2. Unions and intersection of Sets Section 3-8

3. Goals Goal To find the union and intersections of sets. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

4. Vocabulary Union Intersection Disjoint

5. The intersection of sets A and B, symbolized A  B, is the set containing all the elements that are in common to both sets. Definition: The word “and” is generally interpreted to mean intersection. The elements in A “and” in B. If A = { 1, 2, 3 } and B = { 3, 4, 5}, then A  B = { 3 },since 3 is in A and in B.

6. Intersection of Sets The intersection of two sets A and B, denoted A  B, is the set of all elements that belong to both set A and set B. Implies intersection. Let A = {1, 2, 3, 4, 5} and let B = {4, 5, 6, 7, 8}. AB 1 2 3 4 5 6 7 8 A  B = {4, 5}

7. Set Intersection : Venn Diagrams A = {a, b, c, d} B = {a, b} A  B = {a, b} A = {a, b, c, d} B = {x, y, z} A  B =  abab cdcd a b c d x y z

8. {a,b,c}  {2,3} = ___ {2,4,6}  {3,4,5} = ______ Intersection Examples  {4}

9. 8 Find each intersection. a) b) Solution a) b) Your Turn:

10. The union of sets A and B, symbolized by A  B, is the set containing all of the elements that are in set A or in set B, or elements in both sets. Definition: The union is sort of like taking a bag and throwing in set A and set B, then listing what you have in the bag. The word “or” is generally interpreted to mean union. If A = { 1, 2, 3 } and B = { 3, 4, 5 }, then A  B = { 1, 2, 3, 4, 5 }.

11. Union of Sets The union of two sets A and B, denoted A  B, is the set of all elements that are in the set A or in the set B or in both A and B. Implies union. Let A = {1, 2, 3, 4, 5, 6} and let B = {2, 4, 6, 8, 10}. AB 8 10 A  B = {1, 2, 3, 4, 5, 6, 8, 10} 1 2 5 4 6 3

12. Union of Sets: Venn Diagrams A = {a, b, c, d} B = {c, d, e, f} A  B = {a, b, c, d, e, f} A = {a, b, c, d} B = {x, y, z} A  B = {a, b, c, d, x, y, z} Sets overlap Sets are disjoint cdcd a b e f a b c d x y z

13. {a,b,c}  {2,3} = {a,b,c,2,3} {2,3,5}  {3,5,7} = {2,3,5,3,5,7} ={2,3,5,7} Union Examples

14. Find each union. a) b) Solution a) b) Your Turn:

15. Your Turn: Operations with Sets A = { 1, 2, 3,4, 5}B = { 2, 4, 6, 8, 10} C = { 5, 6, 7, 8} Intersection ( ) ( and ) - the elements which are common to both sets. 1) A B = 2) B C = 3) A C = 4) (A B) C { 2, 4} { 6, 8} { 5 } { 2, 4}{ 5, 6, 7, 8} { } or “empty set” or O or “null set”

16. Union ( )( or ) - the combined set of elements from two sets with no duplication of elements. Your Turn: Operations with Sets A = { 1, 2, 3,4, 5}B = { 2, 4, 6, 8, 10} C = { 5, 6, 7, 8} 1) A B = 2) B C = 3) A C = { 1, 2, 3, 4, 5, 6, 8, 10} { 2, 4, 5, 6, 7, 8, 10} { 1, 2, 3, 4, 5, 6, 7, 8}

17. Number Line Graphs of Inequalities IntersectionsUnions x < 5 x < 3 0 1 2 3 4 5 6 { x : x < 3 }{ x : x < 5 } x 3 0 1 2 3 4 5 6 { x : 3 < x < 5 } x 3 0 1 2 3 4 5 6 { x : x = Any Real Number }

18. Number Line Graphs of Inequalities IntersectionsUnions x > 5 x < 3 0 1 2 3 4 5 6 { }{ x : x 5 } x > 5 x > 3 0 1 2 3 4 5 6 { x : x > 5 } x > 5 x > 3 0 1 2 3 4 5 6 { x : x > 3 }

19. The sets can be disjoint which means the two sets have no elements in common. This is represented by two circles which have no points in common. Disjoint Sets:

20. 19 Disjointed Sets Two sets A, B are called disjoint (i.e., unjoined) iff their intersection is empty. (A  B=  ) Example: the set of even integers is disjoint with the set of odd integers. Help, I’ve been disjointed!

