Math 141: Business Mathematics I
Fall 2015
§6.2 The Number of Elements in a Finite Set
Instructor: Yeong-Chyuan Chung
Counting the elements in a set - addition rule
Some problems in mathematics involve counting the number of elements in a set. Counting problems such as these constitute a field of study known as combinatorics. Counting
techniques will also be relevant to us when we study probability.
If we are given a finite set A, then we write n(A) for the number of elements in A.
Example.
1. Recall that the empty set, denoted by ∅, is the set containing no elements,
so n(∅) = 0.
2. If A = {x|x is a letter in the English alphabet}, then n(A) = 26.
3. If B = {a, b}, then n(B) = 2.
Example. Recall that two sets A and B are said to be disjoint if their intersection is empty
(i.e. A ∩ B = ∅). If A and B are both finite sets, and they are disjoint, then what can we
say about n(A ∪ B) in terms of n(A) and n(B)? (Think about the Venn diagram.)
In general, when the two sets A and B are not disjoint, we need to be careful not to double
count elements in the intersection when we are counting the number of elements in A ∪ B.
Addition Rule for Sets
If A and B are finite sets, then
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
You can make sense of this formula by thinking in terms of a Venn diagram.
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§6.2 The Number of Elements in a Finite Set
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Example (Exercise 22 in the text). In a survey of 200 members of a local sports club, 100
members indicated that they plan to attend the next Summer Olympic Games, 60 indicated
that they plan to attend the next Winter Olympic Games, and 40 indicated that they plan to
attend both games. How many members of the club plan to attend
(a) At least one of the two games?
(b) Exactly one of the games?
(c) The Summer Olympic Games only?
(d) None of the games?
Example (Exercise 34 in the text). Let A, B, and C be subsets of a universal set U and
suppose n(U ) = 100, n(A) = 28, n(B) = 30, n(C) = 34, n(A ∩ B) = 8, n(A ∩ C) = 10,
n(B ∩ C) = 15, and n(A ∩ B ∩ C) = 5. Compute n[A ∩ (B ∪ C)] and n[A ∩ (B ∪ C)c ].
(It may be useful to draw a Venn diagram to represent the given information.)
§6.2 The Number of Elements in a Finite Set
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Example (c.f. Exercise 46 in the text). To help plan the number of meals (breakfast, lunch,
and dinner) to be prepared in a college cafeteria, a survey was conducted and the following
data were obtained:
• 8 students ate only breakfast.
• 130 students ate breakfast.
• 80 students ate only lunch.
• 96 students ate exactly 2 meals.
• 68 students ate breakfast and lunch.
• 112 students ate breakfast and dinner.
• 58 students ate all three meals.
• 99 students ate exactly 1 meal.
• 100 students did not eat dinner.
How many students were surveyed?