1. 1 Learning Objectives for Section 7.2 Sets After today’s lesson, you should be able to Identify and use set properties and set notation. Perform set operations. Solve applications involving sets.

2. 2 Set Properties and Set Notation Definition: A set is any collection of objects Notation: e  A means “e is an element of A”, or “e belongs to set A”. e  A means “e is not an element of A”.

3. 3 Set Notation (continued) A  B means “A is a subset of B” A = B means “A and B have exactly the same elements” A  B means “A is not a subset of B” A  B means “A and B do not have exactly the same elements”

4. 4 Set Properties and Set Notation (continued) Example of a set: Let A be the set of all the letters in the alphabet. We write that as A = { a, b, c, d, e, …, z}. This is called the listing method of specifying a set.  We use capital letters to represent sets.  We list the elements of the set within braces, separated by commas.  The three dots (…) indicate that the pattern continues. Question: Is 3 a member of the set A?

5. 5 Set-Builder Notation Sometimes it is convenient to represent sets using set-builder notation. Example: Using set-builder notation, write the letters of the alphabet. A = {x | x is a letter of the English alphabet} This is read, “the set of all x such that x is a letter of the English alphabet.” It is equivalent to A = {a, b, c, d, e, …, z} Note: {x | x 2 = 9} = {3, -3} This is read as “the set of all x such that the square of x equals 9.”

6. 6 Null Set Example: What are the real number solutions of the equation x 2 + 1 = 0? Answer: ________________________________________ ________________________________________________ We represent the solution as the __________, written ____ or ___. It is also called the _______________ set.

7. 7 Subsets A is a subset of B if every element of A is also contained in B. This is written A  B.A  B. For example, the set of integers { …-3, -2, -1, 0, 1, 2, 3, …} is a subset of the set of real numbers. Formal Definition: A  B means “if x  A, then x  B.”

8. 8 Subsets (continued) Note: Every set is a subset of itself. Ø (the null set) is a subset of every set.

9. 9 Number of Subsets Example: List all the subsets of set A = {bird, cat, dog} For convenience, we will use the notation A = {b, c, d} to represent set A.

10. 10 Union of Sets (OR) A  B = { x | x  A or x  B} In the Venn diagram on the left, the union of A and B is the entire region shaded. The union of two sets A and B is the set of all elements formed by combining all the elements of set A and all the elements of set B into one set. It is written A  B. AB

11. 11 Intersection of Sets (AND) A  B = { x | x  A and x  B} The intersection of two sets A and B is the set of all elements that are common to both A and B. It is written A  B. A B In the Venn diagram on the left, the intersection of A and B is the shaded region.

12. 12 Example Example: Given A = {3, 6, 9, 12, 15} and B = {1, 4, 9, 16} find: a)A  B. b)A  B.

13. 13 Disjoint Sets If two sets have no elements in common, they are said to be disjoint. Two sets A and B are disjoint if A  B = . Example: The rational and irrational numbers are disjoint. In symbols:

14. 14 The Universal Set The set of all elements under consideration is called the universal set U. U

15. 15 The Complement of a Set (NOT) The complement of a set A is defined as the set of elements that are contained in U, the universal set, but not contained in set A. The symbolism and notation for the complement of set A are In the Venn diagram on the left, the rectangle represents the universal set. A is the shaded area outside the set A. U

16. 16 Venn Diagram Refer to the Venn diagram below. The indicated values represent the number of elements in each region. How many elements are in each of the indicated sets? AB U 65 12 40 25

17. 17 Application* A marketing survey of 1,000 car commuters found that 600 listen to the news, 500 listen to music, and 300 listen to both. Let N = set of commuters in the sample who listen to news Let M = set of commuters in the sample who listen to music Find the number of commuters in the set The number of elements in a set A is denoted by n(A), so in this case we are looking for

18. The set N (news listeners) consists of 600 elements all together. The middle part has _______, so the other part must have _______ elements. Therefore, 18 Solution (continued) Fill in the remaining blanks. The study is based on 1000 commuters, so n(U)=___________. ______ listen to news but not music. ______ listen to music but not news N M U _____ people listen to neither news nor music _______ listen to both music and news

19. 19 Examples From the Text Page 364 # 2 – 42 even