MM150 Survey of Mathematics

1. MM150Survey of Mathematics Unit 2 Seminar - Sets

2. Section 2.1: Set Concepts • A set is a collection of objects. • The objects in a set are called elements. • Roster form lists the elements in brackets.

3. Section 2.1: Set Concepts Example: The set of months in the year is: M = { January, February, March, April, May, June, July, August, September, October, November, December } Example: The set of natural numbers less than ten is:

4. Section 2.1: Set Concepts • The symbol Є means “is an element of”. Example: March Є { January, February, March, April } Example: Kaplan Є { January, February, March, April }

5. Section 2.1: Set Concepts • Set-builder notation doesn’t list the elements. It tells us the rules (the conditions) for being in the set. Example: M = { x | x is a month of the year } Example: A = { x | x Є N and x < 7 }

6. Section 2.1: Set Concepts Sample: A = { x | x Є N and x < 7 } Example: Write the following using Set Builder Notation. K = { 2, 4, 6, 8 }

7. Section 2.1: Set Concepts Sample : A = { x | x Є N and x < 7 } Example: Write the following using Set Builder Notation. S = { 3, 5, 7, 11, 13 }

8. Section 2.1: Set Concepts • Set A is equal to set B if and only if set A and set B contain exactly the same elements. Example: A = { Texas, Tennessee } B = { Tennessee, Texas } C = { South Carolina, South Dakota } What sets are equal?

9. Section 2.1: Set Concepts • The cardinal number of a set tells us how many elements are in the set. This is denoted by n(A). Example: A = { Ohio, Oklahoma, Oregon } B = { Hawaii } C = { 1, 2, 3, 4, 5, 6, 7, 8 } What is n(A)? n(B)? n(C)?

10. Section 2.1: Set Concepts • Set A is equivalent to set B if and only if n(A) = n(B). Example: A = { 1, 2 } B = { Tennessee, Texas } C = { South Carolina, South Dakota } D = { Utah } What sets are equivalent?

11. Section 2.1: Set Concepts • The set that contains no elements is called the empty set or null set and is symbolized by { } or Ø. This is different from {0} and {Ø}!

12. Section 2.1: Set Concepts • The universal set, U, contains all the elements for a particular discussion. We define U at the beginning of a discussion. Those are the only elements that may be used.

13. Section 2.2: Subsets • Set A is a subset of set B, symbolized by A B, if and only if all the elements of set A are also in set B. orange yellow B = red purple blue green

14. Section 2.2: Subsets Mom B = Dad Sister Brother D =Dad Brother

15. Section 2.2: Subsets 7 3 B = 4 5 1 13 3 1 A =1 C = 6 4 13

16. Section 2.2: Subsets 12 4 B = 8 6 2 10 4 10 A =2 6 C = 6 12 8 8 10

17. Section 2.2: Subsets • Set A is a subset of set B, symbolized by A B, if and only if all the elements of set A are also in set B. Example: A = { Vermont, Virginia } B = { Rhode Island, Vermont, Virginia } Is A B? Is B A?

18. Section 2.2: Subsets • Set A is a propersubset of set B, symbolized by A B, if and only if all the elements of set A are in set B andset A ≠ set B. A =1, 2, 3 B =1, 2, 3, 4, 5 C =1, 2, 3

19. Section 2.2: Subsets • Set A is a propersubset of set B, symbolized by A B, if and only if all the elements of set A are in set B and set A ≠ set B. Example: A = { a, b, c } B = { a, b, c, d, e, f } C = { a, b, c, d, e, f } Is A B? Is B C?

20. Section 2.2: Subsets • The number of subsets of a particular set is determined by 2n, where n is the number of elements. Example: A = { a, b, c } B = { a, b, c, d, e, f } C = { } How many subsets does A have? B? C?

21. Section 2.2: Subsets Example: List the subsets of A. A = { a, b, c }

22. Section 2.3: Venn Diagrams and Set Operations • A Venn diagram is a picture of our sets and their relationships.

23. Section 2.3: Venn Diagrams and Set Operations • The complement of set A, symbolized by A′, is the set of all the elements in the universal set that are not in set A. Example: U = { m | m is a month of the year } A = { Jan, Feb, Mar, Apr, May, July, Aug, Oct, Nov } What is A´ ?

24. Section 2.3: Venn Diagrams and Set Operations • The complement of set A, symbolized by A′, is the set of all the elements in the universal set that are not in set A. Example: U = { 2, 4, 6, 8, 10, 12 } A = { 2, 4, 6 } What is A´ ?

25. Section 2.3: Venn Diagrams and Set Operations • The intersection of sets A and B, symbolized by A ∩ B, is the set of elements containing all the elements that are common to both set A and B. Example: A = { pepperoni, mushrooms, cheese } B = { pepperoni, beef, bacon, ham } C = { pepperoni, pineapple, ham, cheese } What is A ∩ B? B ∩ C? C ∩ A?

26. Section 2.3: Venn Diagrams and Set Operations • The union of sets A and B, symbolized by A U B, is the set of elements that are members of set A or set B or both. Example: A = { Jan, Mar, May, July, Aug, Oct, Dec } B = { Apr, Jun, Sept, Nov } C = { Feb } D = { Jan, Aug, Dec } What is A U B? B U C? C U D?

27. Section 2.3: Venn Diagrams and Set Operations • Special Relationship: n(A U B) = n(A) + n(B) - n(A ∩ B) B = { Max, Buddy, Jake, Rocky, Bailey } G = { Molly, Maggie, Daisy, Lucy, Bailey }

28. Section 2.3: Venn Diagrams and Set Operations • The difference of two sets A and B, symbolized by A – B, is the set of elements that belong to set A but not to set B. Example: A = { n | n Є N, n is odd } B = { n | n Є N, n > 10 } What is A - B?

29. Section 2.4: Venn Diagrams with Three Sets and Verification of Equality of Sets Procedure for Constructing a Venn Diagram with Three Sets: A, B, and C • Determine the elements in A ∩ B ∩ C. • Determine the elements in A ∩ B, B ∩ C, and A ∩ C (not already listed in #1). • Place all remaining elements in A, B, C as needed (not already listed in #1 or #2). • Place U elements not listed.

30. Section 2.4: Venn Diagrams with Three Sets and Verification of Equality of Sets Venn Diagram with Three Sets: A, B, and C U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {2, 4, 6, 8, 10} B = {1, 2, 3, 4, 5} C = {2, 3, 5, 7, 8} • A ∩ B ∩ C • A ∩ B, B ∩ C, and A ∩ C • A, B, C • U U

31. Section 2.4: Venn Diagrams with Three Sets and Verification of Equality of Sets De Morgan’s Laws • (A ∩ B)´ = A´ U B´ • (A U B)´ = A´ ∩ B´

32. Thank You! • Read Your Text • Use the MML Graded Practice • Read the DB • Email: ttacker@kaplan.edu