Exercises

Exercises 8.6 Exercises

1.

Consider the following sets

\(A_1=\set{ x\in\mathbb{Z}\colon x^2 \lt 2 }\text{,}\)
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\(A_2=\set{ x\in\mathbb{N}\colon (3\mid x)\land (x\mid 216) }\text{,}\)
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\(A_3=\set{ x\in\mathbb{Z}\colon \dfrac{x+2}{5}\in\mathbb{Z} }\text{,}\)
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\(A_4=\set{ a\in B\colon 6\leq 4a+1 \lt 17 }\text{,}\) where \(B=\set{1,2,3,4,5,6}\text{,}\)
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\(A_5=\set{ x\in C\colon 50 \lt xd \lt 100 \text{ for some } d\in D}\text{,}\) where \(C=\set{ 2,3,5,7,11,13, \ldots }\) and \(D=\set{ 5,10 }\text{,}\)
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\(A_6=\set{5,10,15,20,25,\ldots}\text{,}\)
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\(A_7=\set{2,3,4,6,8,9,12,16,18, 24\ldots }\text{,}\)
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Write down the sets below by listing their elements.

\(A_2- A_3\text{,}\)
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\(A_5\cap A_6\text{,}\)
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\(\pow{A_1})\text{,}\)
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\(\pow{\pow{A_1-\set{-1} }} \text{,}\)
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\(A_3\cap A_4\text{,}\)
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\(A_3-A_7\text{,}\)
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\(A_5\cup A_2\text{,}\)
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\(A_2\cap A_7\text{.}\)
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\(A_5\cap \overline{A_2}\text{,}\) given the universal set \(U=\mathbb R\text{.}\)
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Verify whether \((A_4\times B)\cap(B\times A_4)=A_4\times A_4\) and \((A_4\times B)\cup(B\times A_4)=B\times B\text{.}\)
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2.

Write down the set \(F\cap G\) where

\begin{equation*} F=\set{(x,x^2-3x+2)\in\mathbb{R}^2: x\in\mathbb R } \qquad \text{and}\qquad G=\set{ (a, a+2)\in\mathbb{R}^2: a\in\mathbb R } \end{equation*}

by listing out all of its elements. Prove your answer.

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3.

Prove or disprove the following statement:

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Suppose \(A\text{,}\) \(B\) and \(C\) are sets. If \(A = B-C\text{,}\) then \(B = A \cup C\text{.}\)
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4.

Suppose \(x, y \in\mathbb{R}\) and \(k \in\mathbb N\) satisfying, \(x,y\gt 0\) and \(x^k = y\text{.}\) Then prove that \(\set{x^a\st a \in \mathbb{Q}}=\set{ y^a\st a\in\mathbb{Q} }\text{.}\)

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5.

Prove or disprove the following statement:

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If \(m,n \in\mathbb{N}\text{,}\) then \(\set{x \in\mathbb{Z} : m\mid x}\cap\set{x \in\mathbb{Z} : n\mid x}\subseteq \set{x \in\mathbb{Z} : mn\mid x} \text{.}\)
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6.

Prove or disprove the following statement:

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Let \(m,n \in\mathbb{Z}\text{.}\) Then \(\set{x \in\mathbb{Z} : mn\mid x}\subseteq \set{x \in\mathbb{Z} : m\mid x}\cap\set{x \in\mathbb{Z} : n\mid x}\text{.}\)
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7.

Let \(A\) be a set. Prove or disprove the following statements. If the statement is false in general, determine if there are any sets for which the statement is true.

\(\displaystyle A\times \emptyset \subseteq A.\)
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\(\displaystyle A\times \emptyset = A.\)
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8.

Suppose that \(A, B\neq \emptyset\text{,}\) and \(C\) are sets such that \(A\subseteq B\text{.}\)

Prove that \(A\times C \subseteq B\times C.\)
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Suppose we have a strict containment \(A\subset B\) instead. What additional constraints do we need (if any) to show that

\begin{equation*} A\times C \subset B\times C? \end{equation*}

Prove your claim.

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9.

Prove or disprove the following statement:

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If \(A\) and \(B\) are sets, then \(\mathcal{P}(A)\cap\mathcal{P}(B) = \mathcal{P}(A\cap B)\text{.}\)
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10.

