lecture20(complete)

lecture20(complete) - MA1100

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MA1100 Lecture 20 Number Theory Greatest Common Divisor Division Algorithm Euclidean Algorithm Linear combination Chartrand: 11.2, 11.3, 11.4
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Lecture 20 2 Lesson Plan (countdown: 28 days) Tue Lecture Fri Lecture Tutorial Week 11 20. Number Th. HW4 21. Number Th. Tut 8 Functions Week 12 22. Number Th. 23. Cardinality Tut 9 Number Th. Week 13 24. Revision lecture HW5 25. Revision lecture (tentative) Tut 10 Number Th./ Cardinality Week 14 READING WEEK Week 15 FINAL EXAM (NOV 24)
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Composition of inverse functions Proposition Let f: A Ø B and g: B Ø C be bijections. Then g o f is a bijection and (g o f) -1 = f -1 o g -1 . A B f C g f -1 g -1 g o f (g o f) -1 3 Lecture 20
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Composition of inverse functions Proof To prove: (g o f) -1 = f -1 o g -1 . f: A Ø B and g: B Ø C be bijections. Recall: To show two functions h and k are equal, we show h(x) = k(x) for all x in the domain. Let c œ C. Show (g o f) -1 (c) = (f -1 o g -1 )(c). (g o f) -1 : C Ø A f -1 o g -1 : C Ø A show: same domain, same codomain, same rule 4 Lecture 20
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Composition of inverse functions Proof Let c œ C. Show (g o f) -1 (c)= (f -1 o g -1 )(c). So $ b œ B such that g(b) = c (1) So $ a œ A such that f(a) = b (2) (g o f)(a) = c (g o f) -1 (c) = a from (1), g -1 (c) = b from (2), f -1 (b) = a so (f -1 o g -1 )(c) = a From (1) and (2), On the other hand, We have shown (g o f) -1 (c)= (f -1 o g -1 )(c) for all c œ C. g is surjective. f is surjective . f(a) = b if and only if f -1 (b) = a 5 Lecture 20
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Summary • f and g injective g o f injective • f and g surjective g o f surjective •g o f injective f injective •g o f surjective g surjective • f bijection f -1 is a bijection • f(a) = b f -1 (b) = a • f bijection f -1 o f = I and f o f -1 = I •g o f = I f is injective and g is surjective •g o f = I and f o g= I f -1 = g •(g o f) -1 = f -1 o g -1 6 Lecture 20
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Number Theory 7 Lecture 20 Mathematics is the Queen of the Sciences Number Theory is the Queen of Mathematics - Carl Friedrich Gauss The Prince of Mathematicians Î
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Common divisor Recall 1 Recall 2 Let a, b be integers and d a nonzero integer. Let n be an integer and m a non-zero integer. We say
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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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lecture20(complete) - MA1100

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