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Exercises 1.4 Exercises
1. Write the following sets by listing their elements.
\(A_1=\set{ x\in\mathbb{N}\st x^2 \lt 2 }\text{.}\)
\(A_2=\set{ x\in\mathbb{Z}\st x^2 \lt 2 }\text{.}\)
\(A_3=\set{ x\in\mathbb{N}\st x=3k=\frac{216}{m} \text{ for some } k,m\in\mathbb N }\text{.}\)
\(A_4=\set{ x\in\mathbb{Z}\st \dfrac{x+2}{5}\in\mathbb{Z} }\text{.}\)
\(A_5=\set{ a\in B\st 6\leq 4a+1 \lt 17 }\text{,}\) where \(B=\set{1,2,3,4,5,6}\text{.}\)
\(A_6=\set{ x\in B\st 50 \lt xd \lt 100 \text{ for some } d\in D }\text{,}\) where \(B=\set{ 2,3,5,7,11,13, \ldots }\) is the set of primes and \(D=\set{ 5,10 }\text{.}\)
\(A_7=\set{n\in \mathbb Z\st n^2-5n-16\leq n}\text{.}\)
2. We are going to write the following sets in set builder notation.
\(A=\set{5,10,15,20,25,\ldots}\text{.}\)
\(B=\set{ 10,11,12,13,\ldots,98,99,100 }\text{.}\)
\(C=\set{0,3,8,15,24,35,..}\text{.}\)
\(D=\set{ \ldots, -\frac{3}{10},-\frac{2}{5}, -\frac{1}{2}, 0,\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17},\ldots }\text{.}\)
\(E=\set{2,4,16,256,65536,4294967296,\ldots}\text{.}\)
\(F=\set{2,3,4,6,8,9,12,16,18,24,27,32,36,\ldots }\text{.}\)
3.
In each of the following parts, a set is defined in one of three ways: (1) listing elements between braces, (2) using set builder notation, or (3) describing in words. Rewrite each set in the two forms of which it is not already given. For example, since the set in part (a) is given by method (1), write the same set using methods (2) and (3). As another example, since the set in part (c) is given by method (2), write the same set using methods (1) and (3).
\(\displaystyle A=\set{0, 2, 4, 6, \dots, 100}\)
\(\displaystyle B=\set{3, 9, 27, 81, \dots}\)
\(\displaystyle C=\set{m \st m\in\mathbb{Z}, \; |m|\leq 3}\)
\(\displaystyle D=\set{4k+1 \st k\in\mathbb{Z}}\)
The set \(E\) of all numbers that are the reciprocal of a natural number.
The set \(F\) of all integers that are two more than a (possibly negative) multiple of 5.
4. Consider the following ill-defined set: \(S=\set{2, 4,\dots}\text{.}\) Show that the definition of \(S\) is ambiguous by providing two different ways that you could interpret its definition.
5.
Consider the set
\begin{equation*}
\set{2n+1: n \in \mathbb{N}}.
\end{equation*}
Explain what is wrong with each of the expressions below and why they should not be used to denote this set.
\(\displaystyle A=\set{2k+1}\)
\(B=\set{2j+1 : j \in N}\text{.}\)
\(\displaystyle c=\set{2\ell+1: \ell \in \mathbb{N}}\)
\(\displaystyle D=\set{2k+1 : n \in \mathbb{N}}\)
\(\displaystyle E=\set{2m+1 : m \oldepsilon \mathbb{N}}\)
\(\displaystyle F=\set{2N+1 : N \in \mathbb{N}}\)
\(\displaystyle G=\set{2m+1 : m \varepsilon \mathbb{N}}\)
H = {2n+1 : n in N}
6.
Are the following statements true or false?
\(\displaystyle \es = \{0\}\)
\(\displaystyle \es = \{\es\}\)
\(\displaystyle |\es| = 0\)
\(\displaystyle \{\{\es \}\} = \{\es\}\)
\(\displaystyle \{\es\} = \{\{\}\}\)
7.
Show that each of the numbers
\begin{equation*}
a=2, \qquad b=8, \qquad \text{and} \qquad c=-12
\end{equation*}
do not belong to any of the following sets:
\begin{align*}
A \amp =\set{-\frac{1}{n}\st n\in\mathbb{N}} \amp \; \amp B=\set{x\in\mathbb{R} \st x\geq0, \; x^2 \gt 100} \\
C \amp =\set{\{2\}, \{8\}, \{-12\}} \amp \; \amp D=\set{4k\st k\in\mathbb{N}, \; k \text{ odd}}
\end{align*}
8.
Are the following sets equal?
\(\mathbb{Z}\) and \(\{ a: a \in \mathbb{N} \text{ or } -a\in\mathbb{N}\}\)
\(\{1,2,2,3,3,3,2,2,1\}\) and \(\{1,1,1,1,1,2,2,2,2,2,3,3,3,3,3\}\)
\(\{d : d \text{ is a day with 40 hours}\}\) and \(\{w: w \text{ is a week with 6 days}\}\)
\(\{p: p \text{ is prime, } p\lt 42\}\) and \(\{1,2,3,5,7,11,13,17,19,23,29,31,37,41\}\)
9.
Determine which of the following sets are equal to the set \(S=\set{\frac{1}{n}\st n\in\mathbb{N}}\text{.}\)
\begin{align*}
A \amp =\set{\frac{1}{n+1}\st n\in\mathbb{N}} \amp \; \amp B=\set{\frac{1}{|n|}\st n\in\mathbb{Z}, \; n\neq0}\\
C \amp =\set{\frac{2}{k}\st k\in\mathbb{N}, \; k \text{ even}} \amp \; \amp D=\set{\frac{a}{b}\st a,b\in\mathbb{N}}\\
E \amp =\set{\frac{1}{n-1}\st n\in\mathbb{N}, \; n>1} \amp \; \amp F=\set{\frac{1}{m} \st m\in\mathbb{Z}, \; m>0 }
\end{align*}
