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Set Theory : Set Theory

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Example Question #3 : Relations, Functions And Cartesian Product

Determine if the following statement is true or false:

If be the set defined as,

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

then $x\subseteq A$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

Given is the set defined as,

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

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

Looking at the elements in is is seen that all three elements in fact do contain therefore $x\subseteq A$ .

Thus, the answer is True.

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Example Question #1 : Relations, Functions And Cartesian Product

For a bijective function from set to set defined by $f:A \rightarrow B$ , which of the following does NOT need to be true?

Possible Answers:

Every element of must map to one or more elements of .

No element of may map to multiple elements of .

Every element of must map to one or more elements of .

All of the conditions here must be true.

Correct answer:

All of the conditions here must be true.

Explanation:

For a bijective function, every element in set must map to exactly one element of set , so that every element in set has exactly one corresponding element in set . All of the conditions presented must be true in order to satisfy this definition.

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Example Question #2 : Relations, Functions And Cartesian Product

For an injective function from set to set defined by $f:A \rightarrow B$ , which of the following does NOT need to be true?

Possible Answers:

No element of may map to multiple elements of .

Every element of must map to one or more elements of .

All of the conditions here must be true.

Correct answer:

Every element of must map to one or more elements of .

Explanation:

For an injective function, every element of must map to exactly one element of . Additionally, every element in must map to a different element in so that no element in has multiple pairings to elements in . It is not necessary for all elements of to be connected to an element in .

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Example Question #1 : Relations, Functions And Cartesian Product

For a surjective function from set to set defined by $f:A \rightarrow B$ , which of the following does NOT need to be true?

Possible Answers:

All of the conditions here must be true.

No element of may map to multiple elements of .

Every element of must map to one or more elements of .

No element of may map to multiple elements of .

Every element of must map to one or more elements of .

Correct answer:

No element of may map to multiple elements of .

Explanation:

For a surjective function, every element of must map to exactly one element of . Additionally, all elements of to be paired with an element in , even if one or more elements of is connected to multiple elements in .

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Example Question #3 : Relations, Functions And Cartesian Product

For which of the following pairs is the cardinality of the two sets equal?

Possible Answers:

$\left \{ x|x\in \mathbb{N}\right \},\left \{ 2y|y\in \mathbb{N}\right \}$

and , where there exists an injective, non-surjective function $f:A\rightarrow B$ .

$\mathbb{R},\mathbb{N}$

$\left \{ 1,2,3\right \},\left \{ 0,1,2,3\right \}$

$\left \{ 0\right \},\varnothing$

Correct answer:

$\left \{ x|x\in \mathbb{N}\right \},\left \{ 2y|y\in \mathbb{N}\right \}$

Explanation:

The cardinality of $\mathbb{R}$ ( $\mathfrak{c}$ ) is greater than that of $\mathbb{N}$ ( $\aleph_{0}$ ,) as established by Cantor's first uncountability proof, which demonstrates that $\mathfrak{c}=2^{\aleph_{0}}>\aleph_{0}$ . The cardinality of the empty set is 0, while the cardinality of $\left \{ 0\right \}$ is 1. $\left | \left \{ 1,2,3\right \} \right |=3$ , while $\left | \left \{ 1,2,3,4\right \} \right |=4$ . For sets and , where there exists an injective, non-surjective function $f:A\rightarrow B$ , must have more elements than , otherwise the function would be bijective (also called injective-surjective). Finally, for $\left \{ x|x\in \mathbb{N}\right \},\left \{ 2y|y\in \mathbb{N}\right \}$ , the cardinality of both sets is equal to the cardinality of $\mathbb{N}$ .

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Example Question #11 : Set Theory

What type of function is $f:\mathbb{Z}\rightarrow \mathbb{N}$ where $x \mapsto \left |x \right |$ ?

Possible Answers:

None of these answers is correct.

Bijective

Injective

Surjective

Correct answer:

Surjective

Explanation:

Because multiple elements of $\mathbb{Z}$ can map to a single element of $\mathbb{N}$ (e.g. -2 and 2 map to 2), this function is surjective.

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Example Question #1 : Relations, Functions And Cartesian Product

What type of function is $f:\mathbb{N}\rightarrow \mathbb{Z}$ where $x \mapsto x^2$ ?

Possible Answers:

Surjective

None of these answers is correct.

Bijective

Injective

Correct answer:

Injective

Explanation:

Because every element of $\mathbb{N}$ maps to a single element of $\mathbb{Z}$ , but there are many elements of $\mathbb{Z}$ that do not pair with any element of $\mathbb{N}$ , this function is injective.

