Chapter 7 Sets and Probability Section 7.1 Sets

What is a Set? A set is a well-defined collection of objects in which it is possible to determine whether or not a given object is included in the collection. Example: The letters of the Alphabet

The Vocabulary of Sets Each object in the set is referred to as an element or member of the set. The symbol  denotes membership in a set, while  is used to show an element is not an element. Example: S = { 2, 4, 6, 8, …} 12  S 25  S It is possible to have a set with no elements. This kind of set is called an empty set and is written as { } or .

Equal sets have exactly the same elements. Equivalent sets have the same number of elements. Example: A = { d, o, g } B = { c, a, t } B = { c, a, t } C = { d, o, g, s } C = { d, o, g, s } D = { a, c, t } D = { a, c, t } Which, if any, of the sets are equal? B = D Equivalent? A, B, and D are equivalent. Not equal? C is not equal to any of the sets.

The cardinality of set A refers to the number of elements in set A and is written as n(A). Example: Set Z is defined as containing all the single digits. List each element in the set, then find n(Z). Z = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } Note: Listing each member of a set one time is called roster, or listing, notation. n(z) = 10 n(z) = 10

Sometimes it is not convenient, or feasible, to list each element of a set. When we are interested in a common property of the elements in a set, we use set-builder notation. { x | x has property P } { x | x has property P } “The set of all x such that x has property P”. “The set of all x such that x has property P”. Example: Use set-builder notation to write the set of elements defined as a number greater than 10. { x | x > 10 } { x | x > 10 }

The universal set is a set that includes all objects being discussed. Sometimes every element of one set also belongs to another set. This is an example of a subset. B = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 2, 4, 6, 8, 10 } A is a subset of B

Set A is a subset of set B (written A  B) if every element of A is also an element of B. Set A is a proper subset (written A  B) if A  B and A  B. The symbol  is used to describe an improper subset in which the subset and set are equal.

For any set A,   A and A  A. Example: List all possible subsets of {x, y}. There are 4 subsets of {x, y} : There are 4 subsets of {x, y} : , proper subset , proper subset {x}, proper subset {y}, proper subset {y}, proper subset {x, y} improper subset {x, y} improper subset A set of n distinct elements has subsets.

Set Operations Given a set A and a universal set U, the set of all elements of U that do not belong to A is called the complement of set A.

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = {2, 4, 7, 9} A = {2, 4, 7, 9} B = {3, 5, 8, 10} B = {3, 5, 8, 10} C = {1, 3, 5, 7, 9} C = {1, 3, 5, 7, 9} Find each of the following sets. Find each of the following sets. 1.) A' 1.) A' A' = {1, 3, 5, 6, 8, 10} A' = {1, 3, 5, 6, 8, 10} 2.) B' 2.) B' B' = {1, 2, 4, 6, 7, 9} B' = {1, 2, 4, 6, 7, 9} 3.) C ' 3.) C ' C ' = {2, 4, 6, 8, 10} C ' = {2, 4, 6, 8, 10}

Given two sets A and B, the set of all elements belonging to both set A and set B is called the intersection of the two sets, written A  B.

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = {2, 4, 7, 9} A = {2, 4, 7, 9} B = {3, 5, 8, 10} B = {3, 5, 8, 10} C = {1, 3, 5, 7, 9} C = {1, 3, 5, 7, 9} Find each of the following sets. Find each of the following sets. 1.) A  C 1.) A  C A  C = {7, 9} A  C = {7, 9} 2.) B  C 2.) B  C B  C = {3, 5} B  C = {3, 5} 3.) A  B 3.) A  B A  B = { } or  A  B = { } or 

Disjoint Sets For any sets A and B, if A and B are disjoint sets, then A  B = . In other words, there are no elements that sets A and B have in common.

The set of all elements belonging to set A, to set B, or to both sets is called the union of the two sets, written A  B.

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = {2, 4, 7, 9} A = {2, 4, 7, 9} B = {3, 5, 8, 10} B = {3, 5, 8, 10} C = {1, 3, 5, 7, 9} C = {1, 3, 5, 7, 9} Find each of the following sets. Find each of the following sets. 1.) A  C 1.) A  C A  C = {1, 2, 3, 4, 5, 7, 9} A  C = {1, 2, 3, 4, 5, 7, 9} 2.) B  C 2.) B  C B  C = {1, 3, 5, 7, 8, 9, 10} B  C = {1, 3, 5, 7, 8, 9, 10} 3.) A  B 3.) A  B A  B = {2, 3, 4, 5, 7, 8, 9, 10} A  B = {2, 3, 4, 5, 7, 8, 9, 10}