lecture06&iuml;&frac14;ˆcomplete&iuml;&frac14;‰

# lecture06&iuml;&frac14;ˆcomplete&iuml;&frac...

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1 MA1100 Lecture 6 Logic Quantified statements Chartrand: section 2.10

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Lecture 6 2 Announcement ± Homework set 1 due next Tuesday ² ² Marking Scheme (next slide) ± Math Clinic ² http://ww1.math.nus.edu.sg/clinic.htm ² Start next week ± Virtual tutorial class ² Tutorial 1 recorded session ± Problem set solutions ² HW solutions will be uploaded ² Solutions for tutorial & additional problems will not be uploaded
Lecture 6 3 Marking Scheme for HW Marks Criteria 5 4 3 2 1 0 done all assigned questions the selected questions correct . (done all assigned questions minor mistakes in selected questions) ¤ ( missed some questions selected questions correct) . (missed some assigned questions minor mistakes in selected questions) ¤ ( done all assigned questions serious mistakes in selected questions). missed some assigned questions serious mistakes in selected questions. missed some assigned questions including all selected questions. missed all questions ¤ did not hand in the homework • Do all assigned questions • Selected questions will be marked

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Lecture 6 4 Summary Logical Equivalence • comparison between statements (involving 2 or more common simpler statements) • same truth values (same mathematical meaning) Contrapositive Commutative Law Associative Law De Morgan’s Law Distributive law Implication as disjunction Negation of implication Implication with disjunction P Q ª ~Q ~P ~(P Q) ª ~ P ¤ ~Q P ¤ Q ª Q ¤ P (P ¤ Q) ¤ R ª P ¤ (Q ¤ R) P ¤ (Q R) ª (P ¤ Q) (P ¤ R) P Q ª ~P ¤ Q ~(P Q) ª P ~Q) P (Q ¤ R) ª (P ~Q) R
Lecture 6 5 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 quantifiers

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Lecture 6 6 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 .
Lecture 6 7 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) inverted A Say: “for all” A

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Lecture 6 8 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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lecture06&iuml;&frac14;ˆcomplete&iuml;&frac...

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