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Example Question #12 : Set Theory

What type of function is $f:\mathbb{Z}\rightarrow \mathbb{R}$ where $x \mapsto \sqrt{x}$ ?

Possible Answers:

Injective

Bijective

Surjective

None of these answers is correct.

Correct answer:

None of these answers is correct.

Explanation:

Because this operation is not defined for the negative elements of $\mathbb{Z}$ , it is either considered to be a partial function or not a function at all! Each element in the sub-domain of $\mathbb{Z}$ for which the operation is defined maps to exactly 1 element in $\mathbb{R}$ ; however, many elements of $\mathbb{R}$ are not paired to any element in this sub-domain of $\mathbb{Z}$ , thus the function could be considered a partial injective function.

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Example Question #11 : Set Theory

Are the sets and equal?

$\\T=\begin{Bmatrix} x|x\ \epsilon\ \mathbb{Z} \end{Bmatrix} \\S=\begin{Bmatrix} 2n|n\ \epsilon\ \mathbb{Z} \end{Bmatrix}$

Possible Answers:

$T\neq S$

T= S, x<0

$T= S, x\geq0$

T= S, x>0

T= S

Correct answer:

$T\neq S$

Explanation:

To determine if two sets are equal to each other it must be proven that each set contains the same elements.

Recall the following terminology,

$\\\mathbb{Z}=\text{Integers}{\begin{Bmatrix} ...,-2,-1,0,1,2,... \end{Bmatrix}} \\\mathbb{R}=\text{Reals} \\\mathbb{N}=\text{Naturals} {\begin{Bmatrix} 0,1,2,... \end{Bmatrix}}$

Now, identify the elements in each set.

$\\T=\begin{Bmatrix} x|x\ \epsilon\ \mathbb{Z} \end{Bmatrix}$

This means all integers are elements of .

Let x=3 ,

$\\S=\begin{Bmatrix} 2n|n\ \epsilon\ \mathbb{Z} \end{Bmatrix}$

is the value for all integers.

Therefore if n=3 the element in is,

2(n)=2(3)=6

This means that all elements in will be divisible by two and thus be an even number therefore, 3 will never be an element in .

Thus, it is concluded that

$T\neq S$

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Example Question #2 : Natural Numbers, Integers, And Real Numbers

Are the sets and equal?

$\\T=\begin{Bmatrix} x|x\ \epsilon\ \mathbb{Z} \end{Bmatrix} \\S=\begin{Bmatrix} 1+n|n\ \epsilon\ \mathbb{Z} \end{Bmatrix}$

Possible Answers:

$T\neq S$

T= S, x<0

$T\neq S, x\geq0$

T= S, x>0

T=S

Correct answer:

T=S

Explanation:

To determine if two sets are equal to each other it must be proven that each set contains the same elements.

Recall the following terminology,

$\\\mathbb{Z}=\text{Integers}{\begin{Bmatrix} ...,-2,-1,0,1,2,... \end{Bmatrix}} \\\mathbb{R}=\text{Reals} \\\mathbb{N}=\text{Naturals} {\begin{Bmatrix} 0,1,2,... \end{Bmatrix}}$

Now, identify the elements in each set.

$\\T=\begin{Bmatrix} x|x\ \epsilon\ \mathbb{Z} \end{Bmatrix}$

This means all integers are elements of .

Let x=3 ,

$\\S=\begin{Bmatrix} 1+n|n\ \epsilon\ \mathbb{Z} \end{Bmatrix}$

is the value 1+n for all integers.

Therefore if n=3 the element in is,

1+n=1+3=4

Since four also belongs to $\mathbb{Z}$ this means that all elements in will be the same as those in .

Thus, it is concluded that

T=S

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Set Theory : Set Theory

Study concepts, example questions & explanations for Set Theory

All Set Theory Resources

Example Questions

Example Question #3 : Relations, Functions And Cartesian Product

Example Question #1 : Relations, Functions And Cartesian Product

Example Question #2 : Relations, Functions And Cartesian Product

Example Question #1 : Relations, Functions And Cartesian Product

Example Question #3 : Relations, Functions And Cartesian Product

Example Question #11 : Set Theory

Example Question #1 : Relations, Functions And Cartesian Product

Example Question #12 : Set Theory

Example Question #11 : Set Theory

Example Question #2 : Natural Numbers, Integers, And Real Numbers

All Set Theory Resources

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