Lecture_6 - MA1100 Lecture 6 Mathematical Proofs Proof by...

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1 MA1100 Lecture 6 Mathematical Proofs Proof by Contrapositive Proof by Contradiction
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MA1100 Lecture 6 2 Calendar September 2008 Sun Mon Tue Wed Thu Fri 4 11 18 25 28 29 30 7 8 9 10 5 12 19 13 26 14 15 16 17 20 21 22 23 24 27 Sat 1 2 3 6
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MA1100 Lecture 6 3 Congruence Definition Let a and b be integers and n a positive integer. If n divides a – b , we say that a is congruent to b modulo n a ª b mod n Notation a ª b mod n Example 24 ª 10 mod 7 -2 ª 8 mod 5 84 ª 0 mod 12 Negation 5 ª 2 mod 6 1 ª 0 mod 4
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MA1100 Lecture 6 4 Congruence Proposition Let a and b be integers and n a positive integer. If a ª b mod n, there exists an integer k such that a = b + nk Proof Given a ª b mod n . This means n | a – b . Hence a - b = nk for some integer k. i.e. a - b = nk for some integer k.
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MA1100 Lecture 6 5 Congruence a ª b mod n Example ¨ ( $ k œ Z ) (a = b + nk) To list integers congruent to b modulo n , We only need to add/subtract multiples of n to/from b. Let b = 3 and n = 4 All integers congruent to 3 modulo 4: All integers congruent to 0 modulo 4:
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MA1100 Lecture 6 6 Congruence Proposition Let a be an integer and n a positive integer. (2) If n | a, then a ª 0 mod n (1) If a ª 0 mod n, then n | a a ª 0 mod n if and only if n | a biconditional statement Need to prove
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MA1100 Lecture 6 7 Congruence Proof Suppose a ª 0 mod n. “Only if” part Suppose n | a. “If” part Proposition Let a be an integer and n a positive integer. a ª 0 mod n if and only if n | a
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MA1100 Lecture 6 8 True or False Statement 1 Let a, b œ Z . If ab ª 0 mod 6 , then a ª 0 or b ª 0 mod 6 . Statement 2 Let n be a positive integer. Then 1 ª 0 mod n .
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MA1100 Lecture 6 9 Properties of Congruence (1) For every integer a, a ª a mod n (2) For all integers a and b, if a ª b mod n, then b ª a mod n (3) For all integers a, b and c, if a ª b mod n and b ª c mod n, then a ª c mod n reflexive property symmetric property transitive property Proposition
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This note was uploaded on 03/19/2012 for the course SCIENCE MA1100 taught by Professor Forgot during the Fall '08 term at National University of Singapore.

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Lecture_6 - MA1100 Lecture 6 Mathematical Proofs Proof by...

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