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Set Theory : Set Theory

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Example Question #1 : Axiomatic Set Theory

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

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

Possible Answers:

$T\subseteq A$

$\text{Neither belong to A}$

$S\subseteq A$

$A\subseteq T$

$S,T \subseteq A$

Correct answer:

$S\subseteq A$

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 (x,y) coordinate pairs that live on the line y=2x+7 .

Since

y=2x+7 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 .

$S\subseteq A$

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Example Question #2 : Axiomatic Set Theory

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

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

Possible Answers:

$A\subseteq T$

$\text{Neither belong to A}$

$S\subseteq A$

$T\subseteq A$

$S,T \subseteq A$

Correct answer:

$S\subseteq A$

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 (x,y) coordinate pairs that live on the parabola y=2x^2 .

Since

y=2x^2 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 y=-x+1 . Thus does not create a parabola in the Cartesian plan therefore does not belong to .

Therefore, answering the question, belongs to .

$S\subseteq A$

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Example Question #1 : 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.

Possible Answers:

False

True

Correct answer:

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.

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Example Question #2 : 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.

Possible Answers:

False

True

Correct answer:

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.

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Example Question #1 : Axiomatic Set Theory

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

Possible Answers:

is a subset of .

and are disjoint.

and intersect.

is a subset of .

and have equal cardinality.

Correct answer:

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.

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Example Question #1 : Axiomatic Set Theory

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 $C=\left \{ 2n|n\in \mathbb{N}\right \}$ ?

Possible Answers:

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

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

$\varnothing$

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

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

Correct answer:

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

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.

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Example Question #1 : Axiomatic Set Theory

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 $C=\left \{ 2,4,6,8\right \}$ ?

Possible Answers:

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

$\varnothing$

Correct answer:

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, $\varnothing \cup C = C$ .

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Example Question #1 : Axiomatic Set Theory

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

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 $B=\left \{ 3,4,5\right \}$ :

$\left | A\right |=3$

$\left | B\right |=3$

$\therefore \left | A\right | + \left | B\right |=6$

and,

$\left | A\cup B\right |=\left | \left \{ 1,2,3,4,5\right \} \right |=5$

$\left | A\cap B\right |=\left | \left \{ 3\right \} \right |=1$

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Example Question #1 : Relations, Functions And Cartesian Product

Determine if the following statement is true or false:

If $D\subseteq E$ and $F\subseteq G$ then $D\times F\subseteq E\times G$ .

Possible Answers:

True

False

Correct answer:

True

Explanation:

Assuming , , , and are classes where $D\subseteq E$ and $F\subseteq G$ .

Then by definition,

the product of and results in the ordered pair (d,f) where is an element is the set and is an element in the set or in mathematical terms,

$D\times F=\begin{Bmatrix} (d,f):d\ \epsilon\ D \ and\ f\ \epsilon\ F \end{Bmatrix}$

and likewise

$E\times G=\begin{Bmatrix} (e,g):e\ \epsilon\ E \ and\ g\ \epsilon\ G \end{Bmatrix}$

Now,

$\\(d,f)\ \epsilon\ D\timesF \\\Rightarrow d\ \epsilon\ D\ and\ f\ \epsilon\ F \\\Rightarrow d\ \epsilon\ E\ and\ f\ \epsilon\ G \\\text{Because } D\subseteq E\ and\ F\subseteq G \\\Rightarrow (d,f)\ \epsilon\ E\timesG$

therefore,

$D\times F\subseteq E \times G$ .

Thus by definition, this statement is true.

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Example Question #2 : Relations, Functions And Cartesian Product

Determine if the following statement is true or false:

If be the set defined as,

$A=\begin{Bmatrix} x,\begin{Bmatrix} x,y,z \end{Bmatrix}, \begin{Bmatrix} y,z \end{Bmatrix} \end{Bmatrix}$

then

$x\subseteq A$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

Given is the set defined as,

$A=\begin{Bmatrix} x,\begin{Bmatrix} x,y,z \end{Bmatrix}, \begin{Bmatrix} y,z \end{Bmatrix} \end{Bmatrix}$

to state that $x\subseteq A$ , every element in must contain .

Looking at the elements in is is seen that the first two elements in fact do contain however, the third element in the set, $\begin{Bmatrix} y,z \end{Bmatrix}$ does not contain therefore $x\nsubseteq A$ .

Therefore, the answer is False.

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Set Theory : Set Theory

Study concepts, example questions & explanations for Set Theory

All Set Theory Resources

Example Questions

Example Question #1 : Axiomatic Set Theory

Example Question #2 : Axiomatic Set Theory

Example Question #1 : Axiomatic Set Theory

Example Question #2 : Axiomatic Set Theory

Example Question #1 : Axiomatic Set Theory

Example Question #1 : Axiomatic Set Theory

Example Question #1 : Axiomatic Set Theory

Example Question #1 : Axiomatic Set Theory

Example Question #1 : Relations, Functions And Cartesian Product

Example Question #2 : Relations, Functions And Cartesian Product

All Set Theory Resources

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