SECTION 13.1 Counting 915 13.1 Assess Your Understanding 2. The of A with B consists of all elements in both A and B. (pp. A1–A3) 1. The of A and B consists of all elements in either A or B or both. (pp. A1–A3) ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 3. True or False The intersection of two sets is always a subset of their union. (pp. A1–A3) 4. True or False If A is a set, the complement of A is the set of all the elements in the universal set that are not in A. (p. A2 ) 5. If each element of a set A is also an element of a set B, we say that A is a of B and write A B. Concepts and Vocabulary 6. If the number of elements in a set is a nonnegative integer, we say that the set is . 7. True or False If a task consists of a sequence of three choices in which there are p selections for the first choice, q selections for the second choice, and r selections for the third choice, then the task of making these selections can be done in ⋅ ⋅ p q r different ways. 8. Multiple Choice The Counting Formula states that if A and B are finite sets, then ( ) ∪ = n A B . (a) ( ) ( ) + n A n B (b) ( ) ( ) ( ) + − ∩ n A n B n A B (c) ( ) ( ) ⋅ n A n B (d) ( ) ( ) − n A n B 9. Write down all the subsets of { } a b c d , , , . 10. Write down all the subsets of { } a b c d e , , , , . Skill Building 11. If ( ) ( ) = = n A n B 15, 20, and ( ) ∩ = n A B 10, find ( ) ∪ n A B . 12. If ( ) ( ) = = n A n B 30, 40, and ( ) ∪ = n A B 45, find ( ) ∩ n A B . 13. If ( ) ( ) ∪ = ∩ = n A B n A B 50, 10, and ( ) = n B 20, find ( ) n A . 14. If ( ) ( ) ∪ = ∩ = n A B n A B 60, 40, and ( ) ( ) = n A n B , find ( ) n A . In Problems 15–22, use the information given in the figure. 15. How many are in set A? 16. How many are in set B? 17. How many are in A or B? 18. How many are in A and B? 19. How many are in A but not C? 20. How many are not in A? 21. How many are in A and B and C? 22. How many are in A or B or C? A B C 15 3 15 10 5 2 4 2 U 23. Shirts and Ties A man has 5 shirts and 3 ties. How many different shirt-and-tie arrangements can he wear? 24. Blouses and Skirts A woman has 5 blouses and 8 skirts. How many different outfits can she wear? 25. Four-digit Numbers How many four-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if the first digit cannot be 0? Repeated digits are allowed. 26. Five-digit Numbers How many five-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if the first digit cannot be 0 or 1? Repeated digits are allowed. 27. Analyzing Survey Data In a consumer survey of 500 people, 200 indicated that they would be buying a major appliance within the next month, 150 indicated that they would buy a car, and 25 said that they would purchase both a major appliance and a car. How many will purchase neither? How many will purchase only a car? 28. Analyzing Survey Data In a student survey, 200 indicated that they would attend Summer Session I, and 150 indicated Summer Session II. If 75 students plan to attend both summer sessions, and 275 indicated that they would attend neither session, how many students participated in the survey? Applications and Extensions 29. Analyzing Survey Data In a survey of 100 investors in the stock market, 50 owned shares in IBM 40 owned shares in AT&T 45 owned shares in GE 20 owned shares in both IBM and GE 15 owned shares in both AT&T and GE 20 owned shares in both IBM and AT&T 5 owned shares in all three (a) How many of the investors surveyed did not have shares in any of the three companies? (b) How many owned just IBM shares? (c) How many owned just GE shares? (d) How many owned neither IBM nor GE? (e) How many owned either IBM or AT&T but no GE? 30. Classifying Blood Types Human blood is classified as either Rh or Rh . + − Blood is also classified by type: A, if it contains an A antigen but not a B antigen; B, if it contains a B antigen but not an A antigen;AB, if it contains both A and B antigens; and O, if it contains neither antigen. Draw a Venn diagram illustrating the various blood types. Based on this classification, how many different kinds of blood are there? 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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