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# Class Note for MATH 1313 with Professor Flagg at UH

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This 8 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 22 views.

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Date Created: 02/06/15
1313isect7601 Page 1 of8 Math 1313 Section 61 Sets and Set Operations In Chapter 6 we will learn to work with sets This chapter sets the stage for everything else we will do in this course so this section is important Definitions and Notation The word set is undefined but we will use the common understanding that a set is a collection of objects From the description of a set you should be able to decide if an object is in the set or not Definition An element is an item or object in a set Notation We name sets with capital letters like A B C and usually eleme are listed rdescribedb some ro ert i 521 Aywmyemjx 3 Roster otation for a set names the set and lists the elements in braces SetBuilder Notation for a set names a set and describes the set as the set ofall elements that have a given property Example 1 Our set is the first five counting numbers and we will call the set A Roster Notation A 7 l A 3 I F I m SetBuilder Notation A t 5 3 1 Q1 L l l 7lt Is ounl39fn V 73 A 65 M at W Example 2 Describe the set Bred white blue in setbuilder notation E B 961 Vlt aco 6r allquot U Example 3 Describe the set gtltgtlt is a natural number less than 13 in roster notation 39 Z7 345 77 llllz 1313isect7601 Page 2 of8 Example 4 Write in set builder notation the set C with elements the people in this class of UM C t X l ch ll lgj MwH 0 59 Chen 7936 Definition Two sets A and B are equal written AB if every element of A is also an element of B and every element of B is also an element of A In other words two sets are equal if they have exactly the same elements Example 5 Suppose F 246810 and G gtltgtlt is an even natural number less than 11 Does F G 1 kw Ll5 a eveVt wot NMltH MF Fl JUMIJF Fag l eg Definition The set A is a subset of the set B if every element ofA is also an element of B This is written A g B Example 6 Suppose G 12346810 H 2468 and J 23510 Which of the following are true amgs was wag His Con lcuMLQ I r G bJgG No W 51 in J W 36 mm A 6 A4 ltcgtGgH N We 5 mH A Q N m dng No 3 6 3 M NOT m H Definition The set A is a proper subset of B if A is a subset of B and there is at least one element of B that is not in A The statement A is a proper subset of B is written A C B 5 llC H N07 1313isect7601 Page 3 of8 Example 7 Suppose A13579 B135 C13567 and D13579 H A Is BCA as B IchD Na problw 16 C ISACD Na bgcmksa are W How to tell if a set A is NOT a subset of the set B lflml ONE elemedgj 05 fl 5 NOT M elm U65 Definition The set with no elements is called the empty set We write the empty set as Q The empty set is a subset of every set Why Try to find an element in the empty set that is not in the other set You CAN T because there are no elements in the empty set So the empty set qualifies by default as a subset of every set p f i Example 8 List all the subsets of the set C 12 I l7 5 oz 73 Z J w l K7 JG PFoPef 1b 5 5 Venn Diagrams A Venn diagram is a rectangle which represents the set of all of the items we are interested in with circles which may overlap inside the rectangle representing subsets of the larger set We use Venn diagrams to picture how sets are related U jfl23l 5 a U 5 Agll f l 4915 151 1313isect7601 Page 4 of8 Definition The universal set is the set of all items of interest In this chapter we will denote the universal set by U Example 9 Suppose A and B are subsets of U Use Venn diagrams to illustrate each of the following statements A The sets A and B are equal U A B B The set A is a proper subset of the set B U 6 C The sets A and B have no elements in common l 6 C53 Pom Fl N at A MW 5 D The sets A and B are not subsets of each other Vellow 39 1r W mag in Il B M NO Tw39A If Suppose U is a universal set and A and B are subsets of U We have three operations that can be performed on these sets Operation Notation Meaning Venn Union A B U gt6 7c 3 n A Q 1 3 g 1313isect7601 Page 5 of8 Intersection AmB x 7 K fij flin H M Z Complement A N W I f Example 10 Suppose U 12345678910 A 13579 B 246810 and C 23467 Find DBoC iiAUB fg l ll 437579 sill637M iii B iv A B j35l7lap AGB E it Km 8 5 Definition Two sets A and B are called disjoint if AmB Q that is A and B have no elements in common Properties of Set Operations Let U be the universal set and let A B and C be subset of U Properties of Set Complements v 39 111543 Ucsltk 06001le 5 1n uIZVSQ V 2 U M 0m 4 m 12 58 U 3 A A 4 AUA U 7 m 5 AHA 1313isect7601 Page 6 of8 Properties of Intersection and Union J k Bk 1 AUBZBUA 7 W M N it 2A BB A amp Inky4 3 AUBUCAUBUC 3 W1 an 4 A B CA B C i 5 AUBmCAUBmAUC arrw 6 A BUCAmBUAmC DeMorgan s Laws 39 3 1 AUB A mB r OmPg imb d Mir C Q g In Sec all 2 AmB A UB I 1 com m 56 Morsech I I Uquot IWiCO 0 a ngx Example 11 Suppose U12345678910 A12357 B23569 and C348910 Find i AnBY UC fine rag53 AA3 j lo may UC e f r f fzSl 7 m a iiA BUC AC 4 61 7 O icfm 5 477 cqev UCc z 5 3919 S 7 3 BUCY39 1 7 A 6UCC 57 1313isect7601 Page 7 of8 Example 12 Let Ugtltgtlt is an enrolled student at the University of Houston Agtltgtlt is a student at UH enrolled in Math 1313 Bgtltgtlt is a student at UH enrolled in a history class Cgtltgtlt is a student at UH enrolled in a business class Describe in words i AmB 39 0cm l bo lxl3l3 at Aisilanglass iiA UC e 71M5 lucyf A 33 ACUC quot slum aky Busyass 073 U04 iii AtB QC Ilg gb A a W lag U D A 539 C mud ammo 7 U AND mm 2mg A sa 39 U N07quot 3 NM UB QR Bl C AND 07 CZSTMV c N66 zl FAK 6 5051 96 W inlay AIS 57 I Act5C No flAGIZ 01 00 A 513707 Describe in set notation v The set of all enrolled UH students who are takinq a history class and are notctalgngmtlLQB 4 C 0 a c b n R e viThe set of all enrolled UH students who are not taking Math 1313 nor are they taking a busin ass v 39 L C an clt vii The set of all enrolled UH students who are taking Math 1313 and a business class but not a history class H W A0570 5 1313isect7601 Page 8 of8 Example 13 Let U be a universal set and let A B and C be subsets of U W Describe using Venn diagrams e a L i A UB WE ii AnBY 74 43 iii AmB iv AnB C V AUB C

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