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Set Theory : Natural Numbers, Integers, and Real Numbers

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Example Question #1 : Natural Numbers, Integers, And Real Numbers

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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Example Question #1 : Natural Numbers, Integers, And Real Numbers

What is the correct expression of the relationships between the sets comprised of natural numbers, real numbers, and integers?

Possible Answers:

$\mathbb{N}=\mathbb{Z}\subseteq \mathbb{R}$

$\mathbb{N}\subseteq\mathbb{Z}\subseteq \mathbb{R}$

$\mathbb{N}\subseteq\mathbb{Z}=\mathbb{R}$

$\mathbb{R}\subseteq\mathbb{Z}\subseteq \mathbb{N}$

$\mathbb{Z}\subseteq\mathbb{N}\subseteq \mathbb{R}$

Correct answer:

$\mathbb{N}\subseteq\mathbb{Z}\subseteq \mathbb{R}$

Explanation:

The natural numbers are defined as $\mathbb{N}=\left \{ 0,1,2,3,...\right \}$ , the integers are defined as $\mathbb{Z}=\left \{ ...-3,-2,-1,0,1,2,3,...\right \}$ , and the real numbers ( $\mathbb{R}$ ) are defined as the set of all non-complex numbers. As such, $\mathbb{N}$ is a subset of $\mathbb{Z}$ , and $\mathbb{Z}$ is a subset of $\mathbb{R}$ .

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Set Theory : Natural Numbers, Integers, and Real Numbers

Study concepts, example questions & explanations for Set Theory

All Set Theory Resources

Example Questions

Example Question #1 : Natural Numbers, Integers, And Real Numbers

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

Example Question #1 : Natural Numbers, Integers, And Real Numbers

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