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2.3 Venn Diagrams and Set Operations Intersection: The intersection of two sets A and B, A B, is the set that consists of the elements that are common to both A and B. If A = {1, 2, 3} and B = {2, 3, 4}, then A B = {2, 3}. Union: The Union of two sets A and B, A B, is the set that consists of all the elements in A or B or both. If A = {1, 2, 3} and B = {2, 3, 4}, then A B = {1, 2, 3, 4}. Two sets that have no elements in common are called disjoint sets. U = {0, 1, 2, …, 20} X = {5, 6, 7, …., 20}, Y = {…, 10, 11, 12}, E = (0, 2, 4, 6, 8, 10}, O = {1, 3, 5, 7, 9}, find: 1. X Y 2. E O 3. XY 4. EO 5. X (E O) 6. (E X)’ Y 6. X 8. Y DeMorgan’s Laws: (A B)’ = A’ B’ The complement of the union is the intersection of the complements. (A B)’ = A’ B’ The complement of the intersection is the union of the complements. Given U = {a, b, c, d, e, f, g}, A = {b, c, g}, and B = {b, c, e}, find: 9. (A B)’ 10. A’ B’ 11. (A B)’ 12. A’ B’ Construct Venn diagrams for each of the following: 13. A B’ 14. A’ B’ 15. B’ A 2.4: SET OPERATIONS AND VENN DIAGRAMS WITH THREE SETS To do set operations involving three sets, begin inside the parentheses. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4, 5}, B = {1, 2, 3, 6, 8}, and C = {2, 3, 4, 6, 7}: 1. Find A (B C) 2. Find (A B) {A C) 3. A (B C') Using Venn Diagrams, three intersecting sets separate the universal set into eight regions. The numbering of the regions is arbitrary; that is, any region can be designated as region I. A C B 1. Which regions represent set B? 2. Which regions represent B C? 3. Which regions represent A B? 4. Which regions represent B’? 5. Which regions represent A B C? 6. Find A (B C) 7. Find (A B) (A C) 8. Which regions represent B (A C) 9. Which regions represent (A C) (B C)’ Class Practice 2.3 Let U = {a, b, c, d, e, f, g, h} A = {a, g, h} B = {b, g, h} C = {b, c, d, e, f} 1. Find B’ C 2. C 3. (A C)’ 4. B U Let U = {xx N and x < 9} A = {xx is odd} B = {xx is even} C = {x1 < x < 6} 5. B C 6. A’ B 7. (A B)’ 2.4 Let U = {a, b, c, d, e, f, g, h} A = {a, g, h} B = {b, g, h} C = {b, c, d, e, f} Fill in the Venn Diagram below for the above sets. Then answer questions 8 – 10. 8. (A B) (A C) 9. (A B C)’ 10. (B C)’ A A B C Determine whether the given sets are equal for all sets A, B, C: 11. A (B C)’; A (B’ C’) 12. B (A C); (A B) (B C) 2.5 SURVEYS AND CARDINAL NUMBERS In surveys, remember that and means intersection, or means union, and not means complement. Formula for the Cardinal Number of the Union of Two Sets: n(A B) = n(A) + n(B) – n(A B). To solve survey problems, [1] use the survey’s description to define sets and draw a Venn diagram; [2] use the survey’s results to determine the cardinality for each region (starting with the intersection of the sets and working outward); [3] use the completed Venn diagram to answer the problem’s questions. 1. If Set A contain 50 elements and Set B contains 30 elements and 15 elements are common to both A and B, how many elements are in A B? 2. In a Gallop poll, 2000 U.S. adults were asked to agree or disagree with the following statement: Job opportunities for women are not equal to those for men. The results shows that 1190 people agreed with the statement, of whom 700 were women. Half the people surveyed were women. Step 1: Define sets; draw diagram. U W A Step 2: Determine cardinality of each region: n(U) = n(W) = n(A) = n(W A) = Step 3: Answer questions from diagram: How many men agreed with the statement? How many men disagreed? On a Venn diagram, shade the following: 3. A’ (B C’) 4. (A B)’ C 5. A poll asks the following question: Do you agree or disagree with the statement: In order to address the trend in diminishing male enrollment, colleges should begin special efforts to recruit men? Construct a Venn diagram that allows the respondent to be identified as man or woman, educational level (college or no college), and whether they agree or disagree with the statement. Draw a Venn diagram and determine how many people fall into each of the eight regions. M I II I C III V A V V VII II VIII Men Women College No College College No College Agree Disagree 6. A survey of 1500 people was taken to determine their preferences for Coke, Sprite, or other carbonated beverages. The results were as follows: Coke (C), 488; Sprite (S), 464; other carbonated beverage (B), 417; Coke and Sprite, 128; Coke and Other, 112; Sprite and Other, 106; all three, 98. 7. The employees in one section of a electric utility company cut down tall trees, climb poles, and splice wire. The section chief reported the following information to management: Of the 100 employees in the section, 45 can cut tall trees (T), 50 can climb poles (P), 57 can splice wire (W), 28 can cut trees and climb poles, 20 can climb poles and splice wire, 25 can cut trees and splice wire, 11 can do all three. How many can’t do any of the three?