lecture07(complete)

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MA1100 Lecture 7 Mathematical Proofs Parity of integers Divisibility of integers irect Proofs Direct Proofs Proof by Contrapositive 1 Chartrand: section 3.2, 3.3, 4.1
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t Announcement ± Homework set 1 due today ² Write your name/student number/tutorial group ² Hand in by end of the lecture ± Tutorial 2 to 4 ² changes in problem sets T2 2.31, 2.32, 2.37, 2.40, 2.48, 2.49, 3.10, 3.16, 3.18, 3.29 3 20 322 327 328 44 412 415 2.62, 2.67, 2.68 T3 3.20, 3.22, 3.27, 3.28, 4.4, 4.12, 4,15, 4.23, 4.33, 4.42 T4 5.4, 5.10, 5.13, 5.17, 5.22, 5.23, 5.26, 5.32, 3.10, 3.18, 3.29 X XX X Lecture 7 2 5.33 4.23, 4.33, 4.42
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t i i t h t t i f i r From Last Lecture Negation with two quantifiers Write down the negation of the following double quantified statements. 1. For all integers x and y, x + y = 0. There exist integers x and y such that x + y 0. 2. There are some integers x and y such that x < y. or every real number y there exists a real number x such For all integers x and y, x ¥ y. 3. For every real number y, there exists a real number x such that y = x 2 . There exists a real number y such that for every real 4. There exists an integer x such that for every integer y, xy = 0. number x, y x 2 . For all integers x, there exists an integer y such that xy 0. Lecture 7 3 g, g y y
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r St t t True Statements • Mathematical statements that are true are known as mathematical facts . •A proposition is a true mathematical statement at as a proof hartrand calls it “Result” that has a proof . • A proposition that is “ important ” is called a theorem . Chartrand calls it Result • A proposition that is used in the proof of a theorem is called a lemma . • A proposition that follows (easily) from a theorem is called a corollary . here are true statements that o not have proofs Lecture 7 4 There are true statements that do not have proofs e.g. definitions , axioms .
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E p l An Example Proposition If x is an even integer, then x 2 is divisible by 4. p We want to give a proof for this (universal) implication. We need to : know the definitions of even integers and divisibility use some basic facts about integers Lecture 7 5 use algebraic manipulation
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f i i t i Definition (mathematical) efinition A (mathematical) definition gives the precise meaning of a word or phrase that represents some object, property or other concepts. Examples (Mathematical) 1. Even and odd integers 2.n is divisible by m Parity of integers Lecture 7 6
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i t r Even integers xample 2 6 Example = 2 ä 1 = 2 ä -14 2 3 = 2 ä (-7) In general, an even integer can be written in the efinition ge e a , a e e tege ca be tte t e form 2n where n is an integer An integer a is an even integer if there exists an integer n such that a = 2n . Definition Lecture 7 7 ( $ n œ Z )(a = 2n)
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dd i t r Odd integers xample 3 7 Example = 2 ä 1 + 1 = 2 ä + 1 -13 2 3 + 1 = 2 ä (-7) + 1 In general, an odd integer can be written in the efinition form 2n + 1 where n is an integer An integer a is an odd integer if there exists an integer n such that a = 2n + 1 .
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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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lecture07&iuml;&frac14;ˆcomplete&iuml;&frac...

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