MATH 105: Finite Mathematics 6-2: The Number of Elements in a Set - - PowerPoint PPT Presentation

Counting with Venn Diagrams Story Problems Conclusion

MATH 105: Finite Mathematics 6-2: 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} are arranged 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} 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 3 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} are arranged in a Venn Diagram. A B 2 3 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} are arranged in a Venn Diagram. A B 2 3 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} are arranged in a Venn Diagram. A B 2 3 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} are arranged in a Venn Diagram. A B 2 3 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} are arranged in a Venn Diagram. A B 2 3 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} are arranged in a Venn Diagram. A B 2 3 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} are arranged in a Venn Diagram. A B 2 3 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. 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 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. Twenty

• f 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

• f 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

• f 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 G

Counting with Venn Diagrams Story Problems Conclusion

Ethnic Foods

Example Fifty people are interviewed about their food preferences. Twenty

• f 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 G I

Counting with Venn Diagrams Story Problems Conclusion

Ethnic Foods

Example Fifty people are interviewed about their food preferences. Twenty

• f 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 G I 12

Counting with Venn Diagrams Story Problems Conclusion

Ethnic Foods

Example Fifty people are interviewed about their food preferences. Twenty

• f 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 G I 12

Counting with Venn Diagrams Story Problems Conclusion

Ethnic Foods

Example Fifty people are interviewed about their food preferences. Twenty

• f 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 G I 12 14

Counting with Venn Diagrams Story Problems Conclusion

Ethnic Foods

Example Fifty people are interviewed about their food preferences. Twenty

• f 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 G I 12 14 6

Counting with Venn Diagrams Story Problems Conclusion

Ethnic Foods

Example Fifty people are interviewed about their food preferences. Twenty

• f 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 G I 12 14 18 6

Counting with Venn Diagrams Story Problems Conclusion

Ethnic Foods

Example Fifty people are interviewed about their food preferences. Twenty

• f 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 G I 12 14 18 6

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

• f 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

• f 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

• f 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

• f 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

• f 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 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

• f 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 37 4

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

• f 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 37 4 59

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

• f 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 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

• f 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 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

• f 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 37 4 59 400

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 Conclusion

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 drive a car at least part of the time 20 use the bus at least part of the time 48 ride a bicycle at least part of the time 38 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?

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 a

single 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 a

single 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 a

single 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 a

single 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 a

single area.

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