Prove It by Velleman, Section 1.4, Problem 4

Problem

Use Venn diagrams to verify that the following identities:

1. 
2. 

Solution

1.

Set A.

This is .

Subtracting from A we get .

2.

Set A.

The combination would be the following:

The commonality between the two is:

Prove It by Velleman, Section 1.4, Problem 3

Problem

Verify that the Venn diagram for and both look like Figure 5, as stated in this section.

Solution

This is .

This is .

Subtracting the bit in the middle from the whole is is naturally this.

This is .

This is .

The combination of the two is naturally this, which is the same.

Prove It by Velleman, Section 1.4, Problem 2

Problem

Let A = {United States, Germany, China, Australia}, B = {Germany France, India, Brazil}, and . List the elements of the following sets. Are any of the sets below disjoint from any of the others? Are any of the sets below subsets of any others?

1. 
2. 
3. 

Solution

1.

{United States, Germany, China, Australia, France, India, Brazil}

2.

{Germany}

3.

{France}

 

Sets 2 and 3 are disjoint. Sets 2 and 2 are subsets of 1.

 

 

Prove It by Velleman, Section 1.4, Problem 1

Problem

Let , , and . List the elements of the following sets. Are any of the sets below disjoint from any of the others? Are any of the sets below subsets of any others?

1. 
2. 
3. 

Solution

1.

{3, 12}

2.

{1, 12, 20, 35}

3.

{1, 3, 12, 20, 35}

 

None of these sets are disjoint. Number 1 and 2 are subsets of 3.

 

 

Prove It by Velleman, Section 1.3, Problem 8

Problem

What are the truth sets of the following statements? List a few elements of the truth set if you can.

1. x is a real number and .
2. x is a real number and .
3. x is a real number and

Solution

1.

x is a real number and .

{-1, -3}

2.

x is a real number and .

3.

x is a real number and

{-4, -3, -2, -1, 0, 1, 2, 3, 4}

 

 

Prove It by Velleman, Section 1.3, Problem 7

Problem

What are the truth sets of the following statements? List a few elements of the truth set if you can.

1. Elizabeth Taylor was once married to x.
2. x is a logical connective studied in Section 1.1.
3. x is the author of this book.

Solution

1.

Elizabeth Taylor was once married to x.

{Conrad Hilton, Michael Wilding, Mike Todd, Eddie Fisher, Richrad Burton, John Warner, Larry Fortensky}

2.

 

x is a logical connective studied in Section 1.1.

3.

x is the author of this book.

{Daniel J. Velleman}

 

 

Prove It by Velleman, Section 1.3, Problem 6

Problem

Simplify the following statements. Which variables are free and which are bound? If the statement has no free variables, say whether it is true or false.

1. 
2. . (It might make this statement easier to read if we let ; using this notation, we could rewrite the statement as .)
3. . (Using the same notation as in part 2, we could write this as

Solution

1.

w and c are free variables.

 

2.

There are no free variables.

It is true.

 

3.

There are no free variables.

It is false.

 

 

Prove It by Velleman, Section 1.3, Problem 5

Problem

Simplify the following statements. Which variables are free and which are bound? If the statement has no free variables, say whether it is true or false.

1. 
2. 
3. 

Solution

1.

So we have

There are no free variables.

It is true.

2.

 

 

There are no free variables.

It is false.

3.

c is a free variable.

 

Prove It by Velleman, Section 1.3, Problem 4

Problem

Write definitions using elementhood tests for the following sets:

1. {1, 4, 9, 16, 25 36, 49, …}
2. {1, 2, 4, 8, 16, 32, 64, …}
3. {10, 11, 12, 13, 14, 15, 16, 17, 18, 19}

Solution

1.

{1, 4, 9, 16, 25 36, 49, …}

2.

{1, 2, 4, 8, 16, 32, 64, …}

3.

{10, 11, 12, 13, 14, 15, 16, 17, 18, 19}

 

Prove It by Velleman, Section 1.3, Problem 3

Problem

Write definitions using elementhood tests for the following sets:

1. [Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto]
2. [Brown, Columbia, Cornell, Dartmouth, Harvard, Princeton, University of Pennsylvania, Yale]
3. [Alabama, Alaska, Arizona, …, Wisconsin, Wyoming]
4. [Alberta, British Columbia, Manitoba, New Brunswick, Newfoundland and Labrador, Northwest Territories, Nova Scotia, Nunavut, Ontario, Prince Edward Island, Quebec, Saskatchewan, Yukon]

Solution

1.

[Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto]

2.

[Brown, Columbia, Cornell, Dartmouth, Harvard, Princeton, University of Pennsylvania, Yale]

3.

[Alabama, Alaska, Arizona, …, Wisconsin, Wyoming]

4.

[Alberta, British Columbia, Manitoba, New Brunswick, Newfoundland and Labrador, Northwest Territories, Nova Scotia, Nunavut, Ontario, Prince Edward Island, Quebec, Saskatchewan, Yukon]