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Set Theory : Natural Numbers, Integers, and Real Numbers

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Example Questions

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

Are the sets $\displaystyle T$ and $\displaystyle S$ equal?

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

Possible Answers:

$T\neq S$ $\displaystyle T\neq S$

T= S, x<0 $\displaystyle T= S, x< 0$

$T= S, x\geq0$ $\displaystyle T= S, x\geq0$

T= S $\displaystyle T= S$

$\displaystyle T= S, x>0$

Correct answer:

$T\neq S$ $\displaystyle T\neq S$

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,

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

Now, identify the elements in each set.

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

This means all integers are elements of $\displaystyle T$.

Let x=3 $\displaystyle x=3$,

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

$\displaystyle S$ is the value $\displaystyle 2n$ for all integers.

Therefore if n=3 $\displaystyle n=3$ the element in $\displaystyle S$ is,

$\displaystyle 2(n)=2(3)=6$

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

Thus, it is concluded that

$T\neq S$ $\displaystyle T\neq S$

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Example Question #11 : Set Theory

Are the sets $\displaystyle T$ and $\displaystyle S$ equal?

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

Possible Answers:

$T\neq S$ $\displaystyle T\neq S$

$\displaystyle T= S, x>0$

T= S, x<0 $\displaystyle T= S, x< 0$

T=S $\displaystyle T=S$

$T\neq S, x\geq0$ $\displaystyle T\neq S, x\geq0$

Correct answer:

T=S $\displaystyle T=S$

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,

Now, identify the elements in each set.

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

This means all integers are elements of $\displaystyle T$.

Let x=3 $\displaystyle x=3$,

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

$\displaystyle S$ is the value 1+n $\displaystyle 1+n$ for all integers.

Therefore if n=3 $\displaystyle n=3$ the element in $\displaystyle S$ is,

$\displaystyle 1+n=1+3=4$

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

Thus, it is concluded that

T=S $\displaystyle T=S$

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Example Question #3 : 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:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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Set Theory : Natural Numbers, Integers, and Real Numbers

Study concepts, example questions & explanations for Set Theory

All Set Theory Resources

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

Example Question #11 : Set Theory

Example Question #3 : Natural Numbers, Integers, And Real Numbers

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