Discrete Mathematics.pptx

Qxyxy3 q12 q30 quantifiers when the variables in a

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Q(x,y)=“x=y+3” Q(1,2), Q(3,0)=?

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Quantifiers When the variables in a propositional function are assigned values I becomes proposition. Propositions can also be created from propositional functions-quantification. All, some, many, none and few are used in quantifications. Universal and existential quantifiers. The area of logic that deals with predicates and quantifiers is called predicate calculus. P(x_1,x_2,x_n), P is called a n-place predicate or n-ary predicate.
Quantifiers D1: The universal quantification of P(x) is the statement “P(x) for all values of x in the domain.” forall x P(x) for all (for every) is universal quantifier. Forall x P(x) is false if there is an z for which P(x) is false. Let P(x) be “ x+1>x.” for all x P(x) is true for all real x. Statement When true? When false? Forall x P(x) P(x) is true for every x There is an x when P(x) is false There exists xP(x) There is an x for which true For every x false

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Quantifiers free. is y variable binding a is x ) 1 ( variable Binding ) 2 0 ( ) 0 0 ( ), 0 x 0 x(x numbers? real of consists case each in domain the where ( 0 and ) 0 ( 0 ), 0 ( 0 statements do What : 17 Ex quantifier s Uniquenes ! x of e every valu for false is P(x) if false quantifier l Existentia ) ( false is P(x) for which x is there if false quantifier Universal ) ( 2 3 2 2 3 2 y x x z z z y y y z z y y x x x xP x xP
Logical Equivalences Involving Quantifiers D3: Statements involving predicates and quantifiers are logically equivalent iff they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for variables.

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Logical Equivalences Involving Quantifiers well. as around other way the show an both true. are ) ( ) in t element every for and true is Q(a) and true is P(a) Hence true. is ) ( then domain, in the is a if that means This true. is )) ( ) ( ( ose e equivalenc establish to versa vice and 2 is so true ) ( ) ( and )) ( ) ( ( that Show x xQ and x a Q x Q x P x x xQ x xP x Q x P x
Negating Quantified Expressions P(x) x xP(x) P(x) x calculus." in course a not taken has who class your in student a is There " means Negation calculus." in course a taken has x " P(x) where ) ( calculus." in course a taken has class your in student Every "  x xP

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Negating Quantified Expressions every x. for ) ( ) ( )) ( ) ( ( . equivalent logically are ))) ( ) ( ( ( and )) ( ) ( ( : Solution equivalent logically are )) ( ) ( ( and )) ( ) ( ( that Show : Ex22 x Q x P x Q x P x Q x P x x Q x P x x Q x P x x Q x P x  
Translating English into logical expr )) ( ) ( ( )) ( ) ( ( )) ( ) ( ( Then coffee. drinks x and fierce is x lion, is x represent R(x) Q(x), P(x), Let coffee." drink not do creatures fierce Some " coffee." drink not do lions Some " fierce." are lions All " x R x Q x x R x P x x Q x P x

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Nested Quantifiers )) 0 ( ) 0 ( ) 0 (( 2 ) ) ( ) ( ( ), ( " 0 " ) , ( y), yP(x, is Q(x) where ) ( as of thought be can ) 0 (
• Fall '12
• Logic, Natural number, Rational number, Quantification, Countable set

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