MATH 100 Survey of Mathematical Ideasbanach.millersville.edu/~bob/math100/Surveys/main.pdfMATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Summer

Surveys and Cardinal NumbersMATH 100 Survey of Mathematical Ideas

J. Robert Buchanan

Department of Mathematics

Summer 2020

Surveys

Today we will learn how to determine the cardinal numbers ofsubsets from information obtained by surveys.There are three ways we can approach such problems:

1. Venn diagrams,2. Cardinal number formulas,3. Tables.

Venn Diagrams (1 of 4)

Paula Story is a fan of the music of Paul Simon and ArtGarfunkel. In her collection of 22 compact disks, she has thefollowing:I 5 on which Simon and Garfunkel singI 8 on which Simon singsI 7 on which Garfunkel singsI 12 on which neither Simon nor Garfunkel sings.

1. How many of her compact disks feature only Paul Simon?2. How many of her compact disks feature only Art

Garfunkel?3. How many feature at least one of these two artists?

Venn Diagrams (1 of 4)

Paula Story is a fan of the music of Paul Simon and ArtGarfunkel. In her collection of 22 compact disks, she has thefollowing:I 5 on which Simon and Garfunkel singI 8 on which Simon singsI 7 on which Garfunkel singsI 12 on which neither Simon nor Garfunkel sings.

1. How many of her compact disks feature only Paul Simon?2. How many of her compact disks feature only Art

Garfunkel?3. How many feature at least one of these two artists?

Venn Diagrams (2 of 4)We can start with a Venn diagram with a universal set of all ofPaula Story’s CDs, a set representing CDs on which Simonsings, and a set on which Garfunkel sings.

U

Simon Garfunkel

Now we must determine the cardinality of each subset. First,on how many CDs do both Simon and Garfunkel sing, and onhow many CDs do neither sing?

Venn Diagrams (2 of 4)We can start with a Venn diagram with a universal set of all ofPaula Story’s CDs, a set representing CDs on which Simonsings, and a set on which Garfunkel sings.

U

Simon Garfunkel

Now we must determine the cardinality of each subset. First,on how many CDs do both Simon and Garfunkel sing, and onhow many CDs do neither sing?

Venn Diagrams (3 of 4)

U

Simon Garfunkel5

12

Venn Diagrams (3 of 4)

U

Simon Garfunkel5

12

Keeping in mind that Simon sings on 8 CDs, how many CDsfeature Simon only?

Venn Diagrams (3 of 4)

U

Simon Garfunkel5

3

12

Venn Diagrams (3 of 4)

U

Simon Garfunkel5

3

12

Keeping in mind that Garfunkel sings on 7 CDs, how many CDsfeature Garfunkel only?

Venn Diagrams (4 of 4)

U

Simon Garfunkel5

23

12

Use your i>clicker to answer the following questions.1. How many of her compact disks feature only Paul Simon?

2. How many of her compact disks feature only ArtGarfunkel?

3. How many feature at least one of these two artists?

Venn Diagrams (4 of 4)

U

Simon Garfunkel5

23

12

Use your i>clicker to answer the following questions.1. How many of her compact disks feature only Paul Simon?

2. How many of her compact disks feature only ArtGarfunkel?

3. How many feature at least one of these two artists?

Venn Diagrams (4 of 4)

U

Simon Garfunkel5

23

12

Use your i>clicker to answer the following questions.1. How many of her compact disks feature only Paul Simon?2. How many of her compact disks feature only Art

Garfunkel?

3. How many feature at least one of these two artists?

Venn Diagrams (4 of 4)

U

Simon Garfunkel5

23

12

Use your i>clicker to answer the following questions.1. How many of her compact disks feature only Paul Simon?2. How many of her compact disks feature only Art

Garfunkel?3. How many feature at least one of these two artists?

Cardinal Number Formula

DefinitionFor any two sets A and B,

n(A ∪ B) = n(A) + n(B)− n(A ∩ B).

ExampleFind the value of n(A ∩ B) if n(A) = 20, n(B) = 14, andn(A ∪ B) = 30.

n(A ∪ B) = n(A) + n(B)− n(A ∩ B)

30 = 20 + 14− n(A ∩ B)

n(A ∩ B) = 4

Cardinal Number Formula

DefinitionFor any two sets A and B,

n(A ∪ B) = n(A) + n(B)− n(A ∩ B).

ExampleFind the value of n(A ∩ B) if n(A) = 20, n(B) = 14, andn(A ∪ B) = 30.

n(A ∪ B) = n(A) + n(B)− n(A ∩ B)

30 = 20 + 14− n(A ∩ B)

n(A ∩ B) = 4

Cardinal Number Formula

DefinitionFor any two sets A and B,

n(A ∪ B) = n(A) + n(B)− n(A ∩ B).

ExampleFind the value of n(A ∩ B) if n(A) = 20, n(B) = 14, andn(A ∪ B) = 30.

n(A ∪ B) = n(A) + n(B)− n(A ∩ B)

30 = 20 + 14− n(A ∩ B)

n(A ∩ B) = 4

Example

Carol Britz who sells college textbooks, interviewed freshmenon a college campus to find out the main goals of today’sstudents.

W : the set of students who want to be wealthyF : the set of students who want to raise a familyE : the set of students who want to become experts in

their fieldsThe findings of the survey included the following cardinalities:

n(W ) = 160n(F ) = 140n(E) = 130

n(W ∩ F ) = 95

n(E ∩ F ) = 90n(W ∩ F ∩ E) = 80

n(E ′) = 95n[(W ∪ F ∪ E)′] = 10

Find the total number of students surveyed.

