Download MATH 105: Finite Mathematics 6-2: The Number of Elements in a Set

Counting with Venn Diagrams
Story Problems
MATH 105: Finite Mathematics
6-2: The Number of Elements in a Set
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
Conclusion
Counting with Venn Diagrams
Outline
1
Counting with Venn Diagrams
2
Story Problems
3
Conclusion
Story Problems
Conclusion
Counting with Venn Diagrams
Outline
1
Counting with Venn Diagrams
2
Story Problems
3
Conclusion
Story Problems
Conclusion
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Set Elements
Number of Elements in a Set
Let A be a set. Then, c(A) is the number of elements in the set A.
Example
Find the number of elements in each set.
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Set Elements
Number of Elements in a Set
Let A be a set. Then, c(A) is the number of elements in the set A.
Example
Find the number of elements in each set.
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Set Elements
Number of Elements in a Set
Let A be a set. Then, c(A) is the number of elements in the set A.
Example
Find the number of elements in each set.
(a) A = {2, 3, 5, a}
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Set Elements
Number of Elements in a Set
Let A be a set. Then, c(A) is the number of elements in the set A.
Example
Find the number of elements in each set.
(a) A = {2, 3, 5, a}
c(A) = 4
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Set Elements
Number of Elements in a Set
Let A be a set. Then, c(A) is the number of elements in the set A.
Example
Find the number of elements in each set.
(a) A = {2, 3, 5, a}
(b) B = {3, x, y }
c(A) = 4
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Set Elements
Number of Elements in a Set
Let A be a set. Then, c(A) is the number of elements in the set A.
Example
Find the number of elements in each set.
(a) A = {2, 3, 5, a}
c(A) = 4
(b) B = {3, x, y }
c(B) = 3
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Set Elements
Number of Elements in a Set
Let A be a set. Then, c(A) is the number of elements in the set A.
Example
Find the number of elements in each set.
(a) A = {2, 3, 5, a}
c(A) = 4
(b) B = {3, x, y }
c(B) = 3
(c) A ∩ B
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Set Elements
Number of Elements in a Set
Let A be a set. Then, c(A) is the number of elements in the set A.
Example
Find the number of elements in each set.
(a) A = {2, 3, 5, a}
c(A) = 4
(b) B = {3, x, y }
c(B) = 3
(c) A ∩ B
c(A ∩ B) = 1
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Set Elements
Number of Elements in a Set
Let A be a set. Then, c(A) is the number of elements in the set A.
Example
Find the number of elements in each set.
(a) A = {2, 3, 5, a}
c(A) = 4
(b) B = {3, x, y }
c(B) = 3
(c) A ∩ B
c(A ∩ B) = 1
(d) A ∪ B
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Set Elements
Number of Elements in a Set
Let A be a set. Then, c(A) is the number of elements in the set A.
Example
Find the number of elements in each set.
(a) A = {2, 3, 5, a}
c(A) = 4
(b) B = {3, x, y }
c(B) = 3
(c) A ∩ B
c(A ∩ B) = 1
(d) A ∪ B
c(A ∪ B) = 6
Counting with Venn Diagrams
Story Problems
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
Conclusion
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
2
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
2
3
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
2
5
3
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
2
5
a
3
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
2
5
a
3
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
2
5
a
3
x
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
2
5
a
3
x
y
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
2
5
a
3
x
y
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
2
5
a
3
x
y
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
c(A ∪ B) = 6
Counting with Venn Diagrams
Story Problems
Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x, y } are
arranged in a Venn Diagram.
A
B
2
5
a
3
x
y
Notice the Relationship. . .
c(A) + c(B) = 4 + 3 = 7
c(A ∪ B) = 6
c(A ∩ B) = 1
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Rules
Counting Formula
c(A ∪ B) = c(A) + c(B) − c(A ∩ B)
Example
Each of the next examples leads to another useful counting rule.
(a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(A ∩ C ).
(b) Let N = {0, 1, 2, . . .} be the set of natural numbers. Find
c(N).
(c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(D).
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Rules
Counting Formula
c(A ∪ B) = c(A) + c(B) − c(A ∩ B)
Example
Each of the next examples leads to another useful counting rule.
(a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(A ∩ C ).
(b) Let N = {0, 1, 2, . . .} be the set of natural numbers. Find
c(N).
(c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(D).
