Set Theory - Axiomatic Set Theory

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

$\S\begin{Bmatrix} (x,y): y=2x^2\end{Bmatrix} \T\begin{Bmatrix} (x,y):y=-x+1 \end{Bmatrix}$

$S\subseteq A$

$S,T \subseteq A$

$T\subseteq A$

$\text{Neither belong to A}$

$A\subseteq T$

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.

$\S\begin{Bmatrix} (x,y): y=2x^2 \end{Bmatrix}$

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 .

$T\begin{Bmatrix} (x,y):y=-x+1 \end{Bmatrix}$

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 .

$S\subseteq A$

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

$\S\begin{Bmatrix} (x,y): y=2x+7 \end{Bmatrix} \T\begin{Bmatrix} 2,4,6,9 \end{Bmatrix}$

$S\subseteq A$

$S,T \subseteq A$

$T\subseteq A$

$\text{Neither belong to A}$

$A\subseteq T$

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.

$\S\begin{Bmatrix} (x,y): y=2x+7 \end{Bmatrix}$

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 .

$T\begin{Bmatrix} 2,4,6,9 \end{Bmatrix}$

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

Therefore, answering the question, belongs to .

$S\subseteq A$

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

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.

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

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.

Which of the following describes the relationship between the inhabited sets and if $X \cap Y = \varnothing$ ?

and are disjoint.

and intersect.

and have equal cardinality.

is a subset of .

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.

Which of the following represents $\left ( A\cup B\right )\cap C$ , where $A=\left { 1,2,5\right }$ , $B=\left { 3,8,10\right }$ , and $C=\left { 2n|n\in \mathbb{N}\right }$ ?

$\left { 2,8,10\right }$

$\left { 1,3,5\right }$

$\varnothing$

$\left { 1,2,3,5,8,10\right }$

$\left { 1,2,3\right }$

Explanation

To solve this problem, we first find the union of and ; this is the set of all elements in both sets, or $A\cup B=\left { 1,2,3,5,8,10\right }$ . 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 $A\cup B$ , which is the set containing the numbers 2, 8, and 10.

Which of the following represents $\left ( A\cap B\right )\cup C$ , where $A=\left { 1,2,5\right }$ , $B=\left { 3,8,10\right }$ , and $C=\left { 2,4,6,8\right }$ ?

$\varnothing$

$\left { 2,8\right }$

Explanation

Because and share no elements, their intersection is $\varnothing$ , such that $\left ( A\cap B\right )\cup C=\varnothing \cup C$ The union of $\varnothing$ and any set is the set itself. Therefore, $\varnothing \cup C = C$ .

For two sets, and , which of the following correctly expresses $\left | A\right | + \left | B\right |$ ?

$\left | A\cap B\right |+\left | A\cup B\right |$

$\left | A\cap B\right |$

$\left | A\cup B\right |$

$\left | A\cap B\right |-\left | A\cup B\right |$

$\left | A\cup B\right |-\left | A\cap B\right |$

Explanation

The sum of the cardinalities of two sets is equal to the sum of the cardinalities of their intersection and union. For instance, if $A=\left { 1,2,3\right }$ and $B=\left { 3,4,5\right }$ :