Solution

Number of students surveyed is n(U).

n(U) = n(E ∪ E ′)

= n(E) + n(E ′)− n(E ∩ E ′)

= 130 + 95− n(∅)= 130 + 95− 0= 225

Solution

Number of students surveyed is n(U).

n(U) = n(E ∪ E ′)

= n(E) + n(E ′)− n(E ∩ E ′)

= 130 + 95− n(∅)= 130 + 95− 0= 225

Solution

Number of students surveyed is n(U).

n(U) = n(E ∪ E ′)

= n(E) + n(E ′)− n(E ∩ E ′)

= 130 + 95− n(∅)= 130 + 95− 0= 225

Solution

Number of students surveyed is n(U).

n(U) = n(E ∪ E ′)

= n(E) + n(E ′)− n(E ∩ E ′)

= 130 + 95− n(∅)= 130 + 95− 0= 225

TablesA study of US Army housing trends categorized personnel ascommissioned officers (C), warrant officers (W ), or enlisted(E), and categorized their living facilities as on-base (B), rentedoff-base (R), or owned off-base (O). The survey yielded thefollowing data.

FacilitiesB R O Totals

C 12 29 54 95Personnel W 4 5 6 15

E 374 71 285 730Totals 390 105 345 840

Use your i>clicker to enter the cardinalities of the following sets:

4. W ∩O

5. C ∪ B6. R′ ∪W ′

7. (C ∩ B) ∪ (E ∩O)

8. (C ∪W ) ∩ (B ∪ R)

9. B ∩ (W ∪ R)′

TablesA study of US Army housing trends categorized personnel ascommissioned officers (C), warrant officers (W ), or enlisted(E), and categorized their living facilities as on-base (B), rentedoff-base (R), or owned off-base (O). The survey yielded thefollowing data.

FacilitiesB R O Totals

C 12 29 54 95Personnel W 4 5 6 15

E 374 71 285 730Totals 390 105 345 840

Use your i>clicker to enter the cardinalities of the following sets:

4. W ∩O

5. C ∪ B6. R′ ∪W ′

7. (C ∩ B) ∪ (E ∩O)

8. (C ∪W ) ∩ (B ∪ R)

9. B ∩ (W ∪ R)′

TablesA study of US Army housing trends categorized personnel ascommissioned officers (C), warrant officers (W ), or enlisted(E), and categorized their living facilities as on-base (B), rentedoff-base (R), or owned off-base (O). The survey yielded thefollowing data.

FacilitiesB R O Totals

C 12 29 54 95Personnel W 4 5 6 15

E 374 71 285 730Totals 390 105 345 840

Use your i>clicker to enter the cardinalities of the following sets:

4. W ∩O5. C ∪ B

6. R′ ∪W ′

7. (C ∩ B) ∪ (E ∩O)

8. (C ∪W ) ∩ (B ∪ R)

9. B ∩ (W ∪ R)′

TablesA study of US Army housing trends categorized personnel ascommissioned officers (C), warrant officers (W ), or enlisted(E), and categorized their living facilities as on-base (B), rentedoff-base (R), or owned off-base (O). The survey yielded thefollowing data.

FacilitiesB R O Totals

C 12 29 54 95Personnel W 4 5 6 15

E 374 71 285 730Totals 390 105 345 840

Use your i>clicker to enter the cardinalities of the following sets:

4. W ∩O5. C ∪ B6. R′ ∪W ′

7. (C ∩ B) ∪ (E ∩O)

8. (C ∪W ) ∩ (B ∪ R)

9. B ∩ (W ∪ R)′

TablesA study of US Army housing trends categorized personnel ascommissioned officers (C), warrant officers (W ), or enlisted(E), and categorized their living facilities as on-base (B), rentedoff-base (R), or owned off-base (O). The survey yielded thefollowing data.

FacilitiesB R O Totals

C 12 29 54 95Personnel W 4 5 6 15

E 374 71 285 730Totals 390 105 345 840

Use your i>clicker to enter the cardinalities of the following sets:

4. W ∩O5. C ∪ B6. R′ ∪W ′

7. (C ∩ B) ∪ (E ∩O)

8. (C ∪W ) ∩ (B ∪ R)

9. B ∩ (W ∪ R)′

TablesA study of US Army housing trends categorized personnel ascommissioned officers (C), warrant officers (W ), or enlisted(E), and categorized their living facilities as on-base (B), rentedoff-base (R), or owned off-base (O). The survey yielded thefollowing data.

FacilitiesB R O Totals

C 12 29 54 95Personnel W 4 5 6 15

E 374 71 285 730Totals 390 105 345 840

Use your i>clicker to enter the cardinalities of the following sets:

4. W ∩O5. C ∪ B6. R′ ∪W ′

7. (C ∩ B) ∪ (E ∩O)

8. (C ∪W ) ∩ (B ∪ R)

9. B ∩ (W ∪ R)′

TablesA study of US Army housing trends categorized personnel ascommissioned officers (C), warrant officers (W ), or enlisted(E), and categorized their living facilities as on-base (B), rentedoff-base (R), or owned off-base (O). The survey yielded thefollowing data.

FacilitiesB R O Totals

C 12 29 54 95Personnel W 4 5 6 15

E 374 71 285 730Totals 390 105 345 840

Use your i>clicker to enter the cardinalities of the following sets:

4. W ∩O5. C ∪ B6. R′ ∪W ′

7. (C ∩ B) ∪ (E ∩O)

8. (C ∪W ) ∩ (B ∪ R)

9. B ∩ (W ∪ R)′