# Set Theory : Axiomatic Set Theory

## Example Questions

### Example Question #1 : Axiomatic Set Theory

Let  denote all straight lines in the Cartesian plane. Does , or both belong to ?

Explanation:

is a set that contains all straight lines that live in the Cartesian plane, this is a vast set. To determine if  , or both belong to , identify if the elements of each set create a straight line, and if so, then that set will be a subset of . In other words, the set will belong to .

Identify the elements in  first.

This statement reads,  contains the  coordinate pairs that live on the line .

Since

is a straight line that lives in the Cartesian plane, that means  belongs to .

Now identify the elements in .

This means that the elements of  are 2, 4, 6, and 9. These are four, individual, values that belong to . They do not create a line in the Cartesian plan and thus  does not belong to .

Therefore, answering the question,  belongs to .

### Example Question #2 : Axiomatic Set Theory

Let  denote all parabolas in the Cartesian plane. Does , or both belong to ?

Explanation:

is a set that contains all parabolas that live in the Cartesian plane, this is a vast set. To determine if  , or both belong to , identify if the elements of each set create a straight line, and if so, then that set will be a subset of . In other words, the set will belong to .

Identify the elements in  first.

This statement reads,  contains the  coordinate pairs that live on the parabola .

Since

is a parabola that lives in the Cartesian plane, that means  belongs to .

Now identify the elements in .

This means that the elements of  are those that live on the straight line . Thus  does not create a parabola in the Cartesian plan therefore  does not belong to .

Therefore, answering the question,  belongs to .

### Example Question #3 : Axiomatic Set Theory

Determine if the following statement is true or false:

In accordance to primitive concepts and notations in set theory, many axioms lead to paradoxes.

False

True

False

Explanation:

First recall the primitive concepts and notations for set theory.

"class", "set", "belongs to"

Now, when deciding what constitutes a primitive concept, it is agreed upon in the math world that four main criteria must be met.

1. Undefined terms and axioms should be few.

2. Axioms should NOT be logically deducible from one another unless clearly expressed.

3. Axioms are able to be proved.

4. Axioms must NOT lead to paradoxes.

Thus, the statement in question is false by criteria four.

### Example Question #4 : Axiomatic Set Theory

Determine if the following statement is true or false:

In accordance with primitive concepts and notations in set theory, many axioms are deducible from other axioms.

False

True

False

Explanation:

First recall the primitive concepts and notations for set theory.

"class", "set", "belongs to"

Now, when deciding what constitutes a primitive concept, it is agreed upon in the math world that four main criteria must be met:

1. Undefined terms and axioms should be few.

2. Axioms should NOT be logically deducible from one another unless clearly expressed.

3. Axioms are able to be proved.

4. Axioms must NOT lead to paradoxes.

Thus, the statement in question is false by criteria two.

### Example Question #5 : Axiomatic Set Theory

Which of the following describes the relationship between the inhabited sets  and  if  ?

and  are disjoint.

is a subset of .

and  intersect.

and  have equal cardinality.

is a subset of .

and  are disjoint.

Explanation:

If the intersection of two sets is equal to the empty set (they do not intersect, i.e. they share no elements), then the two sets are said to be disjoint.

### Example Question #6 : Axiomatic Set Theory

Which of the following represents , where , and  ?

Explanation:

To solve this problem, we first  find the union of  and ; this is the set of all elements in both sets, or   is simply the set of all even natural numbers. The intersection of these two sets is therefore the set of the even numbers present in , which is the set containing the numbers 2, 8, and 10.

### Example Question #7 : Axiomatic Set Theory

Which of the following represents , where , and  ?

Explanation:

Because  and  share no elements, their intersection is , such that  The union of  and any set is the set itself. Therefore, .

### Example Question #8 : Axiomatic Set Theory

For two sets,  and , which of the following correctly expresses  ?