SETS Sets are denoted by Capital letters Sets use “curly” brackets n(A) = | A | = 4 The number of elements in Set A is 4 7 is not an element of A 3 is an element of A

A set is a distinct collection of objects A set is a distinct collection of objects. The objects are called elements. Order does not matter. If a set contains the same elements as another set, the sets are equal. {1, 2, 3, 4} = {2, 3, 1, 4} {1, 3, 2, 5} {1, 3, 2, 3, 5, 2} {1, 2, 3, 5} We never repeat elements in a set. In ascending order This symbol means "is a subset of" A  B This is read "A is a subset of B". A = {1, 2, 3} B = {1, 2, 3, 4, 5}

It is agreed that the empty set is a subset of all other sets so: 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: List all of the subsets of {1, 2, 3}.  {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3} Notice the empty set is NOT in set brackets.

2n Number of Elements in Set Possible Subsets Total Number of Possible Subsets 1. {A} {A}   2  2. {A , B} {A , B} {A} {B}   4  {A , B , C} {A , B} {A , C} {B , C} {A} {B} {C}  8  3. {A , B , C} 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 2n The number of possible subsets of a set of size n is ?

A = {1, 2, 3, 4, 5} B = {1, 3, 5, 7, 9} Remember we do not list elements more than once. A  B = {1, 2, 3, 4, 5, 7, 9} This is the union symbol. It means the set that consists of all elements of set A and all elements of set B. A  B = {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  B A  B 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. A  B =  There are no elements in both A and B. When this is the case they are called disjoint sets. Universal Set A A B A 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 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. Since 24 did not own a dog or cat, there must be 76 that do. 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 24 C D 12 40 24 n(C  D) = 76 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. This n means the number of elements in the set