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Rules
Counting Formula
c(A ∪ B) = c(A) + c(B) − c(A ∩ B)
Example
Each of the next examples leads to another useful counting rule.
(a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(A ∩ C ).
(b) Let N = {0, 1, 2, . . .} be the set of natural numbers. Find
c(N).
(c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(D).
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Rules
Counting Formula
c(A ∪ B) = c(A) + c(B) − c(A ∩ B)
Example
Each of the next examples leads to another useful counting rule.
(a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(A ∩ C ).
(b) Let N = {0, 1, 2, . . .} be the set of natural numbers. Find
c(N).
(c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(D).
Counting with Venn Diagrams
Story Problems
Conclusion
Counting Rules
Counting Formula
c(A ∪ B) = c(A) + c(B) − c(A ∩ B)
Example
Each of the next examples leads to another useful counting rule.
(a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(A ∩ C ).
(b) Let N = {0, 1, 2, . . .} be the set of natural numbers. Find
c(N).
(c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(D).
Counting with Venn Diagrams
Outline
1
Counting with Venn Diagrams
2
Story Problems
3
Conclusion
Story Problems
Conclusion
Counting with Venn Diagrams
Story Problems
Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twenty
of them like Greek food, 32 like Italian food, and 12 like neither
Greek nor Italian food. How many like Greek but not Italian food?
G – Greek food
I – Italian food
G ∪ I – 12
G ∪ I – 50-12 = 38
G ∩ I – 20+32-38 = 14
Counting with Venn Diagrams
Story Problems
Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twenty
of them like Greek food, 32 like Italian food, and 12 like neither
Greek nor Italian food. How many like Greek but not Italian food?
G – Greek food
I – Italian food
G ∪ I – 12
G ∪ I – 50-12 = 38
G ∩ I – 20+32-38 = 14
50
Counting with Venn Diagrams
Story Problems
Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twenty
of them like Greek food, 32 like Italian food, and 12 like neither
Greek nor Italian food. How many like Greek but not Italian food?
50
G – Greek food
I – Italian food
G ∪ I – 12
G ∪ I – 50-12 = 38
G ∩ I – 20+32-38 = 14
G
Counting with Venn Diagrams
Story Problems
Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twenty
of them like Greek food, 32 like Italian food, and 12 like neither
Greek nor Italian food. How many like Greek but not Italian food?
50
G – Greek food
I – Italian food
G ∪ I – 12
G ∪ I – 50-12 = 38
G ∩ I – 20+32-38 = 14
G
I
Counting with Venn Diagrams
Story Problems
Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twenty
of them like Greek food, 32 like Italian food, and 12 like neither
Greek nor Italian food. How many like Greek but not Italian food?
50
G – Greek food
I – Italian food
G
I
G ∪ I – 12
G ∪ I – 50-12 = 38
G ∩ I – 20+32-38 = 14
12
Counting with Venn Diagrams
Story Problems
Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twenty
of them like Greek food, 32 like Italian food, and 12 like neither
Greek nor Italian food. How many like Greek but not Italian food?
50
G – Greek food
I – Italian food
G
I
G ∪ I – 12
G ∪ I – 50-12 = 38
G ∩ I – 20+32-38 = 14
12
Counting with Venn Diagrams
Story Problems
Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twenty
of them like Greek food, 32 like Italian food, and 12 like neither
Greek nor Italian food. How many like Greek but not Italian food?
50
G – Greek food
I – Italian food
G ∪ I – 12
G ∪ I – 50-12 = 38
G ∩ I – 20+32-38 = 14
G
I
14
12
Counting with Venn Diagrams
Story Problems
Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twenty
of them like Greek food, 32 like Italian food, and 12 like neither
Greek nor Italian food. How many like Greek but not Italian food?
50
G – Greek food
I – Italian food
G ∪ I – 12
G ∪ I – 50-12 = 38
G ∩ I – 20+32-38 = 14
G
I
6
14
12
Counting with Venn Diagrams
Story Problems
Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twenty
of them like Greek food, 32 like Italian food, and 12 like neither
Greek nor Italian food. How many like Greek but not Italian food?
50
G – Greek food
I – Italian food
G ∪ I – 12
G ∪ I – 50-12 = 38
G ∩ I – 20+32-38 = 14
G
I
6
14
18
12
Counting with Venn Diagrams
Story Problems
Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twenty
of them like Greek food, 32 like Italian food, and 12 like neither
Greek nor Italian food. How many like Greek but not Italian food?
