MATH 105: Finite Mathematics 6-2: The Number of …math.wallawalla.edu/.../math105/winter06/slides/finite_chapter_6-2.pdf · Counting with Venn Diagrams Story Problems Conclusion

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


Outline

1 Counting with Venn Diagrams

2 Story Problems

3 Conclusion


Outline


2 Story Problems

3 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.





Example






Example


(a) A = {2, 3, 5, a}





Example


(a) A = {2, 3, 5, a} c(A) = 4





Example


(a) A = {2, 3, 5, a} c(A) = 4

(b) B = {3, x , y}





Example


(a) A = {2, 3, 5, a} c(A) = 4

(b) B = {3, x , y} c(B) = 3





Example


(a) A = {2, 3, 5, a} c(A) = 4

(b) B = {3, x , y} c(B) = 3

(c) A ∩ B





Example


(a) A = {2, 3, 5, a} c(A) = 4

(b) B = {3, x , y} c(B) = 3

(c) A ∩ B c(A ∩ B) = 1





Example


(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





Example


(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


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




A B


c(A) + c(B) = 4 + 3 = 7




A B

2


c(A) + c(B) = 4 + 3 = 7




A B

23


c(A) + c(B) = 4 + 3 = 7




A B

23

5


c(A) + c(B) = 4 + 3 = 7




A B

23

5

a


c(A) + c(B) = 4 + 3 = 7




A B

23

5

a


c(A) + c(B) = 4 + 3 = 7




A B

23

5

a

x


c(A) + c(B) = 4 + 3 = 7




A B

23

5

a

x

y


c(A) + c(B) = 4 + 3 = 7




A B

23

5

a

x

y


c(A) + c(B) = 4 + 3 = 7




A B

23

5

a

x

y


c(A) + c(B) = 4 + 3 = 7

c(A ∪ B) = 6




A B

23

5

a

x

y


c(A) + c(B) = 4 + 3 = 7

c(A ∪ B) = 6

c(A ∩ B) = 1


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 Rules

Counting Formula

c(A ∪ B) = c(A) + c(B)− c(A ∩ B)

Example






Counting Rules

Counting Formula

c(A ∪ B) = c(A) + c(B)− c(A ∩ B)

Example






Counting Rules

Counting Formula

c(A ∪ B) = c(A) + c(B)− c(A ∩ B)

Example






Counting Rules

Counting Formula

c(A ∪ B) = c(A) + c(B)− c(A ∩ B)

Example






Outline


2 Story Problems

3 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


Ethnic Foods

Example


G – Greek food

I – Italian food

G ∪ I – 12

G ∪ I – 50-12 = 38

G ∩ I – 20+32-38 = 14

50


Ethnic Foods

Example


G – Greek food

I – Italian food

G ∪ I – 12

G ∪ I – 50-12 = 38

G ∩ I – 20+32-38 = 14

50

G


Ethnic Foods

Example


G – Greek food

I – Italian food

G ∪ I – 12

G ∪ I – 50-12 = 38

G ∩ I – 20+32-38 = 14

50

G I


Ethnic Foods

Example


G – Greek food

I – Italian food

G ∪ I – 12

G ∪ I – 50-12 = 38

G ∩ I – 20+32-38 = 14

50

G I

12


Ethnic Foods

Example


G – Greek food

I – Italian food

G ∪ I – 12

G ∪ I – 50-12 = 38

G ∩ I – 20+32-38 = 14

50

G I

12


Ethnic Foods

Example


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


Ethnic Foods

Example


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


Ethnic Foods

Example


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


Ethnic Foods

Example


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


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



Example



R – Rolling Stone

R ∩W – 37

R ∩W – 41 - 37 = 4

R ∩W – 63 - 4 = 59

R ∪W – 500 - 37 - 4 - 59 = 400

500



Example



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



Example



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



Example



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



Example



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



Example



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



Example



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



Example



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



Example



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


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?


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?


Outline


2 Story Problems

3 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.


Important Concepts







Important Concepts







Important Concepts







Important Concepts







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


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

Documents

MATH 105: Finite Mathematics 6-2: The Number of …math.wallawalla.edu/.../math105/winter06/slides/finite_chapter_6-2.pdf · Counting with Venn Diagrams Story Problems Conclusion