lecture06 - MA1100 Lecture 6 Logic Quantified statements...

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1 MA1100 Lecture 6 Logic Quantified statements Chartrand: section 2.10
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Lecture 6 2 Quantified Statements There are integers x and y : existential quantifier For all integers x and y : universal quantifier When a quantifier is attached to an open sentence, the sentence becomes a statement . We call this a quantified statement 1)There are integers x and y such that 2x + 5y = 7 2)For all integers x and y, 2x + 5y = 7 Examples Two types of quantifiers
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Lecture 6 3 Universal quantifier For each real number x, x 2 > 0 This phrase quantifies the variable x The whole sentence claims: P(x) is true when x is substituted with every real number. If every real number substituted makes P(x) true , then this quantified statement is true . P(x): x 2 > 0 If ( at least) one real number substituted makes P(x) false , then this quantified statement is false .
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Lecture 6 4 Universal quantifier For each real number x, x 2 > 0 This phrase quantifies the variable x We call this sentence a universally quantified statement . The phrase “for each”, “for every”, “for all” is called a universal quantifier Notation : " ( " x œ R )(x 2 > 0) Say: “for all”
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Lecture 6 5 Universal quantifier For each real number x, x 2 > 0 ( " x œ R )(x 2 > 0) Other forms The square of every real number is greater than 0 The square of a real number is greater than 0 If x is in R , then x 2 > 0 (hidden quantifier) (hidden variable)
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Lecture 6 6 Universal quantifier For each real number x, x 2 ¥ 0 For each real number x, x 2 >0
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Lecture 6 7 Existential quantifier There exists an integer x such that x 2 = 2 This phrase quantifies the variable x The whole sentence claims: P(x) is true when x is substituted with some integers. If one (or more) integer substituted makes P(x) true , then this quantified statement is true . P(x): x 2 = 2 If no integer substituted makes P(x) true , then this quantified statement is false .
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Lecture 6 8 Existential quantifier We call this sentence an existentially quantified statement . The phrase “there exists”, “there is”, is called an existential quantifier Notation : $ ( $ x œ Z )(x 2 = 2) There exists an integer x such that x 2 = 2 This phrase quantifies the variable x Say: “there exist”
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Lecture 6 9 Existential quantifier Other forms ( $ x œ Z )(x 2 = 2)
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This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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lecture06 - MA1100 Lecture 6 Logic Quantified statements...

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