50
G – Greek food
I – Italian food
G ∪ I – 12
G ∪ I – 50-12 = 38
G ∩ I – 20+32-38 = 14
G
I
6
14
18
12
Counting with Venn Diagrams
Story Problems
Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribe
to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37
of the families who subscribe to Rolling Stone do not subscribe to
the Wall Street Journal. How many families subscribe to both, and
how many subscribe to neither?
W – Wall Street Journal
R – Rolling Stone
R ∩ W – 37
R ∩ W – 41 - 37 = 4
R ∩ W – 63 - 4 = 59
R ∪ W – 500 - 37 - 4 - 59 = 400
Counting with Venn Diagrams
Story Problems
Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribe
to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37
of the families who subscribe to Rolling Stone do not subscribe to
the Wall Street Journal. How many families subscribe to both, and
how many subscribe to neither?
W – Wall Street Journal
R – Rolling Stone
R ∩ W – 37
R ∩ W – 41 - 37 = 4
R ∩ W – 63 - 4 = 59
R ∪ W – 500 - 37 - 4 - 59 = 400
500
Counting with Venn Diagrams
Story Problems
Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribe
to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37
of the families who subscribe to Rolling Stone do not subscribe to
the Wall Street Journal. How many families subscribe to both, and
how many subscribe to neither?
W – Wall Street Journal
R – Rolling Stone
R ∩ W – 37
R ∩ W – 41 - 37 = 4
R ∩ W – 63 - 4 = 59
R ∪ W – 500 - 37 - 4 - 59 = 400
500
W
Counting with Venn Diagrams
Story Problems
Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribe
to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37
of the families who subscribe to Rolling Stone do not subscribe to
the Wall Street Journal. How many families subscribe to both, and
how many subscribe to neither?
W – Wall Street Journal
R – Rolling Stone
R ∩ W – 37
R ∩ W – 41 - 37 = 4
R ∩ W – 63 - 4 = 59
R ∪ W – 500 - 37 - 4 - 59 = 400
500
W
R
Counting with Venn Diagrams
Story Problems
Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribe
to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37
of the families who subscribe to Rolling Stone do not subscribe to
the Wall Street Journal. How many families subscribe to both, and
how many subscribe to neither?
W – Wall Street Journal
R – Rolling Stone
500
W
R
R ∩ W – 37
R ∩ W – 41 - 37 = 4
R ∩ W – 63 - 4 = 59
R ∪ W – 500 - 37 - 4 - 59 = 400
37
Counting with Venn Diagrams
Story Problems
Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribe
to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37
of the families who subscribe to Rolling Stone do not subscribe to
the Wall Street Journal. How many families subscribe to both, and
how many subscribe to neither?
W – Wall Street Journal
R – Rolling Stone
500
W
R
R ∩ W – 37
R ∩ W – 41 - 37 = 4
R ∩ W – 63 - 4 = 59
R ∪ W – 500 - 37 - 4 - 59 = 400
4
37
Counting with Venn Diagrams
Story Problems
Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribe
to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37
of the families who subscribe to Rolling Stone do not subscribe to
the Wall Street Journal. How many families subscribe to both, and
how many subscribe to neither?
W – Wall Street Journal
R – Rolling Stone
500
W
R
R ∩ W – 37
R ∩ W – 41 - 37 = 4
R ∩ W – 63 - 4 = 59
R ∪ W – 500 - 37 - 4 - 59 = 400
59
4
37
Counting with Venn Diagrams
Story Problems
Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribe
to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37
of the families who subscribe to Rolling Stone do not subscribe to
the Wall Street Journal. How many families subscribe to both, and
how many subscribe to neither?
W – Wall Street Journal
R – Rolling Stone
500
W
R
R ∩ W – 37
R ∩ W – 41 - 37 = 4
59
4
37
R ∩ W – 63 - 4 = 59
R ∪ W – 500 - 37 - 4 - 59 = 400
400
Counting with Venn Diagrams
Story Problems
Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribe
to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37
of the families who subscribe to Rolling Stone do not subscribe to
the Wall Street Journal. How many families subscribe to both, and
how many subscribe to neither?
W – Wall Street Journal
R – Rolling Stone
500
W
R
R ∩ W – 37
R ∩ W – 41 - 37 = 4
59
4
37
R ∩ W – 63 - 4 = 59
R ∪ W – 500 - 37 - 4 - 59 = 400
400
Counting with Venn Diagrams
Story Problems
Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribe
to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37
of the families who subscribe to Rolling Stone do not subscribe to
the Wall Street Journal. How many families subscribe to both, and
how many subscribe to neither?
