1. Sets Digital Lesson

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

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

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Find Subsets Example: Let set A consist of the following pizza toppings: pepperoni (P), sausage (S), onions (O), and mushrooms (M). The subsets of A consist of {P, S, O, M}, {P},{S}, { }. Find the subsets of set A. There will be 2 4 = 16 subsets {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}, A = {P, S, O, M}

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Roster & Set Notation Set notation uses a defining property to describe a set. 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} The general form of set notation is { x | x has some property } The set of all elements x such that x has the given property Roster notation lists all of the elements in a set.

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

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

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Definition of Union of Sets 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}B = {2, 4, 6, 8, 10} A 1 2 3 4 5 6 8 10 B A  B = {1, 2, 3, 4, 5, 6, 8, 10}

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Definition of Intersection of Sets 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 5 6 7 8 ABC 9 10 11 A  B = {4, 5} A  C = { } A and C are disjoint sets.

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Complement of a Set The complement of set A, denoted A', is the set of elements that are not in A. Let A = {1, 2, 3, 4, 5}. Let U represent the universal set of all the natural numbers. U 1 2 34 5 A' = {6, 7, 8, …}