Set Theory - Natural Numbers, Integers, and Real Numbers

Are the sets and equal?

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

$T= S, x\geq0$

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 ,

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

is the value for all integers.

Therefore if the element in is,

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

Are the sets and equal?

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

$T eq S, x\geq0$

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 ,

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

is the value for all integers.

Therefore if the element in is,

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

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

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

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

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

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

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

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 $\mathbb{R}$ .

Natural Numbers, Integers, and Real Numbers

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