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# MATH 103 WEEK 4 Math 103

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This 3 page Class Notes was uploaded by Ariel Harris on Monday February 8, 2016. The Class Notes belongs to Math 103 at James Madison University taught by Debra Warne in Spring 2016. Since its upload, it has received 10 views. For similar materials see THE NATURE OF MATHEMATICS in General at James Madison University.

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Date Created: 02/08/16

CLASS NOTES FEBRUARY 1 2016 CONTINUATION FROM END OF LAST NOTES Note that b E Z by closure of the integers under addition What is closure The idea that you don39t leave the set the integers are closed under addition Integers represent a set Addition represents and operation This means that if we add integers the result will still be an integer CLOSURE involves a set and an operation E PROVE the sum of two odd integers is even PROOF let m and n be odd integers therefore 39 m2k1 and n2j1 where m n E Z adding nm 2k1 2j1 2k12j1 2k2j2 we can now pull a factor of two from this equation 2kj1 2b where b kj1 note that b E Z by de nition of even mn is evenl I Means quotend of proofquot PROVE the products indicating multiplication of two odd integers are odd IE is also an odd integer must have the form 2integer 1 PROOF let m and n be odd integers m2k1 and n2j1 where kj E Z We are multiplying in this proof mn 2k12j1 we now need to FOIL First Outer Inner Last this equation FO lL Fl OL 2k 12j 1 F 2k2j o 2k1 1ZJ L11 We then add all of these equations together because the operation within The nal FOILed equation looks like this 2kZJ2k112111 This can then be simpli ed after combining like terms to look like this 4kj 2k 2j 1 What is exciting about this simpli ed equation is the plus one at the end because the end result is hopefully going to look like the odd equation 2k 1 we can then factor out a 2 from the terms that contain a factor of 2 22kj k j 1 2b1 Note that b is an integer under addition and multiplication by de nition of odd mn is odd CLASS NOTES FEBRUARY 3 2016 CLASS NOTES Day of the rst quiz Rational numbers 2 Ratio examples 2 to 1 21 T l 2 Z 2 IE 3 1 So it seems that every integer is also a rational number N 1 2 3 Whoe 0123 Z3210 1 2 3 Rational Are there rational numbers that aren39t integers YES 2 i l 3 6 2 2 2 2 4 l E2 66 There are in nitely many ways to write the same rational number We39ve seen lots of examples let39s get a de nition and try to visualize and understand these 2 q Rational numbers I this long line symbolizes quotsuch that where it is true I that pq E Z the set 0 all numbers that have the form where q 0 so the whole Rational set looks like this I 9 Rational numbers g I pq E Z the set of all numbers that have the form where q 0 This set is countably in nite and can be found to have a 11 correspondence

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