Sets Finite 7-1

Definitions Set: A set is an unordered collection of (unique) objects The objects in a set are called elements or members of a set. A set is said to contain its elements Two sets, A and B, are equal is they contain the same elements. We write A=B. Notation, for a set A: x  A: x is an element of A x  A: x is not an element of A Definitions

Example Are the following sets equal? {2,3,5,7} & {3,2,7,5}, Yes, because a set is unordered {2,3,5,7} & {2,2,3,5,3,7} Yes, because a set contains unique elements {2,3,5,7} & {2,3} No Example

Example Are these sets equal? A = {9, 2, 7, -3}, B = {7, 9, -3, 2} : A = {dog, cat, horse}, B = {cat, horse, squirrel, dog} : A  B A = {dog, cat, horse}, B = {cat, horse, dog, dog} : A = B Example

Standard Sets “Standard” Sets: Natural numbers N = {0, 1, 2, 3, …} Integers Z = {…, -2, -1, 0, 1, 2, …} Positive Integers Z+ = {1, 2, 3, 4, …} Real Numbers R = {47.3, -12, , …} Rational Numbers Q = {1.5, 2.6, -3.8, 15, …} Standard Sets

Terminology The set-builder notation O={ x | (xZ)  (x=2k) for some kZ} reads: O is the set that contains all x such that x is an integer and x is even A set is defined in intension when you give its set-builder notation O={ x | (xZ)  (0x8)  (x=2k) for some k  Z } A set is defined in extension when you enumerate all the elements: O={0,2,4,6,8} Terminology

U Venn Diagrams x y B A z a C A set can be represented graphically using a Venn Diagram U x y B A z a C Venn Diagrams

A set that has no elements is called the empty set or null set and is denoted  A is said to be a subset of B, and we write A  B, if and only if every element of A is also an element of B That is, we have the equivalence: A  B   x (x  A  x  B) 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} Subsets

Subsets Subsets Useful rules: A = B  (A  B)  (B  A) (A  B)  (B  C)  A  C (see Venn Diagram) U C B A Subsets

Example Is A a subset of B? A = {3, 9}, B = {5, 9, 1, 3}, A  B ? true false Example

Null Set Notice the empty set is NOT in set brackets. 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. Null Set

Arithmetic operators (+,-,  ,) can be used on pairs of numbers to give us new numbers Similarly, set operators exist and act on two sets to give us new sets Union Intersection Set Operations

The union of two sets A and B is the set that contains all elements in A, B, r both. We write: AB = { x | (a  A)  (b  B) } U A B Union of Sets

U Intersection of Sets A B The intersection of two sets A and B is the set that contains all elements that are element of both A and B. We write: A  B = { x | (a  A)  (b  B) } U A B Intersection of Sets

Two sets are said to be disjoint if their intersection is the empty set: A  B =  U A B Disjoint Sets

A  B A  B Example 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. Example

The complement of a set A, denoted A ($\bar$), consists of all elements not in A. That is the difference of the universal set and U: U\A A= AC = {x | x  A } U A A Complement

The complement of a set A contains exactly those elements under consideration that are not in A: Ac = U-A Example: U = N, B = {250, 251, 252, …} Bc = {0, 1, 2, …, 248, 249} Complement

n(A  B) = n(A) + n(B) - n(A  B) 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 Counting Formula: n(A  B) = n(A) + n(B) - n(A  B)

Pages 290 – 292 1 – 19 odd, 25 – 31 odd, 35 – 41 odd, 59, 61, 79 Homework