The Number of Elements in a Set 2.4

The Cardinality of a Set Number of Elements in A ∪ B If A is any set, the number of elements in A is denoted by n(A). The number n(A) is called the cardinal number of A. For example, if A = {1,2,3,4}, then n(A) = 4. Number of Elements in A ∪ B n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

Exercises 5. On checking with 100 families, it was found that 75 subscribe to Time, 55 to Newsweek, and 10 to neither magazine. How many subscribe to both? n(T ∪ N) = n(T) + n(N) – n(T ∩ N) 90 = 75 + 55 – n(T ∩ N) 90 = 130 – n(T ∩ N) T N n(T ∩ N) = 40 35 40 15 10 Ų

Exercises 8. Mr. N. Roll, the registrar at a university, has observed that, of the students, 45% have a 9am class 45% have a 10am class 40% have an 11am class 20% have a 9 and a 10am class 10% have a 9 and an 11am class 15% have a 10 and an 11amclass 5% have a 9, 10, and an 11am class What percent of the students have only a 9am class? What percent of the students have no classes at these times?

20% of students have only a 9am class. 10% of the students have no classes at these times. A B 15 20 15 5 10 5 20 10 C U

Exercises 12. In a survey of 100 investors, it was found that 5 owned utilities stock only 15 owned transportation stock only 70 owned bonds 13 owned utilities and transportation stock 23 owned transportation stock and bonds 10 owned utilities stock and bonds 3 owned all three kinds How many investors owned bonds only? How many investors owned utilities and/or transportation stock? How many investors owned neither bonds nor utilities stock?

40 investors owned bonds only. 60 investors owned utilities and/or transportation stock. 15 investors owned neither bonds nor utilities stock. U T 10 5 15 3 20 7 40 B U

Exercises 23. If the total number of students surveyed to obtain the data for problem 21 is 200, find a. n(A’)=80 b. n(C’)=120 c. n(A’ ∩ C’)=50 A C 70 50 30 50 Ų