Finite Math with Applications
Finite Math with Applications MATH 1313
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This 7 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1313 at University of Houston taught by Mary Flagg in Fall. Since its upload, it has received 73 views. For similar materials see /class/208406/math-1313-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
13137563017601 Page 1 of7 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 De nitions and Notation The word set is unde ned 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 elements are listed or described by some property Roster Notation 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 of all elements that have a given property Example 1 Our set is the first ve counting numbers and we will call the set A Roster Notation SetBuilder Notation Example 2 Describe the set Bred white blue in setbuilder notation Example 3 Describe the set xlx is a natural number less than 13 in roster notation 1313isect7601 Page 2 of7 Example 4 Write in set builder notation the set C with elements the people in this class 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 xlx is an even natural number less than 11 Does F G Definition The set A is a subset of the set B if every element of A 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 a H gG bJgG GgH 1ng 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 c3 Example 7 Suppose A13579 B135 C13567 and D 1 3 579 A Is BcA B Is C cD C Is AcD 1313isect7601 Page 3 of7 How to tell if a set A is NOT a subset of the set B 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 CANT because there are no elements in the empty set So the empty set quali es by default as a subset of every set Example 8 List all the subsets of the set C 12 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 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 1313isect7601 Page 4 of7 B The set A is a proper subset of the set B C The sets A and B have no elements in common D The sets A and B are not subsets of each other Set Operations 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 AuB Intersection A n B Complement A Example 10 Suppose U 12345678910 A 13579 B 246810 and C 23467 Find i BnC ii AuB iii B iv AnB 1313isect7601 Page 5 of7 Definition Two sets A and B are called disjoint if AnB 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 1 U Q 2 Q U 3 A A 4 A UA U 5 A nA Q Properties of Intersection and Union 1 AUBBUA AnBBnA AUBUCAUBUC AnBnCAnBnC AUBnCAUBnAUC AnBUCAnBUAnC 01019905 DeMorgan s Laws 1 AUB A MB 2 A03 A UB Example 11 Suppose U12345678910 A12357 B23569 and C348910 Find i AnBY UC ii A nBUC iii BUCY MA 13137563017601 Page 6 of7 Example 12 Let Uxx is an enrolled student at the University of Houston Axx is a student at UH enrolled in Math 1313 Bxx is a student at UH enrolled in a history class Cxx is a student at UH enrolled in a business class Describe in words i AnB ii A UC iii AUBY ac iv A we Describe in set notation v The set of all enrolled UH students who are taking a history class and are not taking Math 1313 vi The set of all enrolled UH students who are not taking Math 1313 nor are they taking a business class vii The set of all enrolled UH students who are taking Math 1313 and a business class but not a history class Example 13 Let U be a universal set and let A B and C be subsets of U Describe using Venn diagrams i A DB 1313isect7601 Page 7 of7 ii A n B iii AnB iv AanC V AUBmC Vi BmCnA