Chapter 5.1, Problem 3E

a.

To determine

Explain the given set is finite.

Answer to Problem 3E

The set ${\mathrm{N}}_{480}$ is finite.

Explanation of Solution

Given information:

  $\mathrm{A}=\left\{1,2,3,\mathrm{......},479,480\right\}$

Calculation:

Consider, definition and result for finite set,

Definition: the set $\mathrm{S}$ is infinite if $\mathrm{S}=\mathrm{\varphi }$ or $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some $\mathrm{k}\in \mathrm{N}$ . Further if $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ , then $\mathrm{A}$ has cardinal number $\mathrm{k}$ , and we write $\mathrm{S}=\mathrm{k}$ .

Result:

If a set $\mathrm{A}$ has finite cardinality $\mathrm{m}$ , then $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{m}}$

The given set is as,

  $\mathrm{A}=\left\{1,2,3,\mathrm{......},479,480\right\}$

So cardinality of set $\mathrm{A}$ is $480$ . It implies that $\mathrm{A}\approx {\mathrm{N}}_{480}$ .

Hence, set $\mathrm{A}$ is finite.

b.

To determine

Explain the given set is finite.

Answer to Problem 3E

Set $\mathrm{A}$ is finite.

Explanation of Solution

Given information:

  $\mathrm{A}=\left[3,5\right]-\left(3,5\right)$

Calculation:

Consider, definition and result for finite set,

Definition: the set $\mathrm{S}$ is infinite if $\mathrm{S}=\mathrm{\varphi }$ or $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some $\mathrm{k}\in \mathrm{N}$ . Further if $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ , then $\mathrm{A}$ has cardinal number $\mathrm{k}$ , and we write $\mathrm{S}=\mathrm{k}$ .

Result:

If a set $\mathrm{A}$ has finite cardinality $\mathrm{m}$ , then $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{m}}$

The given set is as,

  $\begin{array}{l}\mathrm{A}=\left[3,5\right]-\left(3,5\right)\\ =\left\{3,5\right\}\end{array}$

So the cardinality of set $\mathrm{A}\mathrm{i}\mathrm{s}2.$ it implies that $\mathrm{A}\approx {\mathrm{N}}_2.$

Hence, set $\mathrm{A}$ is finite.

c.

To determine

Explain the given set is finite.

Answer to Problem 3E

Set ${\mathrm{N}}_4$ is finite.

Explanation of Solution

Given information:

  $\mathrm{A}=\left\{3,5\right\}\times \left\{1,4\right\}$

Calculation:

Consider, definition and result for finite set,

Definition: the set $\mathrm{S}$ is infinite if $\mathrm{S}=\mathrm{\varphi }$ or $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some $\mathrm{k}\in \mathrm{N}$ . Further if $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ , then $\mathrm{A}$ has cardinal number $\mathrm{k}$ , and we write $\mathrm{S}=\mathrm{k}$ .

Result:

If a set $\mathrm{A}$ has finite cardinality $\mathrm{m}$ , then $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{m}}$

The given set is as,

  $\begin{array}{l}\mathrm{A}=\left\{3,5\right\}\times \left\{1,4\right\}\\ =\left(3,1\right),\left(3,4\right),\left(5,1\right),\left(5,4\right)\end{array}$

So the cardinality of set $\mathrm{A}$ is $4.$ it implies that $\mathrm{A}\approx {\mathrm{N}}_4.$

Hence, set $\mathrm{A}$ is finite.

d.

To determine

Explain the given set is finite.

Answer to Problem 3E

Set $\mathrm{A}$ is finite.

Explanation of Solution

Given information:

  $\mathrm{A}=\left\{3^{-1},3^{-2},3^{-3},\mathrm{...},3^{-479},3^{-480}\right\}$

Calculation:

Consider, definition and result for finite set,

Definition: the set $\mathrm{S}$ is infinite if $\mathrm{S}=\mathrm{\varphi }$ or $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some $\mathrm{k}\in \mathrm{N}$ . Further if $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ , then $\mathrm{A}$ has cardinal number $\mathrm{k}$ , and we write $\mathrm{S}=\mathrm{k}$ .

Result:

If a set $\mathrm{A}$ has finite cardinality $\mathrm{m}$ , then $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{m}}$

The given set is as,

  $\mathrm{A}=\left\{3^{-1},3^{-2},3^{-3},\mathrm{...},3^{-479},3^{-480}\right\}$

So the cardinality of set $\mathrm{A}$ is $480$ .it implies that $\mathrm{A}\approx {\mathrm{N}}_{480}.$

Hence, set $\mathrm{A}$ is finite.

e.

To determine

Explain the given set is finite.

Answer to Problem 3E

Set $\mathrm{A}$ is finite.

