Section 2.3 Set Operations and Cartesian Products - PowerPoint PPT Presentation

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Section 2.3 Set Operations and Cartesian Products

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  1. Intersections of Sets The intersection of Set A and B, written is the set of elements common to both A and B. Section 2.3 Set Operations and Cartesian Products

  2. Example Suppose we have two candidates, Mr. Brown and Mr. Green running for a city office. A voter decided for whom she should vote by recalling their campaign promises.

  3. Lets look at each candidates promises as a set. Mr. Green’s set = { m,t,s} Mr. Brown’s set = { m,p,c} The only element common to both sets is m, this is the intersection of both sets.

  4. To represent the sets as a Venn Diagram Mr. Brown Mr. Green T S P C M

  5. Find the intersection of the given sets • A) { 3,4,5,6,7} and {4,6,8,10} • Elements common to both sets: { 3,4,5,6,7}{4,6,8,10} = {4,6} B) { 9,14,25,30} {10,17,19,38,52} { 9,14,25,30} {10,17,19,38,52} =

  6. DOGS CATS • C){ 5,9,11} and • { 5,9,11} = • Sets with no elements in common are called disjoint sets A set of dogs and a set of cats are disjoint sets

  7. Union of Sets • Form the union of sets A and B by taking all the elements of set A and including all the elements of set B. Set B Set A P C T S M

  8. Union of Sets

  9. Find the union of the sets • A) { 2,4,6} and {4,6,8,10,12} • Answer: { 2,4,6} {4,6,8,10,12} = {2,4,6,8,10,12} • B) { a,b,c,d} and { c, f, g} • Answer: {a,b,c,d,f,g} • C) {3,4,5} and • Answer: {3,4,5}

  10. More examples • U = { 1,2,3,4,5,6,9} • A = { 1,2,3,4} • B = { 2,4,6} • C = { 1,3,6,9} • Find

  11. Find

  12. Answer:

  13. Try to describe the following sets in words: • 1. The set of all elements that are in A, and are in B or not in C. The set of all elements that are not in A or not in C, and are not in B.

  14. Differences of Sets • Let Set A = { 1,2,3,4,5,6,7,8,9,10} • Let Set B = { 2,4,6,8,10} • If the elements from B are taken away from Set A • then Set C = {1,3,5,7,9} • Set C is the difference of sets A and B.

  15. Difference of Sets The difference of sets A and B, written A – B is the set of all elements belonging to set A and not to set B, or

  16. A B A - B

  17. Examples • Let U = {1,2,3,4,5,6,7} • A = {1,2,3,4,5,6} • B = { 2,3,6} • C = {3,5,7} • Find • A – B • B – A • (A – B)

  18. Let U = {1,2,3,4,5,6,7}A = {1,2,3,4,5,6}B = { 2,3,6}C = {3,5,7} Find A – B • Begin with set A and exclude any elements found also in set B. • So A – B = {1,2,3,4,5,6} – { 2,3,6 } = {1,4,5}

  19. B Let U = {1,2,3,4,5,6,7}A = {1,2,3,4,5,6}B = { 2,3,6}C = {3,5,7} Find B – A For B-A an element must be in set B and not in set A. But all elements of B are in A, so B-A = • {2,3,6} – {1,2,3,4,5,6} = 2,3,6 A 1,4,5

  20. Let U = {1,2,3,4,5,6,7}A = {1,2,3,4,5,6}B = { 2,3,6}C = {3,5,7} Find (A – B) We know A – B = {1,4,5} And C’ = { 1,2,4,6} So (A – B) = { 1,2,4,5,6} In general A – B does not equal B - A

  21. Writing ordered pairs • In set notation {4,5} = {5,4} • There are many instances in math where order matters. So we write ordered pairs using parentheses. • Ordered Pairs: • In the ordered pair (a,b),a is called the firstcomponent and b is called the second component. • In general (a,b) (b,a).

  22. Ordered Pairs • Two ordered pairs (a,b) and (c,d) are equal if their first components are equal and if their second components are equal. • So (a,b) = (c,d) if and only if a = c and b=d. • True or false: (4,7) = (7,4)

  23. Cartesian Product of Sets • A set may contain ordered pairs as elements. • If A and B are sets, then each element of A can be paired with an element of B. • The set of all ordered pairs is known as the Cartesian Product of A and B. • Written A X B.

  24. Exercises • Find • A X B = • B X A = • A X A = For Set A = {1,5,9} and B = { 6,7} {(1,6),(1,7),(5,6),(5,7),(9,6),(9,7)} {(6,1),(6,5),(6,9),(7,1),(7,5),(7,9)} {(1,1),(1,5),(1,9),(5,1),(5,5),(5,9),(9,1),(9,5),(9,9)}

  25. Cardinal Number of a Cartesian Product For example: Set A = {1,2,3}, then n(A) = Set B = {4,5}, then n(B)= 3 2 So, n(A) X n(B) = 3x2 = 6 AXB= {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)}

  26. Set Operations • Let A and B be any sets, with U the universal set. • The complement of A, written A’ is A’ A U

  27. The intersection of A and B is A B U

  28. The Union of A and B is A B U

  29. The Difference of A – B is B A U

  30. The Cartesian Product of A and B is

  31. Region 1: Elements outside of set A and Set B. Region 2: Elements belong to A but not to B Region 3: Elements belonging to both A and B Region 4: Elements belong to B but not to A 1 A B 3 2 4 U Regions of Venn Diagrams 1 A B 3 2 4 U

  32. Let U = {q,r,s,t,u,v,w,x,y,z} Let A = {r,s,t,u,v} Let B = {t,v,x} Place the elements in their proper regions. 1 A B 3 2 4 U Example q, w, y, z x t, v First find intersection points r, s, u What elements belong to B and A? What elements belong to U?

  33. Venn Diagram • Represent three sets. • Let A, B and C be sets 1 B A 6 4 2 5 7 3 C 8 U

  34. Exercises • Shade the set Is the statementtrue

  35. Use the Venn Diagram 1 Region 3 B A Region 1,2,4 3 3 2 4 Region 1,4 U Region 1,2 Region 1,2,4

  36. De Morgan’s Laws • For any sets A and B,

  37. Section 2.4Cardinal Numbers and Surveys • Suppose we have this data from a survey • 33 people like Tim McGraw • 32 favor Celine Dion • 28 favor Britney Spears • 11 favor Tim and Celine • 15 favor Tim and Britney • 5 like all performers • 7 like non of the performers • Can we determine the total number of people surveyed from the data.

  38. a c h d b e g f First look at intersection region d – 5 who like all three singers 7 7 like non Region a Region b 33-10-5-6=12 Britney Spears Tim Mc Graw 10 11 like Tim and Celine Put them in regions d and e 11-5=6 for region e Region g 14 like Britney and Celine 14-5=9 12 4 5 9 6 Region h 28-10-5-9= 4 15 like Britney and Tim Regions c and d 15 -5 = 10 region c 12 Region f 32-6-5-9=12 Celine Dion U To find out how many students were surveyed – Add numbers in all the regions - 65

  39. Cardinal Number Formula • For any two sets A and B

  40. Find n(A) • Try this Use the formula: Rearrange the formula to find n(A)

  41. Some utility company has 100 employees with • T = set of employees who can cut trees • P = set of employees who can climb poles • W = set of employees who can splice wires T 3 P 17 11 13 14 9 W 23 U

  42. Homework • Section 2.3 odd only • 7-28,41-54,55-96,97,101,105,109,117,127 • Section 2.4 odd • 1-16,17