W – Wall Street Journal
R – Rolling Stone
500
W
R
R ∩ W – 37
R ∩ W – 41 - 37 = 4
59
4
37
R ∩ W – 63 - 4 = 59
R ∪ W – 500 - 37 - 4 - 59 = 400
400
Counting with Venn Diagrams
Story Problems
Conclusion
Car Sales
Example
Of the cars sold during the month of July, 90 had air conditioning,
100 had automatic transmissions, and 75 had power steering. Five
cars had all three of these extras. Twenty cars had none of these
extras. Twenty cars had only air conditioning; 60 cars had only
automatic transmissions; and 30 cars had only power steering. Ten
cars had both automatic transmission and power steering.
(a) How many cars had both power steering and air conditioning?
(b) How many had both automatic transmission and air
conditioning?
(c) How many cars were sold in July?
Counting with Venn Diagrams
Story Problems
Student Transportation
Example
The transportation and Parking Committee at Gigantic State
University collects data from 100 students on how they commute
to campus. The following data is obtained:
8
20
48
38
drive a car at least part of the time
use the bus at least part of the time
ride a bicycle at least part of the time
do none of these
no student who drives a care also uses the bus
How many students who ride a bicycle also dirve a car or use the
bus?
Conclusion
Counting with Venn Diagrams
Outline
1
Counting with Venn Diagrams
2
Story Problems
3
Conclusion
Story Problems
Conclusion
Counting with Venn Diagrams
Story Problems
Important Concepts
Things to Remember from Section 6-2
1
Do not double count elements in a union.
2
Counting Formula #1:
c(A ∪ B) = c(A) + c(B) − c(A ∩ B)
3
Counting Formula #2:
c(U) = c(A) + c(A)
4
Only place numbers on Venn Diagrams if they belong to a
single area.
Conclusion
Counting with Venn Diagrams
Story Problems
Important Concepts
Things to Remember from Section 6-2
1
Do not double count elements in a union.
2
Counting Formula #1:
c(A ∪ B) = c(A) + c(B) − c(A ∩ B)
3
Counting Formula #2:
c(U) = c(A) + c(A)
4
Only place numbers on Venn Diagrams if they belong to a
single area.
Conclusion
Counting with Venn Diagrams
Story Problems
Important Concepts
Things to Remember from Section 6-2
1
Do not double count elements in a union.
2
Counting Formula #1:
c(A ∪ B) = c(A) + c(B) − c(A ∩ B)
3
Counting Formula #2:
c(U) = c(A) + c(A)
4
Only place numbers on Venn Diagrams if they belong to a
single area.
Conclusion
Counting with Venn Diagrams
Story Problems
Important Concepts
Things to Remember from Section 6-2
1
Do not double count elements in a union.
2
Counting Formula #1:
c(A ∪ B) = c(A) + c(B) − c(A ∩ B)
3
Counting Formula #2:
c(U) = c(A) + c(A)
4
Only place numbers on Venn Diagrams if they belong to a
single area.
Conclusion
Counting with Venn Diagrams
Story Problems
Important Concepts
Things to Remember from Section 6-2
1
Do not double count elements in a union.
2
Counting Formula #1:
c(A ∪ B) = c(A) + c(B) − c(A ∩ B)
3
Counting Formula #2:
c(U) = c(A) + c(A)
4
Only place numbers on Venn Diagrams if they belong to a
single area.
Conclusion
Counting with Venn Diagrams
Story Problems
Conclusion
Next Time. . .
Venn Diagrams are useful for organizing known information about
set sizes, but we don’t always know that information.
In the next section we look at the first of several counting rules
used to determine set sizes.
For next time
Read Section 6-3 (pp 332-335)
Prepare for quiz on 6-1 and 6-2
Do Problem Sets 6-1 A; 6-2 A,B
Counting with Venn Diagrams
Story Problems
Conclusion
Next Time. . .
Venn Diagrams are useful for organizing known information about
set sizes, but we don’t always know that information.
In the next section we look at the first of several counting rules
used to determine set sizes.
For next time
Read Section 6-3 (pp 332-335)
Prepare for quiz on 6-1 and 6-2
Do Problem Sets 6-1 A; 6-2 A,B