Explanation of Solution

Given information:

  $\mathrm{A}=\left\{1,2,3,\mathrm{...},479,480,\mathrm{z}\right\}$

Calculation:

Consider, definition and result for finite set,

Definition: the set $\mathrm{S}$ is infinite if $\mathrm{S}=\mathrm{\varphi }$ or $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some $\mathrm{k}\in \mathrm{N}$ . Further if $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ , then $\mathrm{A}$ has cardinal number $\mathrm{k}$ , and we write $\mathrm{S}=\mathrm{k}$ .

Result:

If a set $\mathrm{A}$ has finite cardinality $\mathrm{m}$ , then $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{m}}$

The given set is as,

  $\mathrm{A}=\left\{1,2,3,\mathrm{...},479,480,\mathrm{z}\right\}$

So the cardinality of set $\mathrm{A}$ is $\left(480+1\right)$ . It implies that $\mathrm{A}\approx {\mathrm{N}}_{481}.$

Hence, set $\mathrm{A}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}.$

f.

To determine

Explain the given set is finite.

Answer to Problem 3E

Set $\mathrm{A}$ is finite.

Explanation of Solution

Given information:

  $\mathrm{A}=\left\{2,4,6,\mathrm{...},478,480,\right\}$

Calculation:

Consider, definition and result for finite set,

Definition: the set $\mathrm{S}$ is infinite if $\mathrm{S}=\mathrm{\varphi }$ or $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some $\mathrm{k}\in \mathrm{N}$ . Further if $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ , then $\mathrm{A}$ has cardinal number $\mathrm{k}$ , and we write $\mathrm{S}=\mathrm{k}$ .

Result:

If a set $\mathrm{A}$ has finite cardinality $\mathrm{m}$ , then $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{m}}$

The given set is as,

  $\mathrm{A}=\left\{2,4,6,\mathrm{...},478,480,\right\}$

So the cardinality of set $\mathrm{A}$ is $\left(480/2=240\right)$ .it implies that $\mathrm{A}\approx {\mathrm{N}}_{240}.$

Hence, set $\mathrm{A}$ is finite.

g.

To determine

Explain the given set is finite.

Answer to Problem 3E

Set $\mathrm{A}$ is finite.

Explanation of Solution

Given information:

  $\mathrm{A}=\left\{1,2,3,\mathrm{...},479,480,\right\}\cup \left\{3^{-1},3^{-2},3^{-3},{\mathrm{...3}}^{-479},3^{-480}\right\}$

Calculation:

Consider, definition and result for finite set,

Definition: the set $\mathrm{S}$ is infinite if $\mathrm{S}=\mathrm{\varphi }$ or $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some $\mathrm{k}\in \mathrm{N}$ . Further if $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ , then $\mathrm{A}$ has cardinal number $\mathrm{k}$ , and we write $\mathrm{S}=\mathrm{k}$ .

Result:

If a set $\mathrm{A}$ has finite cardinality $\mathrm{m}$ , then $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{m}}$

The given set is as,

  $\mathrm{A}=\left\{1,2,3,\mathrm{...},479,480,\right\}\cup \left\{3^{-1},3^{-2},3^{-3},{\mathrm{...3}}^{-479},3^{-480}\right\}$

So the cardinality of the set $\mathrm{A}$ is $\left(480+4802=960\right)$ . It implies that $\mathrm{A}\approx {\mathrm{N}}_{960}.$

Hence, set $\mathrm{A}$ is finite.

h.

To determine

Explain the given set is finite.

Answer to Problem 3E

Set $\mathrm{A}$ is finite.

Explanation of Solution

Given information:

  $\mathrm{A}=\left\{4,5,479,480,3^{-20},3^{-21},\mathrm{...},3^{-479}\right\}$

Calculation:

Consider, definition and result for finite set,

Definition: the set $\mathrm{S}$ is infinite if $\mathrm{S}=\mathrm{\varphi }$ or $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some $\mathrm{k}\in \mathrm{N}$ . Further if $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ , then $\mathrm{A}$ has cardinal number $\mathrm{k}$ , and we write $\mathrm{S}=\mathrm{k}$ .

Result:

If a set $\mathrm{A}$ has finite cardinality $\mathrm{m}$ , then $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{m}}$

The given set is as,

  $\begin{array}{l}\mathrm{A}=\left\{4,5,479,480,3^{-20},3^{-21},\mathrm{...},3^{-479}\right\}\\ =\left\{4,5,479,480\right\}\cup \left\{3^{-20},3^{-21},\mathrm{...},3^{-479}\right\}\end{array}$

So the cardinality of set $\mathrm{A}$ is $\left(4+479-20+1=464\right)$ . It implies that $\mathrm{A}\approx {\mathrm{N}}_{464}.$

Hence, set $\mathrm{A}$ is finite.

i.

To determine

Explain the given set is finite.

Answer to Problem 3E

Set $\mathrm{A}$ is finite.

