L01 - First Order Logic Propositional Logic A proposition...

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First Order Logic
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Propositional Logic A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false , but not both. Are the following sentences propositions? Toronto is the capital of Canada. Read this carefully. 1+2=3 x+1=2 What time is it? Propositional Logic the area of logic that deals with propositions (No) (No) (No) (Yes) (Yes) 2
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Propositional Variables Propositional Variables –var iab lesthat represent propositions: p, q, r, s E.g. Proposition p–“Today± is ±Fr iday .” Truth values –T , ±F 3
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Negation Examples Find the negation of the proposition “Today is Friday.” and express this in simple English. Find the negation of the proposition “At least 10 inches of rain fell today in Miami.” and express this in simple English. DEFINITION 1 Let p be a proposition. The negation of p, denoted by ¬p, is the statement “It is not the case that p.” The proposition ¬p is read “not p.” The truth value of the negation of p, ¬pis the opposite of the truth value of p. Solution : The negation is “It is not the case that today is Friday. In simple English, “Today is not Friday.” or “It is not Friday today.” Solution : The negation is “It is not the case that at least 10 inches of rain fell today in Miami.” In simple English, “Less than 10 inches of rain fell today in Miami.” 4
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Truth Table Truth table: Logical operators are used to form new propositions from two or more existing propositions. The logical operators are also called connectives. The Truth Table for the Negation of a Proposition . p ¬ p T F F T 5
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Conjunction Examples Find the conjunction of the propositions p and q where p is the proposition “Today is Friday.” and q is the proposition “It is raining today.”, and the truth value of the conjunction. DEFINITION 2 Let p and q be propositions. The conjunction of p and q, denoted by p Λ q, is the proposition “pand q”. The conjunction p Λ qis true when both p and q are true and is false otherwise. Solution : The conjunction is the proposition “Today is Friday and it is raining today.” The proposition is true on rainy Fridays. 6
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Disjunction Note: inclusive or: The disjunction is true when at least one of the two propositions is true. E.g. “Students who have taken calculus or computer science can take this class.” those who take one or both classes. exclusive or : The disjunction is true only when one of the proposition is true. E.g. “Students who have taken calculus or computer science, but not both, can take this class.” only those who take one of them. Definition 3 uses inclusive or . DEFINITION 3 Let p and q be propositions. The disjunction of p and q, denoted by p ν q, is the proposition “por q”. The conjunction p ν qis false when both p and q are false and is true otherwise.
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This note was uploaded on 04/22/2010 for the course CS 23022 taught by Professor Staff during the Spring '08 term at Kent State.

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L01 - First Order Logic Propositional Logic A proposition...

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