math 105: finite mathematics 6-2: the number of...
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Counting with Venn Diagrams Story Problems Conclusion
MATH 105: Finite Mathematics6-2: The Number of Elements in a Set
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
Counting with Venn Diagrams Story Problems Conclusion
Outline
1 Counting with Venn Diagrams
2 Story Problems
3 Conclusion
Counting with Venn Diagrams Story Problems Conclusion
Outline
1 Counting with Venn Diagrams
2 Story Problems
3 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} c(A) = 4
(b) B = {3, x , y}
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 Conclusion
Placing Elements in a Venn Diagram
Note how the elements of A = {2, 3, 5, a} and B = {3, x , y} arearranged in a Venn Diagram.
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} arearranged 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} arearranged 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} arearranged in a Venn Diagram.
A B
23
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} arearranged in a Venn Diagram.
A B
23
5
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} arearranged in a Venn Diagram.
A B
23
5
a
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} arearranged in a Venn Diagram.
A B
23
5
a
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} arearranged in a Venn Diagram.
A B
23
5
a
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} arearranged in a Venn Diagram.
A B
23
5
a
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} arearranged in a Venn Diagram.
A B
23
5
a
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} arearranged in a Venn Diagram.
A B
23
5
a
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} arearranged in a Venn Diagram.
A B
23
5
a
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. Findc(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. Findc(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. Findc(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. Findc(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. Findc(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
Outline
1 Counting with Venn Diagrams
2 Story Problems
3 Conclusion
Counting with Venn Diagrams Story Problems Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twentyof them like Greek food, 32 like Italian food, and 12 like neitherGreek 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. Twentyof them like Greek food, 32 like Italian food, and 12 like neitherGreek 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. Twentyof them like Greek food, 32 like Italian food, and 12 like neitherGreek 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
G
Counting with Venn Diagrams Story Problems Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twentyof them like Greek food, 32 like Italian food, and 12 like neitherGreek 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
G I
Counting with Venn Diagrams Story Problems Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twentyof them like Greek food, 32 like Italian food, and 12 like neitherGreek 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
G I
12
Counting with Venn Diagrams Story Problems Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twentyof them like Greek food, 32 like Italian food, and 12 like neitherGreek 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
G I
12
Counting with Venn Diagrams Story Problems Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twentyof them like Greek food, 32 like Italian food, and 12 like neitherGreek 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
G I
12
14
Counting with Venn Diagrams Story Problems Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twentyof them like Greek food, 32 like Italian food, and 12 like neitherGreek 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
G I
12
146
Counting with Venn Diagrams Story Problems Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twentyof them like Greek food, 32 like Italian food, and 12 like neitherGreek 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
G I
12
14 186
Counting with Venn Diagrams Story Problems Conclusion
Ethnic Foods
Example
Fifty people are interviewed about their food preferences. Twentyof them like Greek food, 32 like Italian food, and 12 like neitherGreek 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
G I
12
14 186
Counting with Venn Diagrams Story Problems Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribeto the Wall Street Journal, 41 subscribe to Rolling Stone, and 37of the families who subscribe to Rolling Stone do not subscribe tothe Wall Street Journal. How many families subscribe to both, andhow 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 subscribeto the Wall Street Journal, 41 subscribe to Rolling Stone, and 37of the families who subscribe to Rolling Stone do not subscribe tothe Wall Street Journal. How many families subscribe to both, andhow 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 subscribeto the Wall Street Journal, 41 subscribe to Rolling Stone, and 37of the families who subscribe to Rolling Stone do not subscribe tothe Wall Street Journal. How many families subscribe to both, andhow 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 subscribeto the Wall Street Journal, 41 subscribe to Rolling Stone, and 37of the families who subscribe to Rolling Stone do not subscribe tothe Wall Street Journal. How many families subscribe to both, andhow 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 subscribeto the Wall Street Journal, 41 subscribe to Rolling Stone, and 37of the families who subscribe to Rolling Stone do not subscribe tothe Wall Street Journal. How many families subscribe to both, andhow 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
37
Counting with Venn Diagrams Story Problems Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribeto the Wall Street Journal, 41 subscribe to Rolling Stone, and 37of the families who subscribe to Rolling Stone do not subscribe tothe Wall Street Journal. How many families subscribe to both, andhow 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
374
Counting with Venn Diagrams Story Problems Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribeto the Wall Street Journal, 41 subscribe to Rolling Stone, and 37of the families who subscribe to Rolling Stone do not subscribe tothe Wall Street Journal. How many families subscribe to both, andhow 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
37459
Counting with Venn Diagrams Story Problems Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribeto the Wall Street Journal, 41 subscribe to Rolling Stone, and 37of the families who subscribe to Rolling Stone do not subscribe tothe Wall Street Journal. How many families subscribe to both, andhow 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
37459
400
Counting with Venn Diagrams Story Problems Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribeto the Wall Street Journal, 41 subscribe to Rolling Stone, and 37of the families who subscribe to Rolling Stone do not subscribe tothe Wall Street Journal. How many families subscribe to both, andhow 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
37459
400
Counting with Venn Diagrams Story Problems Conclusion
Newspaper Subscriptions
Example
A survey of 500 families provided the following data: 63 subscribeto the Wall Street Journal, 41 subscribe to Rolling Stone, and 37of the families who subscribe to Rolling Stone do not subscribe tothe Wall Street Journal. How many families subscribe to both, andhow 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
37459
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. Fivecars had all three of these extras. Twenty cars had none of theseextras. Twenty cars had only air conditioning; 60 cars had onlyautomatic transmissions; and 30 cars had only power steering. Tencars 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 airconditioning?
(c) How many cars were sold in July?
Counting with Venn Diagrams Story Problems Conclusion
Student Transportation
Example
The transportation and Parking Committee at Gigantic StateUniversity collects data from 100 students on how they commuteto campus. The following data is obtained:
8 drive a car at least part of the time20 use the bus at least part of the time48 ride a bicycle at least part of the time38 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 thebus?
Counting with Venn Diagrams Story Problems Conclusion
Outline
1 Counting with Venn Diagrams
2 Story Problems
3 Conclusion
Counting with Venn Diagrams Story Problems Conclusion
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 asingle area.
Counting with Venn Diagrams Story Problems Conclusion
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 asingle area.
Counting with Venn Diagrams Story Problems Conclusion
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 asingle area.
Counting with Venn Diagrams Story Problems Conclusion
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 asingle area.
Counting with Venn Diagrams Story Problems Conclusion
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 asingle area.
Counting with Venn Diagrams Story Problems Conclusion
Next Time. . .
Venn Diagrams are useful for organizing known information aboutset sizes, but we don’t always know that information.
In the next section we look at the first of several counting rulesused 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 aboutset sizes, but we don’t always know that information.
In the next section we look at the first of several counting rulesused 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