# Theory of Positive Integers : Theory of Positive Integers

## Example Questions

### Example Question #1 : Sets

Let

if  is some condition of  such that it can be described as  what is  when ?

Explanation:

First, identify what is given.

and  can be described in the following format

Since  contains the elements in  that are greater than zero,  can be written as follows.

### Example Question #2 : Sets

Let

if  is some condition of  such that it can be described as  what is  when ?

Explanation:

First, identify what is given.

and  can be described in the following format

Since  contains the elements in  that are less than or zero,  can be written as follows.

### Example Question #1 : Logic

over the domain

For all  which  is true?

Explanation:

This question is giving a subset  who lives in the domain  and it is asking for the partition or group of elements that live in both  and .

Looking at what is given,

it is seen that both four and seven live in  and  therefore both these elements will be in the partition of . Another element that also exists in both sets is the empty set.

Thus the final solution is,

### Example Question #2 : Logic

Negate the following statement.

is a prime number.

is not a prime number

is a prime number

is not a prime number

is an odd number

is an even number

is not a prime number

Explanation:

Negating a statement means to take the opposite of it.

To negate a statement completely, each component of the statement needs to be negated.

The given statement,

is a prime number.

contains to components.

Component one:

Component two: "is a prime number"

To negate component one, simply take the compliment of it. In mathematical terms this looks as follows,

To negate component two, simply add a "not" before the phrase "a prime number".

Now, combine these two components back together for the complete negation.

is not a prime number.

### Example Question #3 : Logic

Determine which statement is true giving the following information.

is a prime number  is odd

Explanation:

To determine which statement is true first state what is known.

The first component of this statement is:

is a prime number

This is a true statement since only one and seventeen are factors of seventeen.

The second component of this statement is:

is odd

This statement is false since .

Therefore, the only true statement is the one that uses the "or" operator because only one component is true.

### Example Question #4 : Logic

over the domain

For all  which  is true?

Explanation:

This question is giving a subset  who lives in the domain  and it is asking for the partition or group of elements that live in both  and .

Looking at what is given,

it is seen that only ten lives in  and  therefore both these elements will be in the partition of . Another element that also exists in both sets is the empty set.

Thus the final solution is,

### Example Question #1 : Theory Of Positive Integers

Which of the following is a property of a relation?

Equivalency Property

Partition Property

Non-symmetric Property

All are properties of a relation

Symmetric Property

Symmetric Property

Explanation:

For a relation to exist there must be a non empty set present. If a non empty set is present then there are three relation properties.

These properties are:

I. Reflexive Property

II. Symmetric Property

III. Transitive Property

When all three properties represent a specific set, then that set is known to have an equivalence relation.

### Example Question #2 : Function & Equivalence Relations

What is an equivalency class?

Explanation:

An equivalency class is a definitional term.

Suppose  is a non empty set and  is an equivalency relation on . Then  belonging to  is a set that holds all the elements that live in  that are equivalent to .

In mathematical terms this looks as follows,

### Example Question #2 : Theory Of Positive Integers

Which of the following is a property of a relation?

Equivalency Property

Associative Property

Non-symmetric Property

All are relation properties

Reflexive Property

Reflexive Property

Explanation:

For a relation to exist there must be a non empty set present. If a non empty set is present then there are three relation properties.

These properties are:

I. Reflexive Property

II. Symmetric Property

III. Transitive Property

When all three properties represent a specific set, then that set is known to have an equivalence relation.

### Example Question #3 : Theory Of Positive Integers

Which of the following is a property of a relation?

All are properties of relations.

Non-symmetric Property

Equivalency Property

Partition Property

Transitive Property

Transitive Property

Explanation:

For a relation to exist there must be a non empty set present. If a non empty set is present then there are three relation properties.

These properties are:

I. Reflexive Property

II. Symmetric Property

III. Transitive Property

When all three properties represent a specific set, then that set is known to have an equivalence relation.