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Flashcards: Natural Numbers, Integers, and Real Numbers

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

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

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

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

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

T= S $\displaystyle T= S$

$\displaystyle T= S, x>0$

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Flashcards: Natural Numbers, Integers, and Real Numbers

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

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

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

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

T= S $\displaystyle T= S$

$\displaystyle T= S, x>0$

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Flashcards: Natural Numbers, Integers, and Real Numbers

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

Your answer:

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

Your answer:

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

Your answer:

T= S $\displaystyle T= S$

Your answer:

$\displaystyle T= S, x>0$

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

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

Flashcards: Natural Numbers, Integers, and Real Numbers

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

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

$\displaystyle T= S, x>0$

T=S $\displaystyle T=S$

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

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

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

Report an Error Copyright Notice

Flashcards: Natural Numbers, Integers, and Real Numbers

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

Your answer:

$\displaystyle T= S, x>0$

Your answer:

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

Your answer:

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

Your answer:

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

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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