DS_Lecture_2.pdf - Lecture 2 Propositional Logic Dr Chengjiang Long Computer Vision Researcher at Kitware Inc Adjunct Professor at SUNY at Albany Email

DS_Lecture_2.pdf - Lecture 2 Propositional Logic Dr...

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Lecture 2: Propositional Logic 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 2 August 30, 2018 2 ICEN/ICSI210 Discrete Structures Introduction: Logic? Logic 1) is the study of the logic relationships between objects 2) forms the basis of all mathematical reasoning and all automated reasoning
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C. Long Lecture 2 August 30, 2018 3 ICEN/ICSI210 Discrete Structures Introduction: PL? In Propositional Logic (a.k.a Propositional Calculus or Sentential Logic), the objects are called propositions Definition : A proposition is a statement that is either true or false , but not both We usually denote a proposition by a letter: p , q , r , s ,
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C. Long Lecture 2 August 30, 2018 4 ICEN/ICSI210 Discrete Structures Outline Defining Propositional Logic Precedence of Logical Operators Usefulness of Logic Logical Equivalences
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C. Long Lecture 2 August 30, 2018 5 ICEN/ICSI210 Discrete Structures Outline Defining Propositional Logic Precedence of Logical Operators Usefulness of Logic Logical Equivalences
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C. Long Lecture 2 August 30, 2018 6 ICEN/ICSI210 Discrete Structures Introduction: Proposition Definition : The value of a proposition is called its truth value ; denoted by T or 1 if it is true or F or 0 if it is false Opinions , interrogative , and imperative are not propositions Truth table p 0 1
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C. Long Lecture 2 August 30, 2018 7 ICEN/ICSI210 Discrete Structures Propositions Propositional logic operates with statements. Statements could be true or false and are called propositions . Is the sentence proposition? Richmond is the capital of Virginia. 2 + 3 = 7. Open the door. 5 + 7 < 10. The moon is a satellite of the earth. x + 5 = 7. x + 5 > 9 for every real number x . Yes (True) Yes (False) No Yes (False) Yes (True) No Yes (False)
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C. Long Lecture 2 August 30, 2018 8 ICEN/ICSI210 Discrete Structures Propositions: Examples The following are propositions Today is Monday The grass is wet It is raining The following are not propositions C++ is the best language Opinion When is the pretest? Interrogative Do your homework Imperative
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C. Long Lecture 2 August 30, 2018 9 ICEN/ICSI210 Discrete Structures Are these propositions? 2+2=5 Every integer is divisible by 12 Microsoft is an excellent company
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C. Long Lecture 2 August 30, 2018 10 ICEN/ICSI210 Discrete Structures Logical connectives Connectives are used to create a compound proposition from two or more propositions Negation (denote ~ or ¬ or !) $\neg$ And or logical conjunction (denoted Ù ) $\wedge$ Or or logical disjunction (denoted Ú ) $\vee$ XOR or exclusive or (denoted Å ) $\xor$ Implication (denoted Þ or ® ) $\Rightarrow$, $\rightarrow$ Biconditional (denoted Û or « ) $\LeftRightarrow$, $\leftrightarrow$ We define the meaning (semantics) of the logical connectives using truth tables
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C. Long Lecture 2 August 30, 2018 11 ICEN/ICSI210 Discrete Structures Logical Connective: Negation ¬ p , the negation of a proposition p , is also a proposition Examples: Today is not Monday It is not the case that today is Monday, etc.
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