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# 1proposition.pdf 1proposition

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This 4 page Class Notes was uploaded by Yue YU on Tuesday October 6, 2015. The Class Notes belongs to 1proposition at University of Texas at Dallas taught by Ernest Hannig Dennis Miller in Fall 2015. Since its upload, it has received 28 views.

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Date Created: 10/06/15

A proposition is a declarative sentence that is a sentence that declares a fact that is either true or false but not both We use letters to denote propositional variables or statement variables that is variables that represent propositions just as letters are used to denote numerical variables The conventional letters used for propositional variables are p q I s The truth value of a proposition is true denoted by T if it is a true proposition and the truth value of a proposition is false denoted by F if it is a false proposition DEFINITION 1 Let p be a proposition The negation of p denoted by p also denoted by p is the statement It is not the case that p The proposition p is read not 9 The truth value of the negation of p p is the opposite of the truth value of 9 DEFINITION 2 Let p and q be propositions The conjunction of p and q denoted by p q is the proposition 9 and q The conjunction p q is true when both 9 and q are true and is false otherwise DEFINITION 3 Let p and q be propositions The disjunction of p and q denoted by 9 V q is the proposition 9 or q The disjunction p q is false when both 9 and q are false and is true otherwise TABLE 2 Tlme T j j Tellile fee TABLE 3 The Tellile fee the Eenjmneiien If the ejiijielimzi elf Twe Pmpeeiiie e Prepeei eije F 1139 F E 939 F 1 F V 939 T T T T T T T F F T F T F T F F T T F F F F F F DEFINITION 4 Let p and q be propositions The exclusive or of p and q denoted by p 63 q is the proposition that is true when exactly one of p and q is true and is false otherwise DEFINITION 5 Let p and q be propositions The conditional statement 9 gt q is the proposition if p then q The conditional statement 9 gt q is false when p is true and q is false and true otherwise In the conditional statement 9 gt q p is called the hypothesis or antecedent or premise and q is called the conclusion or consequence TABLE 4 The T j lj Tattle for TA ELE 5 Trur i Tattle fur the EHI ire t all Tm the Canal Statement Pmti ma p 1 age p 1139 F E at F If F 1 9 T T F T T T T F T T F F F T T F T T F F F F F T p gt q can also be expressed in following ways below if p then q p implies q Cifp q33 66p q33 33 66 p is sufficient for q a suf cient condition for q is p 33 66 q if p q whenever p 33 66 q when p q is necessary for p 33 66 a necessary condition for p is q q follows from p q unless p DEFINITION 6 Let p and q be propositions The biconditional statement 9 lt gt q is the proposition 9 if and only if q The biconditional statement 9 lt gt q is true when p and q have the same truth values and is false otherwise Biconditional statements are also called biimplications TABLE TheTrii39lilm Table rm the F 5 are P I PHquot quotF quotF T quotF F F F quotF F F F T DEFINITION 7 A bit string is a sequence of zero or more bits The length of this string is the number of bits in the string TABLE 9 Table rm the it HE AMI and I m 1 J39 I v r x n y x E 3 3 3 II II 1 l l J l l H I 1 l l l I II

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