Prove or disprove the following statement:

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If \(A\) and \(B\) are sets, then \(\mathcal{P}(A)\cup\mathcal{P}(B) = \mathcal{P}(A\cup B)\text{.}\)
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11.

Let \(A\) be a finite set with \(|A|=n\text{.}\) Prove that \(|\pow A|=2^n\text{.}\)

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12.

Let \(A\) and \(B\) be sets. Prove or disprove the following statements:

\(\pow{A - B} \subseteq \pow{A} - \pow{B}\text{,}\) and
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\(\pow{A} - \pow{B} \subseteq \pow{A - B}\text{.}\)
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13.

Prove that

\(\displaystyle \ds \bigcup_{n=3}^\infty \left(\frac{1}{n},1-\frac{1}{n}\right) = (0,1) \)
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\(\displaystyle \ds \bigcap_{n=1}^\infty \left(-\frac{1}{n}, 1+\frac{1}{n}\right)=[0,1] \)
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14.

Determine what each of the following unions is equal to, and prove your answer.

\begin{equation*} \bigcup_{n\in\mathbb{N}}[-n,n] \end{equation*}

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\begin{equation*} \bigcup_{r\in\mathbb{R}, r\gt0} B_r \end{equation*}

where

\begin{equation*} B_r = \{(x,y)\in \mathbb{R}\times\mathbb{R}:x^2+y^2\lt r\}. \end{equation*}

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15.

Let \(S\subset\mathbb{R}\text{.}\) We say \(b\in\mathbb{R}\) is an upper bound of \(S\) if \(s\leq b\) for every \(s\in S\text{.}\) Further, we say \(a\in\mathbb{R}\) is the supremum (or the least upper bound) of \(S\text{,}\) denoted by \(\sup(S)\text{,}\) if

\(a\) is an upper bound for \(S\text{,}\) and
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if \(b\) is an upper bound for \(S\text{,}\) then \(a\leq b\text{.}\)
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We also call \(c\in S\) the maximum element of \(S\text{,}\) denoted by \(\max(S)\text{,}\) if it is the largest element in \(S\text{.}\) So, \(\max(S)\) belongs to \(S\text{,}\) and is an upper bound of \(S\text{.}\)

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For each of the following sets, determine its maximum and supremum, if they exist. Justify your answers.

\(\displaystyle [1,3]= \set{x \in \mathbb{R} \;:\; 1\leq x \leq 3}\)
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\(\displaystyle (1,3)= \set{x \in \mathbb{R} \;:\; 1\lt x \lt 3}\)
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\(\displaystyle \set{m\in\mathbb{Z}\;:\; |2(m-4)|\leq 15}\)
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\(\displaystyle \set{2-\frac{1}{n}\;:\;n\in\mathbb{N}}\)
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\(\displaystyle \set{x\in\mathbb{R}\;:\;\cos(2x)=1}\)
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16.

This question involves the supremum, which we first introduced in a previous exercise, Exercise 8.6.15. We recommend that you complete that question before you attempt this one.

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\(a\) is an upper bound for \(S\text{,}\) and
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if \(b\) is an upper bound for \(S\text{,}\) then \(a\leq b\text{.}\)
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Suppose that \(S,T\) are non-empty subsets of \(\mathbb{R}\text{,}\) and \(s=\sup(S)\text{,}\) \(t=\sup(T)\text{,}\) where \(s,t \in\mathbb{R}\text{.}\)

Show that \(\sup(S\cup T)=\max\set{s,t}\text{.}\)
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Can you determine \(\sup(S\cap T)\text{?}\)
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Define \(S+T=\set{s+t:s\in S, t\in T}\text{.}\) Show that \(\sup(S+T)=s+t\text{.}\)
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17.

Before completing this question you should look at Exercise 8.6.15 and Exercise 8.6.16. Let \(\{a_n\}_{n\in\in\mathbb{N}}\) be a sequence such that \(a_{n+1}\geq a_n\) for all \(n\in \mathbb{N}\text{,}\) and such that

\begin{equation*} a = \sup\{a_n:n\in\mathbb{N}\} \end{equation*}

exists as a real number. Show that

\begin{equation*} \lim_{n\to\infty}a_n = a. \end{equation*}

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