21. Disjoint Sets -Example Two sets are called disjoint if their intersection is the empty set.  A = {1,3,5}, B = {1,2,3}, C = {6,7,8}  Are A and B disjoint?  NO, A ∩ B = {1,3}  Are A and C are disjoint?  YES, A ∩ C = ∅ A C

22. Solve practical problems involving sets by using Venn diagrams. Applications

23. When there are two sets, the sets are represented by two circles. There are then 4 different regions within the venn diagram. We can shade selected regions of the diagram to illustrate a particular set. The center is the intersection of both sets, and when illustrating two sets with Venn diagrams, we generally start with this region and work outward. Two Set Venn Diagrams:

24. Regions – Two Set Venn Diagrams 12 3 4 3 1 Region 1 A only 2 Region 2 A and B Region 3 B only 4 Region 4 Universal Set only

25. Draw Venn Diagrams to represent... A = { 1, 2, 3,4, 5}B = { 2, 4, 6, 8, 10} 2) A B 1) A B A 1 3 5 4 B 6 8 10 2 Example:

26. Draw Venn Diagrams to represent... A = { 1, 2, 3,4, 5}B = { 2, 4, 6, 8, 10} 2) A B 1) A B A 1 3 5 2 4 B 6 8 10 A 1 3 5 2 4 B 6 8 Example:

27. Draw Venn Diagrams to represent... A = { 1, 2, 3,4, 5}B = { 2, 4, 6, 8, 10} 2) A B 1) A B A 1 3 5 2 4 B 6 8 10 A 1 3 5 2 4 B 6 8 Note: The only difference is the shading. Example:

28. The word “and” means intersection, the word “or” means union, and the word “not” means complement. The following problems involve two or three sets, and are all solved in a similar manner. We will draw and label the Venn diagram, fill in the regions of the diagram with the number of elements that fall in that region, and then we will answer questions about the completed diagram. A Few Things to Remember:

29. Remember, in solving Venn diagram problems, we generally start in the middle region (the intersection of the two or three sets) and then work our way out. Remember:

30. Suppose that a group of 180 people were questioned about their travel preferences and the following information was collected; –120 prefer airplane –90 prefer automobile –50 prefer airplane and automobile a)How many people preferred traveling by only airplane? b)How many people preferred traveling by only automobile ? c)How many people preferred not traveling by automobile or airplane? First notice that there are two sets being discussed in the problem, traveling by airplane and traveling by automobile. Example - Travel Preferences

31. First notice that there are two sets being discussed in the problem, traveling by airplane and traveling by automobile. We draw an initial Venn diagram with two sets, labeling one set P (plane) and one set A (auto). Example - Travel Preferences U PA 120 prefer airplane 90 prefer automobile 50 prefer airplane and automobile

32. We place the number 50 in the middle since 50 people preferred to travel by both plane and automobile (the intersection- note that this number is at the bottom of the list of information). Example - Travel Preferences U PA 50 120 prefer airplane 90 prefer automobile 50 prefer airplane and automobile

33. Now working our way out, since 120 people preferred airplanes (which has to be the total of the entire plane circle) we calculate 120 – 50 (people who liked both) = 70. Seventy is the number of people who prefer to travel only by airplane. We place a 70 in the diagram. Example - Travel Preferences U PA 5070 120 prefer airplane 90 prefer automobile 50 prefer airplane and automobile

34. Since 90 people preferred automobiles (so 90 is the total of all the values in the auto circle), we calculate 90 – 50 = 40. Forty is the number of people who preferred to travel only by automobiles. We place the 40 in our diagram. Example - Travel Preferences U PA 507040 120 prefer airplane 90 prefer automobile 50 prefer airplane and automobile

35. Next we total the values in the two circles, (70 + 50 + 40 = 160). Since there were 180 people surveyed and only 160 in P  A, then 180 – 160 or 20 people are placed outside the circles. These people preferred not to travel by airplane (P) or automobile (A). Example - Travel Preferences U PA 507040 20 120 prefer airplane 90 prefer automobile 50 prefer airplane and automobile

36. The diagram is complete! Now you can answer the questions (a – c) by referring to the diagram? Example - Travel Preferences U PA 507040 20

37. Solution: a)How many people preferred traveling by only airplane? a)70 b)How many people preferred traveling by only automobile ? a)40 c)How many people preferred not traveling by automobile or airplane? a)20 Example - Travel Preferences U PA 507040 20

38. Note that when there are three sets, the sets are represented by three circles. We can shade selected regions of the diagram to illustrate a particular set. The center is the intersection of all three sets, and when illustrating three sets with Venn diagrams, we generally start with this region and work outward. Three Set Venn Diagrams:

39. Regions in 3 Set Venn Diagram Name the regions 1 – A only 2 – A and B only 3 – B only 4 – B and C only 5 – C only 6 – A and C only 7 – A and B and C 8 – Universal set only Illustrates the 8 possible regions between Sets, A, B and C

40. 40 countries were surveyed as to which gain crops, wheat, rice or maize, they grew. The following data resulted.  18 countries grew wheat  12 countries grew rice  16 countries grew maize  3 countries grew wheat and rice  9 countries grew wheat and maize  3 countries grew maize and rice  2 countries grew all three grains a)How many countries grew wheat and maize only? b)How many countries grew only one grain crop? c)How many countries grew exactly two grain crops? d)How many countries grew none of the three grain crops? e)How many countries grew only rice or maize? Example - Raising Grain:

