The Number of Elements in a Set
It is possible for a set to be finite or infinite. If a set is finite, then the
number of elements in the set is a non-negative integer. In this class, we will
only be concerned with finite sets.
Notation: n(A) = number of elements in the (finite) set A.
Examples: C = {0, 1, . . . , 10}
D = {3, 4, 9, a}
E = {a, b, f, e}
Rules for Counting Sets
Rule #1: If A and B are finite sets, then n(A∪B) = n(A)+n(B)−n(A∩B).
We have to subtract off n(A ∩ B) to avoid “over-counting.”
Example 1: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is n(A ∪ B)?
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Counting Elements of Disjoint Sets
What is n(∅)?
If A ∩ B = ∅, then we notice that
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
= n(A) + n(B) − n(∅)
= n(A) + n(B) − 0
= n(A) + n(B)
Rule #2 (special case of Rule #1): If A and B are disjoint sets, then
n(A ∪ B) = n(A) + n(B)
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Example
Of 80 TAMU students recently surveyed, 78 were on Facebook, and 64 were
on Twitter.
(a.) How many were on Facebook and on Twitter?
(b.) How many were only on Facebook?
(c.) How many were only on Twitter?
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Counting Rule for Three Sets
Rule #3: If A, B, and C are finite sets, then
n(A∪B∪C) = n(A)+n(B)+n(C)−n(A∪B)−n(B∪C)−n(A∪C)+n(A)+n(B)+n(C).
Example: based on an example by Kathryn Bollinger A survey of 50 students was conducted.
Let G be the set of students who watch Game of Thrones.
Let H be the set of students who watch House of Cards.
Let N be the set of students who watch New Girl.
Use the survey results below to find the number of people in each section of
the given Venn diagram.
• 20 like Game of Thrones
• 35 like House of Cards
• 16 like New Girl
• 13 like Game of Thrones and House of Cards
• 5 like Game of Thrones and New Girl
• 7 like House of Cards and New Girl
• 3 like all three of these shows
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Example, continued
How many of the students surveyed liked
• none of the shows?
• House of Cards and Game of Thrones, but not New Girl?
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Another Example
(based on an example by Kathryn Bollinger )
A survey of 85 teenagers asked which type of pet they owned.
Let C be the set of people who own a cat.
Let D be the set of people who own a dog.
Let O be the set of people who own some other kind of pet (fish, rabbit, etc.)
(a) Use the survey results below to find the number of people in each section
of the given Venn diagram.
• 6 own all three kinds of pet
• 22 own at least two kinds of pet
• 40 own cats
• 4 participate own a cat and a dog, but not an other
• 5 do not own a pet of any kind
• 8 own a dog and an other
• 55 own a cat or an other, but not a dog
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Another Example, continued
Using set notation, write
(a.) the set of all people who own a pet (of any type)
(b.) the set of all people who own only a dog
(c.) the set of all people who own a cat or an other
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