equivalences Predicate Logic The Language of Quantifiers Logical Equivalences Nested Quantifiers Proofs Rules of Inference Proof Methods Proof Strategy
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Proposi1onal Logic Summary The Language of Propositions Connectives Truth Values Truth Tables Applications Translating English Sentences System Specifications Logic Puzzles Logic Circuits Logical Equivalences Important Equivalences Showing Equivalence Satisfiability
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Section 1.1
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Sec1on Summary Propositions Connectives Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional Truth Tables
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Proposi1ons A proposition is a declarative sentence that is either true or false. Examples of propositions: a) The Moon is made of green cheese. b) Trenton is the capital of New Jersey. c) Toronto is the capital of Canada. d) 1 + 0 = 1 e) 0 + 0 = 2 Examples that are not propositions. a) Sit down! b) What time is it? c) x + 1 = 2 d) x + y = z
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Proposi1onal Logic Constructing Propositions Propositional Variables: p , q, r , s , … The proposition that is always true is denoted by T and the proposition that is always false is denoted by F . Compound Propositions; constructed from logical connectives and other propositions Negation ¬ Conjunction Disjunction Implication Biconditional
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Compound Proposi1ons: Nega1on The negation of a proposition p is denoted by ¬ p and has this truth table: Example : If p denotes “The earth is round.”, then ¬ p denotes “It is not the case that the earth is round,” or more simply “The earth is not round.” p ¬ p T F F T
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Conjunc1on The conjunction of propositions p and q is denoted by p q and has this truth table: Example : If p denotes “I am at home.” and q denotes “It is raining.” then p q denotes “I am at home and it is raining.” p q p q T T T T F F F T F F F F
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Disjunc1on The disjunction of propositions p and q is denoted by p q and has this truth table: Example : If p denotes “I am at home.” and q denotes “It is raining.” then p q denotes “I am at home or it is raining.” p q p q T T T T F T F T T F F F
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The Connec1ve Or in English In English “or” has two distinct meanings. “Inclusive Or” ‐ In the sentence “Students who have taken CS 202 or Math 120 may take this class,” we assume that students need to have taken one of the prerequisites, but may have taken both. This is the meaning of disjunction. For p q to be true, either one or both of p and q must be true.
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