41. Notice that there are three sets being discussed; wheat, maize, and rice. Draw a Venn diagram with three sets and label them W (wheat), M (maize), and R (rice). Example - Raising Grain:

42. W (wheat), M (maize), and R (rice). Example - Raising Grain: W R M U

43. Since 2 countries raised all three crops, place 2 in the middle of the diagram (note that this is at the bottom of the list of information given in the problem). Example - Raising Grain: W R M U 2 18 grew wheat 12 grew rice 16 grew maize 3 grew wheat and rice 9 grew wheat and maize 3 grew maize and rice 2 grew all three grains

44. Since 3 raised wheat and rice, then 3 – 2 = 1. Note the placement of the 1 in the diagram. Example - Raising Grain: W R M U 2 1 18 grew wheat 12 grew rice 16 grew maize 3 grew wheat and rice 9 grew wheat and maize 3 grew maize and rice 2 grew all three grains

45. Following in the same manner we can fill in the two intersections of the other two sets 3 – 2 = 1, and 9 – 2 = 7. Example - Raising Grain: W R M U 2 1 1 7 18 grew wheat 12 grew rice 16 grew maize 3 grew wheat and rice 9 grew wheat and maize 3 grew maize and rice 2 grew all three grains

46. Now to find out which countries only grew one crop. Twelve raised rice, thus 12 should be the total for the entire rice circle. Example - Raising Grain: Since we already have 1 + 2 + 1 = 4 countries in the rice circle, then 12 – 4 = 8. Eight countries raised only rice. W R M U 2 11 7 18 grew wheat 12 grew rice 16 grew maize 3 grew wheat and rice 9 grew wheat and maize 3 grew maize and rice 2 grew all three grains

47. Now to find out which countries only grew one crop. Twelve raised rice, thus 12 should be the total for the entire rice circle. Example - Raising Grain: Since we already have 1 + 2 + 1 = 4 countries in the rice circle, then 12 – 4 = 8. Eight countries raised only rice. W R M U 2 11 7 8 18 grew wheat 12 grew rice 16 grew maize 3 grew wheat and rice 9 grew wheat and maize 3 grew maize and rice 2 grew all three grains

48. The other two can be calculated in the same manner. Sixteen countries raised maize, thus 16 – 7 – 2 – 1 = 6. Six countries raised only maize. Eighteen countries raised wheat, so 18 – 7 – 2 – 1 = 8. Example - Raising Grain: Eight countries raised only wheat. Place the 6 and 8 in the appropriate regions in the diagram. W R M U 2 11 7 8 18 grew wheat 12 grew rice 16 grew maize 3 grew wheat and rice 9 grew wheat and maize 3 grew maize and rice 2 grew all three grains

49. The other two can be calculated in the same manner. Sixteen countries raised maize, thus 16 – 7 – 2 – 1 = 6. Six countries raised only maize. Eighteen countries raised wheat, so 18 – 7 – 2 – 1 = 8. Example - Raising Grain: Ten countries raised only wheat. Place the 6 and 8 in the appropriate regions in the diagram. W R M U 2 11 7 8 6 8 18 grew wheat 12 grew rice 16 grew maize 3 grew wheat and rice 9 grew wheat and maize 3 grew maize and rice 2 grew all three grains

50. Adding the regions within all the circles yields 8 + 7 + 6 + 1 + 2 + 1 + 8 = 33. Since 40 countries were surveyed, 40 – 33 = 7, or seven countries grew none of these three crops. Example - Raising Grain: W R M U 2 11 7 8 6 8 18 grew wheat 12 grew rice 16 grew maize 3 grew wheat and rice 9 grew wheat and maize 3 grew maize and rice 2 grew all three grains

51. Adding the regions within all the circles yields 8 + 7 + 6 + 1 + 2 + 1 + 8 = 33. Since 40 countries were surveyed, 40 – 33 = 7, or seven countries grew none of these three crops. Example - Raising Grain: The diagram is complete! W R M U 2 11 7 8 6 8 7

52. Solution a)How many countries grew wheat and maize only? a)7 b)How many countries grew only one grain crop? b)22 (8 + 8 + 6) c)How many countries grew exactly two grain crops? c)9 (7 + 1 + 1) d)How many countries grew none of the three grain crops? d)7 e)How many countries grew only rice or maize? e)15 (8 + 1 +6) Example - Raising Grain: W R M U 2 11 7 8 6 8 7

53. Joke Time What’s a quick way to double your money? You fold it! What does a pickle say when he wants to play cards? Dill me in! What do you call a hippo in a phone booth? Stuck!

54. Assignment 3-8 Exercises Pg. 234 – 236: #10 – 52 even