Chapter 12, Transition to the Twentieth Century: Cantor and Kronecker Video Solutions, The History of Mathematics An Introduction

Problem 1

The argument involved in establishing Russell's paradox can be used to show that $P(N)$ is an uncountable set. That is, suppose to the contrary that $N \sim P(N)$ via the function $f: N \rightarrow P(N)$. Put $B=\{n \in N \mid n g f(n)\}$. Because $B \in P(N)$, it follows that $B=f(b)$ for some $b \in N$. Complete the argument by reasoning until a contradiction is obtained.

Doruk Isik

Numerade Educator

07:00

Problem 1

Prove that the following sets are countable:
(a) $\left\{2,2^2, 2^3, \ldots, 2^2, \ldots\right\}$.
(b) $\left(1, \frac{1}{2}, \frac{1}{3}, \ldots, 1 / n, \ldots\right)$.
(c) $\{5,10,15, \ldots, 5 n, \ldots\}$.
(d) $\left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots, \frac{n}{n+1}, \ldots\right\}$.

Oswaldo Jiménez

Numerade Educator

03:27

Problem 1

A schoolmaster being asked how many scholars he had said, "If 1 had as many more as I now have, half as many, one-third, and one-fourth as many, I should then have 148 ." How many scholars had he?

Christopher Stanley

Numerade Educator

03:47

Problem 2

A purse of 100 dollars is to be divided among four men, $A, B, C$, and $D$, so that $B$ may have 4 dollars more than $A$, and $C 8$ dollars more than $B$, and $D$ twice as many as $C$. What is each one's share of the money?

Swati Agarwal

Numerade Educator

Problem 2

Verify that $P\left(R^t\right)$ is uncountable, where $R^t$ is the set of real numbers.

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01:11

Problem 2

Verify that the set $N_e$ of all even natural numbers and the set $N_{\text {e. }}$ of all odd natural numbers are countable; do the same for the sets $Z_c$ and $Z_e$ of all even and odd integers.

Carson Merrill

Numerade Educator

01:15

Problem 3

A man driving geese to market is met by another who said, "Good morning, master, with your 100 geese." Says he, "I have not 100 ; but if $I$ had half as many as I have now and 2 geese and a half, beside the number I now have already, then there would be 100 ." How many geese did he have?

Shahab Ullah

Numerade Educator

03:18

Problem 3

For any sets $A$ and $B$, either prove or give a counterexample for each of the following assertions:
(a) If $A \subseteq B$, then $P(A) \subseteq P(B)$.
(b) $P(A \cup B)=P(A) \cup P(B)$.
(c) $P(A \cap B)=P(A) \cap P(B)$.
(d) $P(A \times B)=P(A) \times P(B)$.

Mohamed Mohamed

Numerade Educator

20:30

Problem 3

Prove, by confirming that the function $f: Q \rightarrow N$ defined by $f(m / n)=2^m 3^n$ is one-to-one, that the set $Q$ of positive rational numbers is countable.

Chris Trentman

Numerade Educator

01:04

Problem 4

Three men $A, B$, and $C$ built a house that cost 500 dollars, of which $A$ paid a certain sum. $B$ paid 10 dollars more then $A$, and $C$ paid as much as $A$ and $B$ both. How much did each man pay?

Jennifer Stoner

Numerade Educator

01:38

Problem 4

Prove that if $A$ and $B$ are sets for which $A \sim B$, then $P(A) \sim P(B)$.

Mohamed Mohamed

Numerade Educator

07:00

Problem 4

Use the theorems in this section to show that the set of prime numbers is a countable set.

Oswaldo Jiménez

Numerade Educator

01:00

Problem 5

If $N$ denotes the set of natural numbers, establish that
(a) The set of all infinite subsets of $N$ is uncountable.
(b) The set of all finite subsets of $N$ is countable.

Angelo Rendina

Numerade Educator

07:00

Problem 5

Establish that the Cartesian product $A \times B$ [that is, the set of all ordered pairs $(a, b)$ with $a$ in $A$ and $b$ in $B$ ] of two countable sets $A$ and $B$ is countable; in particular, conclude that $N \times Z, Z \times Z$, and $Q \times Q$ are countable sets.

Oswaldo Jiménez

Numerade Educator

03:26

Problem 5

The sum of the ages of a father and son is 100 years. Also $1 / 10$ of the product of their ages, in years, exceeds the father's by 180 . How old are they?

Anas Venkitta

Numerade Educator

02:35

Problem 6

If $S$ is the set of all right triangles whose sides have integral lengths, then $S$ is a countable set. Prove this statement.

Angelo Rendina

Numerade Educator

Problem 6

Let $a$ and $b$ be cardinal numbers and $A$ and $B$ be sets such that $a=o(A)$ and $b=o(B)$. Then the sum and product of $a$ and $b$ can be defined as follows:
$$
\begin{aligned}
& a+b=o(A \cup B), \quad \text { provided } A \cap B=\emptyset ; \\
& a+b=o(A \times B) .
\end{aligned}
$$
Using these definitions, prove that
(a) $$x+0=x, x-0=0, x-1=x$$.
(b) $$\mathfrak{K}_0+\mathrm{K}_0=\mathfrak{K}_0$$
(c) $$\aleph_0-\aleph_0=\aleph_0$$.
(d) $$c+\kappa_0=c$$. $$\left[H i n t ; c+K_0=o(I \cup Q)\right.$$, where $I$ and $Q$ are the irrational and rational numbers, respectively.]
(e) $$c+c=c \cdot
(f) $$c \cdot \mathbb{K}_0=c$$. $\mid$
(g) $$c \cdot c=c$$.

