Digital Lesson Sets A set is a collection

Digital Lesson Sets

A set is a collection of objects called elements or members of the set. A = {1, 2, 3, . . . } is the set of natural numbers. 4 A – 99 A “is an element of ” “is not an element of ” Set A is an example of an infinite set. B = {2, 4, 6} is an example of a finite set. A set with no elements is called the empty set, or null set. C = { } or C = both denote the empty set. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

B A “is a subset of ” A = {1, 2, 3, 4, 5, 6} B A B = {2, 4, 6} The subsets of B are {2}, {4}, {6}, {2, 4}, {4, 6}, {2, 4, 6}, and { }. The set itself is always a subset. The empty set is always a subset. Note that there are 23 = 8 subsets. A set of n elements has 2 n subsets. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

Example: Let set A consist of the following pizza toppings: pepperoni (P), sausage (S), onions (O), and mushrooms (M). Find the subsets of set A. A = {P, S, O, M} There will be 24 = 16 subsets The subsets of A consist of {P}, {S}, {O}, {M}, {P, S}, {P, O}, {P, M}, {S, O}, {S, M}, {O, M}, {P, S, O}, {P, S, M}, {P, O, M}, {S, O, M}, {P, S, O, M}, { }. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4

Roster notation lists all of the elements in a set. Set notation uses a defining property to describe a set. The general form of set notation is { x | x has some property } The set of such that all elements x x has the given property Let set A consist of all natural numbers greater than or equal to 6. Roster notation: A = {6, 7, 8, …} Set notation: A = {x | x 6} Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

Example: Let set B consist of all real numbers. Represent the set using roster notation and set notation. Roster notation: B = {. . . , – 2, – 1, 0, 1, 2, . . . } The set continues to negative infinity. The set continues to positive infinity. Set notation: B = {x | x real numbers} Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6

Sets can be represented using overlapping circles called a Venn Diagram. A 1 3 5 2 4 6 B 10 8 7 C A = {1, 2, 3, 4, 5, 6} B = {2, 4, 6, 8, 10} C = {5, 6, 7, 8} Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7

The union of two sets, A B, is the set of all elements that belong to either A or B or both. A = {1, 2, 3, 4, 5, 6} A 1 3 5 B = {2, 4, 6, 8, 10} 2 4 6 8 B 10 A B = {1, 2, 3, 4, 5, 6, 8, 10} Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

The intersection of two sets, A B, is the set of all elements that are common to both A and B. A = {1, 2, 3, 4, 5} B = {4, 5, 6, 7, 8} C = {7, 8, 9, 10, 11} 1 2 3 4 6 5 A A B = {4, 5} 7 8 B 9 10 11 C A C={} A and C are disjoint sets. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

The complement of set A, denoted A', is the set of elements that are not in A. Let U represent the universal set of all the natural numbers. Let A = {1, 2, 3, 4, 5}. 1 2 3 4 5 U A' = {6, 7, 8, …} Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10