1. The Number of Elements in a Set 2.4

2. 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)

3. 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 = – n(T ∩ N)
90 = – n(T ∩ N) T N
n(T ∩ N) = 40 35 40 15 10 Ų

4. 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?

5. 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

6. 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?

7. 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

8. 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