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Set Theory : Set Theory

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Example Questions

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:

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

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

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

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

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

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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Example Question #21 : Set Theory

Determine if the following statement is true or false:

Let A_1 and A_2 be well ordered and order isomorphic sets. If B_1 and B_2 are also order isomorphic sets, then $A_1\times B_1$ and $A_2\times B_2$ are also order isomorphic.

Possible Answers:

True

False

Correct answer:

True

Explanation:

This is a theorem for well ordered sets and the proof is as follows.

First identify what is given in the statement.

1. The sets are onto order isomorphisms

$f:A_1\rightarrow A_2$ and $g:B_1\rightarrow B_2$

2. The goal is to make an onto order isomorphism. Let us call it .

$t:A_1\times B_1 \rightarrow A_2\times B_2$

Thus, can be defined as the function,

$\\t:A_1\times B_1\rightarrow A_2\times B_2 \\t(a,b)=(f(a),g(b))$

To show is a well defined, one-to-one, function on $A_1\times B_1$ since and are one-to-one, the following is performed.

$t(a,b)=t(u,v)\Rightarrow(f(a),g(b))=(f(u),g(v))$

$\\\Rightarrow f(a)=f(u), g(b)=g(v) \\\Rightarrow a=u, b=v \\\Rightarrow (a,b)=(u,v)$

Thus, is one-to-one.

Now to prove in onto $A_2\times B_2$ if $(a,b)\ \epsilon\ A_2\times B_2$

a=f(u), b=g(v)

for some

$u\ \epsilon\ A_1, v\ \epsilon\ B_1$

thus

t(u,v)=(a,b)

proving is onto $A_2 \times B_2$ .

Lastly prove ordering.

$(a_1,b_1)\leq_{A\times B}(a_2,b_2)\Leftrightarrow \left\{\begin{matrix} a_1<_aa_2\\ a_1=a_2, b+1\leq_{r}b_2 \end{matrix}\right.$

$\Leftrightarrow\left\{\begin{matrix} f(a_1)<_{A}f(a_2)\\ f(a_1)=f(a_2), g(b_1\leq_{r}g(b_2)) \end{matrix}\right.$

$\Leftrightarrow\left\{\begin{matrix} (f(a_1),g(b_1))<_{A\times B}(f(a_2),g(b_2))\\ (f(a_1),g(b_1))\leq(f(a_2), g(b_2)) \end{matrix}\right.$

Thus proving is isomorphic. Therefore the statement is true.

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Example Question #1 : Cardinal Numbers

Given the set , calculate the cardinality.

$A=\begin{Bmatrix} 3,7,5,4,9,11,24 \end{Bmatrix}$

Possible Answers:

|A|=21

|A|=8

|A|=7

|A|=9

|A|=11

Correct answer:

|A|=7

Explanation:

Given a set , the cardinality is the number of elements in . This is also written as |W| .

Looking at this particular problem,

$A=\begin{Bmatrix} 3,7,5,4,9,11,24 \end{Bmatrix}$

Therefore the number of elements in a is,

|A|=7

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Example Question #2 : Cardinal Numbers

Given the set , calculate the cardinality.

$A=\begin{Bmatrix} a,a,b,c,d,e,e \end{Bmatrix}$

Possible Answers:

$|A|=\frac{a+e}{7}$

|A|=c

|A|=a

|A|=7

|A|=5

Correct answer:

|A|=5

Explanation:

Given a set , the cardinality is the number of elements in . This is also written as |W| .

Looking at this particular problem,

$A=\begin{Bmatrix} a,a,b,c,d,e,e \end{Bmatrix}$

First, rewrite the set to depict a more accurate description of the space.

$A=\begin{Bmatrix} a,b,c,d,e \end{Bmatrix}$

Therefore, counting up the number of elements, results in the following cardinality,

|A|=5

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Set Theory : Set Theory

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 #21 : Set Theory

Example Question #1 : Cardinal Numbers

Example Question #2 : Cardinal Numbers

All Set Theory Resources

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