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01:26

Problem 6

In a certain family, 11 times the number of children is greater by 12 than twice the square of the number. How many children are there?

Naomi Motes

Numerade Educator

00:43

Problem 7

Let $\mathcal{C}$ be the set of all circles in the Cartesian plane that have rational radii and centers at points whose coordinates are both rational. Show that $\mathcal{C}$ forms a countable set.

Sriparna Bhattacharjee

Numerade Educator

01:00

Problem 7

(a) Could there exist a town in which the barber shaves all men who do not shave themselves? that barbers are male.]
(b) Could there exist a book that lists in its bibliography exactly those books that do not list themselves in their bibliographies?

Nick Johnson

Numerade Educator

00:56

Problem 7

One man and two boys can do in 12 days certain work that could be done in 6 days by three men and one boy. How long would it take one man to do it?

Ankit Gupta

Numerade Educator

02:13

Problem 8

Prove the following:
(a) If $Z_n[x]$ is the set of all polynomials of degree $n$ with integral coefficients, then $Z_N[x]$ is countable.
(b) The set $Z[x]$ of all polynomials with integral coefficients is countable.

Hoan Nguyen

Numerade Educator

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Problem 8

Suppose that a lexicon is drawn up containing all the words that occur in the text of this book; names and punctuation marks and also mathematical symbols are counted as words. Let $$S$$ be the set of natural numbers that are defined by sentences containing at most 50 words (a word being counted each time it occurs), all of them chosen from our lexicon; then the set $$S$$ is finite. Consider the natural number defined as follows:
Let $$n$$ be the smallest natural number, in accordance with the usual ordering of the natural numbers, that cannot be defined by means of a sentence containing at most 50 words, all taken from our lexicon.
Show that $$n$$ is defined in no more than 50 words, but is not a member of $$S$$ (this is Berry's paradox).

Ariana Nash

Numerade Educator

02:12

Problem 8

A man walking from town $A$ to another town $B$ at the rate of 4 miles an hour, starts one hour before a coach that goes 12 miles an hour, and is picked up by the coach. On arriving at $B$, he observes that his coach joumey last two hours. Find the distance from $A$ to $B$.

Akshaya Rs

Numerade Educator

09:04

Problem 9

Let $S$ be the set of all decimals that are defined by sentences containing a finite number of words, all taken from our lexicon; then $S$ is a countable set of real numbers $r_1, r_2, r_3, \ldots$. Consider the real number $r$ defined as follows:
If the digit in the $n$th decimal place of $r_n$ is denoted by $r_{\text {an. }}$, then construct the real number $r=0 . a_1 a_1 a_3 \ldots$ so that in $n$th digit $a_n=1$ if $r_{\text {an }} \neq 1$, and $a_n=2$ if $r_m=1$.
Show that $r$ is defined in a finite number of words but is not a member of $S$ (this is Richard's paradox).

Angelo Rendina

Numerade Educator

Problem 9

Use Cantor's diagonal argument to show that the set of all infinite sequences of $0 \mathrm{~s}$ and $1 \mathrm{~s}$ (that is, of all expressions such as 11010001 ...) is uncountable.

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02:17

Problem 10

Determine whether each of the following sets is countable or uncountable:
(a) The set of all numbers of the form $m / 2^4$, where $m$ is an integer and $n$ is a natural number.
(b) The set of all straight lines in the Cartesian plane, each of which passes through the origin.
(c) The set of all straight lines in the Cartesian plane, each of which passes through the origin and a point having both coordinates rational.
(d) The set of all intervals on the real line having both endpoints rational.
(e) Any infinite set of nonoverlapping intervals on the real line.

Angelo Rendina

Numerade Educator

01:37

Problem 10

Show, as an example of a nonconstructive existence proof, that there exists a solution of the equation $x^y=z$ with $x, y$ irrational and $z$ rational.

Prashant Bana

Numerade Educator

06:53

Problem 11

Prove that the set $L$ of Liouville numbers, and hence the set of transcendental numbers, has cardinality $c$.

Mohamed Mohamed

Numerade Educator

01:03

Problem 11

(a) Consider the number $n=(-1)^k$, where $k$ is the number of the first decimal place in the decimal expansion of $\pi$ where the sequence of consecutive digits 01234567890 begins; or if no such number $k$ exists, then $n=0$. Would the intuitionist accept the statement that $n$ is either positive, negative, or zero?
(b) The intuitionist views the following as a situation in which the statement $p$ is not the same as "not not $p . "$ Write $r=0.3333 \ldots$..., breaking this off as soon as a sequence of consecutive digits 01234567890 has appeared in the decimal expansion of $\pi$. Thus, if the 9 of the first sequence 01234567890 in $\pi$ is the $k$ th digit after the decimal point, then
$$
\begin{aligned}
r & =0.33333 \ldots 3(k \text { decimals }) \\
& =\frac{10^4-1}{3 \cdot 10^k} .
\end{aligned}
$$
Let $p$ be the statement " $r$ is a rational number." Show that "not $p$ " leads to a contradiction, so that "not not $p$ " must hold. On the other hand, the intuitionist would not conclude that $r$ is rational, because there is no effective way of calculating $a$ and $b$ for which $r=a / b$.

Carson Merrill

Numerade Educator

01:59

Problem 12

Verify that the function $f: R \rightarrow[0,1]$ defined by
$$
f(x)=\frac{1}{2}\left(1+\frac{x}{1+|x|}\right)
$$
is one-to-one, so that $R \sim[0,1]$.

Rukhmani Jain

Numerade Educator

06:19

Problem 13

Establish that any nondegenerate interval in $R$ has cardinality $c_{.}

Mohamed Mohamed

Numerade Educator