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SETS Sets are denoted by Capital letters Sets use “curly” brackets A = {1, 3, 2, 5} n(A) = | A | = 4 The number of elements in Set A is 4 3 A 7 A 3 is an element of A 7 is not an element of A A set is a distinct collection of objects. The objects are called elements. Order does not matter. If a set {1, 2, 3, 4} = {2, 3, 1, 4} contains the same elements as another set, the sets are equal. {1,2,3,2,3,2,3,5,5} 5} {1, {1, 3, 2} In ascending order We never repeat elements in a set. This symbol means "is a subset of" AB This is read "A is a subset of B". A = {1, 2, 3} B = {1, 2, 3, 4, 5} If a set doesn't contain any elements it is called the empty set or the null set. It is denoted by or { }. NOT {} It is agreed that the empty set is a subset of all other sets so: A A where A is any set. List all of the subsets of {1, 2, 3}. {1} {2} {3} {1, 2} Notice the empty set is NOT in set brackets. {1, 3} {2, 3} {1, 2, 3} Number of Elements in Set Possible Subsets Total Number of Possible Subsets 1. {A} {A} 2 2. {A , B} {A , B} {A} {B} 4 3. {A , B , C} {A , B , C} {A , B} {A , C} {B , C} {A} {B} {C} 8 4. {A , B , C, D} {A , B , C , D} {A , B , C} {A , B , D} {A , C , D} {B , C , D} {A , B} {A , C} {A , D} {A , B} …… {D} ? 16 The number of possible subsets of a set of size n is ? n 2 A = {1, 2, 3, 4, 5} B = {1, 3, 5, 7, 9} AB = {1, 2, 3, 4, 5, 7, 9} Remember we do not list elements more than once. This is the union symbol. It means the set that consists of all elements of set A and all elements of set B. AB = {1, 3, 5} This is the intersect symbol. It means the set containing all elements that are in both A and B. These sets can be visualized with circles in what is called a Venn Diagram. A A B B A B AB AB Everything that is in A or B. Everything that is in A AND B. Often will have a set that contains all elements that we wish to consider. This is called the universal set. All other sets are subsets of this set. AB= Universal Set AA A BA There are no elements in both A and B. When this is the case they are called disjoint sets. This means the complement of A, and means the set of all elements in the universal set that are not in A. 100 people were surveyed. 52 people in a survey owned a cat. 36 people owned a dog. 24 did not own a dog or cat. Draw a Venn diagram. 52 + 36 = 88 so there must be 88 - 76 = 12 people that own both a dog and a cat. universal set is 100 people surveyed C 40 12 D 24 24 Set C is the cat owners and Set D is the dog owners. The sets are NOT disjoint. Some people could own both a dog and a cat. Counting Formula: n(A B) = n(A) + n(B) - n(A B) Since 24 did not own a dog or cat, there must be 76 that do. n(C D) = 76 This n means the number of elements in the set Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au