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How to Prove It: A Structured Approach

Daniel J. Velleman

Chapter 7

Infinite Sets - all with Video Answers

Educators

Section 1

Equinumerous sets

01:30

Problem 1

Show that the following sets are denumerable.
(a) $\mathbb{N}$.
(b) The set of all even integers.

Nick Johnson
Nick Johnson
Numerade Educator
00:34

Problem 2

In this problem we'll use the following notation for intervals of real numbers. If $a$ and $b$ are real numbers and $a<b$, then
$$
\begin{aligned}
& {[a, b]=\{x \in \mathbb{R} \mid a \leq x \leq b\}} \\
& (a, b)=\{x \in \mathbb{R} \mid a<x<b\} \\
& (a, b]=\{x \in \mathbb{R} \mid a<x \leq b\} \\
& {[a, b)=\{x \in \mathbb{R} \mid a \leq x<b\} .}
\end{aligned}
$$
(a) Show that $[0,1] \sim[0,2]$.
(b) Show that $(-\pi / 2, \pi / 2) \sim \mathbb{R}$.
(c) Show that $(0,1) \sim \mathbb{R}$.
(d) Show that $(0,1] \sim(0,1)$.

Krishna Kumar
Krishna Kumar
Numerade Educator

Problem 3

Justify your answer to each question with either a proof or a counterexample.
(a) Suppose $A \sim B$ and $A \times C \sim B \times D$. Must it be the case that $C \sim D$ ?
(b) Suppose $A \sim B, A$ and $C$ are disjoint, $B$ and $D$ are disjoint, and $A \cup C \sim B \cup D$. Must it be the case that $C \sim D$ ?

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04:11

Problem 4

Prove that if $A \sim B$ then $\mathscr{P}(A) \sim \mathscr{P}(B)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:37

Problem 5

(a) Prove that for all natural numbers $n$ and $m$, if $\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\} \sim$ $\left\{i \in \mathbb{Z}^{+} \mid i \leq m\right\}$, then $n=m$. (Hint: Use induction on $n$.)
(b) Prove that if $A$ is finite, then there is exactly one natural number $n$ such that $\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\} \sim A$.

Bobby Barnes
Bobby Barnes
University of North Texas
01:14

Problem 6

Prove that if $A \sim B$ and $A$ is finite, then $B$ is also finite, and the number of elements in $B$ is the same as the number of elements in $A$.

Doruk Isik
Doruk Isik
Numerade Educator

Problem 7

(a) Prove that if $n \in \mathbb{N}$ and $A \subseteq\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\}$, then $A$ is finite and $A$ has at most $n$ elements. Furthermore, if $A \neq\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\}$, then $A$ has fewer than $n$ elements.
(b) Suppose $B \subseteq A, B \neq A$, and $B \sim A$. Prove that $A$ is infinite.

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04:10

Problem 8

Prove that if $n \in \mathbb{N}, f:\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\} \rightarrow B$, and $f$ is onto, then $B$ is finite.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
03:31

Problem 9

Show that the function $f: \mathbb{Z}^{+} \times \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$defined by the formula $$
f(i, j)=\frac{(i+j-2)(i+j-1)}{2}+i
$$
is one-to-one and onto.

Kenwa Nandi
Kenwa Nandi
Numerade Educator

Problem 10

Complete the proof of part 2 of Theorem 7.1.2 by showing that if $f$ : $A \rightarrow B$ and $g: C \rightarrow D$ are one-to-one, onto functions, $A$ and $C$ are disjoint, and $B$ and $D$ are disjoint, then $f \cup g$ is a one-to-one, onto function from $A \cup C$ to $B \cup D$.

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Problem 11

In this exercise you will complete the proof of $3 \rightarrow 1$ of Theorem 7.1.5. Suppose $B \subseteq \mathbb{Z}^{+}$and $B$ is infinite. We now define a function $f: \mathbb{Z}^{+} \rightarrow B$ by recursion as follows:
For all $n \in \mathbb{Z}^{+}$,
$$
f(n)=\text { the smallest element of } B \backslash\left\{f(m) \mid m \in \mathbb{Z}^{+}, m<n\right\} .
$$
Of course, the definition is recursive because the specification of $f(n)$ refers to $f(m)$ for all $m<n$.
(a) Suppose $n \in \mathbb{Z}^{+}$. The definition of $f(n)$ only makes sense if we can be sure that $B \backslash\left\{f(m) \mid m \in \mathbb{Z}^{+}, m<n\right\} \neq \varnothing$, in which case the well-ordering principle guarantees that it has a smallest element. Prove that $B \backslash\left\{f(m) \mid m \in \mathbb{Z}^{+}, m<n\right\} \neq \varnothing$. (Hint: See exercise 8.)
(b) Prove that for all $n \in \mathbb{Z}^{+}, f(n) \geq n$.
(c) Prove that $f$ is one-to-one and onto.

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01:54

Problem 12

Prove that if $B \subseteq A$ and $A$ is countable, then $B$ is countable.

Angelo Rendina
Angelo Rendina
Numerade Educator

Problem 13

Suppose $A$ is denumerable and $R$ is a partial order on $A$. Prove that $R$ can be extended to a total order on $A$. In other words, prove that there is a total order $T$ on $A$ such that $R \subseteq T$. Note that we proved a similar theorem for finite $A$ in Example 6.2.2. (Hint: Since $A$ is denumerable, we can write the elements of $A$ in a list: $A=\left\{a_1, a_2, a_3, \ldots\right\}$. Now recursively define partial orders $R_n$, for $n \in \mathbb{N}$, so that $R=R_0 \subseteq R_1 \subseteq R_2 \subseteq \ldots$. and $\forall i \leq n \forall j\left(\left(a_i, a_j\right) \in R_n \vee\left(a_j, a_i\right) \in R_n\right)$. For a hint on how to do this, see exercise 2 in Section 6.2. Finally, let $T=\cup_{n \in \mathbb{N}} R_n$.)

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