Explanation of Solution

Given information:

  $\mathrm{A}=\underset{\mathrm{i}=1}{\overset{9}{{\displaystyle \cup }}}{\mathrm{A}}_{\mathrm{i}}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}1\le \mathrm{i}\le 9{\mathrm{A}}_{\mathrm{i}}=\left\{10\mathrm{i},\mathrm{i}+10\mathrm{i}\right\}$

Calculation:

Consider, definition and result for finite set,

Definition: the set $\mathrm{S}$ is infinite if $\mathrm{S}=\mathrm{\varphi }$ or $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some $\mathrm{k}\in \mathrm{N}$ . Further if $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ , then $\mathrm{A}$ has cardinal number $\mathrm{k}$ , and we write $\mathrm{S}=\mathrm{k}$ .

Result:

If a set $\mathrm{A}$ has finite cardinality $\mathrm{m}$ , then $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{m}}$

The given set is as,

  $\begin{array}{l}\mathrm{A}=\underset{\mathrm{i}=1}{\overset{9}{{\displaystyle \cup }}}\mathrm{A}={\mathrm{A}}_1\cup {\mathrm{A}}_2\cup \mathrm{....}\cup {\mathrm{A}}_9\\ =\left\{\left(10\times 1\right),\left(1+\left(10\times 1\right)\right)\right\}\cup \left\{\left(10\times 2\right),\left(2+\left(10\times 2\right)\right)\right\}\cup \mathrm{...}\left\{\left(10\times 9\right),\left(9+\left(10\times 9\right)\right)\right\}\\ =\left\{10,11\right\}\cup \left\{20,22\right\}\cup \mathrm{...}\cup \left\{90,99\right\}\end{array}$

So the cardinality of set ${\mathrm{A}}_{\mathrm{i}}$ is $2$ for $0\le \mathrm{i}\le 10$ and when $\mathrm{i}\ne \mathrm{i},{\mathrm{A}}_{\mathrm{i}}\cap {\mathrm{A}}_{\mathrm{j}}=\mathrm{\varphi }$ for $0\le \mathrm{i},\mathrm{j}\le 9$ .

It implies that cardinality of set $\mathrm{A}$ is $\left(9\times 2=18\right)$ . So $\mathrm{A}\approx {\mathrm{N}}_{18}$ .

Hence, set $\mathrm{A}$ is finite.

j.

To determine

Explain the given set is finite.

Answer to Problem 3E

Set $\mathrm{A}$ is finite.

Explanation of Solution

Given information:

  $\mathrm{A}=\underset{\mathrm{i}=1}{\overset{9}{{\displaystyle \cup }}}{\mathrm{A}}_{\mathrm{i}}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}1\le \mathrm{i}\le 9{\mathrm{A}}_{\mathrm{i}}={\mathrm{N}}_{\mathrm{i}}.$

Calculation:

Consider, definition and result for finite set,

Definition: the set $\mathrm{S}$ is infinite if $\mathrm{S}=\mathrm{\varphi }$ or $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some $\mathrm{k}\in \mathrm{N}$ . Further if $\mathrm{S}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ , then $\mathrm{A}$ has cardinal number $\mathrm{k}$ , and we write $\mathrm{S}=\mathrm{k}$ .

Result:

If a set $\mathrm{A}$ has finite cardinality $\mathrm{m}$ , then $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{m}}$

The given set is as,

  $\mathrm{A}=\underset{\mathrm{i}=1}{\overset{9}{{\displaystyle \cup }}}{\mathrm{A}}_{\mathrm{i}}$ where for each $1\le \mathrm{i}\le 9$ .

So ${\mathrm{A}}_{\mathrm{i}}={\mathrm{N}}_{\mathrm{i}}\mathrm{f}\mathrm{o}\mathrm{r}1\le \mathrm{i}\le 9$ it implies that cardinality of ${\mathrm{A}}_{\mathrm{i}}={\mathrm{N}}_{\mathrm{i}}{}_{}$

  ${\mathrm{A}}_{\mathrm{i}}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}1\le \mathrm{i}\le 9$ . Note that for $1\le \mathrm{i}\le 9$ it is not necessarily true that ${\mathrm{A}}_{\mathrm{i}}\cap {\mathrm{A}}_{\mathrm{j}}=\mathrm{\varphi }\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\ne \mathrm{j}$

The maximum cardinality of set $\mathrm{A}$ occur for $1+2+\mathrm{...}+9=\frac{9\left(10\right)}{2}=45$ The minimum cardinality of set $\mathrm{A}$ occur for $\subseteq \mathrm{...}\subseteq {\mathrm{A}}_9.$ it implies that $\mathrm{A}=\underset{\mathrm{i}=1}{\overset{9}{{\displaystyle \cup }}}{\mathrm{A}}_{\mathrm{i}}={\mathrm{A}}_9$ that is minimum possible cardinality for set $\mathrm{A}$ is $9$ .

Hence, the cardinality for given set $\mathrm{A}$ is a natural number greater than or equal to $9$ and less than or equal to $45$ . It implies that $\mathrm{A}\approx {\mathrm{N}}_{\mathrm{k}}$ for some natural number $\mathrm{k}$ such that $9\le \mathrm{k}\le 45$ .

Hence, set $\mathrm{A}$ is finite.