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hw7
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hw/hw7/hw7.aux
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\@writefile{toc}{\contentsline {section}{Unit 2.3}{2}{}\protected@file@percent }
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\gdef \@abspage@last{8}
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# Fdb version 4
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|
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|
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|
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|
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|
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|
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|
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|
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\normalsize\text{Tuesday/Thursday 11:00-12:15, Phillips 383}\\
|
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|
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|
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|
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|
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|
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|
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\newcommand{\dx}{\mathrm{d}x}
|
||||
|
||||
% Alias for the Solution section header
|
||||
\newcommand{\solution}{\textbf{\large Solution}}
|
||||
|
||||
\newcommand{\unit}[1]{\section{Unit #1}}
|
||||
\newcommand{\problem}[1]{\textbf{\##1}}
|
||||
\newcommand{\prob}[1]{\problem{#1}}
|
||||
|
||||
|
||||
% Probability commands: Expectation, Variance, Covariance, Bias
|
||||
\newcommand{\E}{\mathrm{E}}
|
||||
\newcommand{\Var}{\mathrm{Var}}
|
||||
\newcommand{\Cov}{\mathrm{Cov}}
|
||||
\newcommand{\Bias}{\mathrm{Bias}}
|
||||
|
||||
\renewcommand{\And}{\wedge}
|
||||
\newcommand{\Or}{\vee}
|
||||
\newcommand{\Xor}{\oplus}
|
||||
\newcommand{\Not}{\neg}
|
||||
\newcommand{\Implies}{\rightarrow}
|
||||
\newcommand{\Iff}{\leftrightarrow}
|
||||
\newcommand{\union}{\cup}
|
||||
\newcommand{\intersection}{\cap}
|
||||
|
||||
\newcommand{\AllIntegers}{\mathbb{Z}}
|
||||
\newcommand{\AllNaturals}{\mathbb{N}}
|
||||
\newcommand{\AllRationals}{\mathbb{Q}}
|
||||
\newcommand{\AllReals}{\mathbb{R}}
|
||||
\newcommand{\AllComplexes}{\mathbb{C}}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
|
||||
\pagebreak
|
||||
|
||||
\unit{2.3}
|
||||
\prob{44}
|
||||
Let \(f\) be the function from \(\AllReals\) to \(\AllReals\) defined by
|
||||
\(f(x) = x^2\). Find
|
||||
|
||||
\begin{enumerate}[a)]
|
||||
\item \(f^{-1}(\{1\})\)
|
||||
\item \(f^{-1}(\{x \mid 0 < x < 1\})\)
|
||||
\item \(f^{-1}(\{x \mid x > 4\})\)
|
||||
\end{enumerate}
|
||||
\textbf{Note: } \(f^{-1}(x) = \{\pm\sqrt{x}\}\)
|
||||
\begin{enumerate}[a)]
|
||||
\item \{1, -1\}
|
||||
\item \(\{x \neq 0 \mid -1 < x < 1\}\)
|
||||
\item \(\{x \mid (x > 2) \Or (x < -2)\}\)
|
||||
\end{enumerate}
|
||||
|
||||
\pagebreak
|
||||
|
||||
\prob{46}
|
||||
Let \(f\) be a function from \(A\) to \(B\). Let \(S\) and \(T\) be subsets of \(B\). Show that
|
||||
|
||||
\begin{enumerate}[a)]
|
||||
\item \(f^{-1}(S \union T) = f^{-1}(S) \union f^{-1}(T)\)
|
||||
\begin{proof}We will use the definition of \(f^{-1}(S)\) to prove this.
|
||||
\begin{enumerate}
|
||||
\item[\(\rightarrow\)] Let \(x \in f^{-1}(S \union T)\). Then \(f(x) \in S \union T\). Thus \(f(x) \in S\) or \(f(x) \in T\). Thus \(x \in f^{-1}(S)\) or \(x \in f^{-1}(T)\). Thus \(x \in f^{-1}(S) \union f^{-1}(T)\).
|
||||
\item[\(\leftarrow\)] Let \(x \in f^{-1}(S) \union f^{-1}(T)\). Then \(x \in f^{-1}(S)\) or \(x \in f^{-1}(T)\). Thus \(f(x) \in S\) or \(f(x) \in T\). Thus \(f(x) \in S \union T\). Thus \(x \in f^{-1}(S \union T)\).
|
||||
\end{enumerate}
|
||||
Therefore, \(f^{-1}(S \union T) = f^{-1}(S) \union f^{-1}(T)\).
|
||||
\end{proof}
|
||||
\item \(f^{-1}(S \intersection T) = f^{-1}(S) \intersection f^{-1}(T)\)
|
||||
\begin{proof}
|
||||
We will use the same method as part a) to prove this.
|
||||
\begin{enumerate}
|
||||
\item [\(\rightarrow\)] Let \(x \in f^{-1}(S \intersection T)\). Then \(f(x) \in S \intersection T\). Thus \(f(x) \in S\) and \(f(x) \in T\). Thus \(x \in f^{-1}(S)\) and \(x \in f^{-1}(T)\). Thus \(x \in f^{-1}(S) \intersection f^{-1}(T)\).
|
||||
\item [\(\leftarrow\)] Let \(x \in f^{-1}(S) \intersection f^{-1}(T)\). Then \(x \in f^{-1}(S)\) and \(x \in f^{-1}(T)\). Thus \(f(x) \in S\) and \(f(x) \in T\). Thus \(f(x) \in S \intersection T\). Thus \(x \in f^{-1}(S \intersection T)\).
|
||||
\end{enumerate}
|
||||
Therefore, \(f^{-1}(S \intersection T) = f^{-1}(S) \intersection f^{-1}(T)\).
|
||||
\end{proof}
|
||||
\end{enumerate}
|
||||
\pagebreak
|
||||
|
||||
|
||||
\unit{4.1}
|
||||
\prob{12}
|
||||
Prove that if \(a\) is a positive integer, then \(4\) does not divide \(a^2 + 2\).
|
||||
\begin{proof}
|
||||
We will conduct a proof by contradicition. Assume \(4 \mid a^2 + 2\). This means that \(\exists k \in \AllIntegers, 4k = a^2 + 2\). We can rewrite this as \(a^2 = 4k - 2 = 2(2k - 1)\). This means that \(a^2\) is even which means that \(a\) is even. This means that \(\exists j \in \AllIntegers, a = 2j\). We can rewrite this as \(a^2 = 4j^2\). This means that \(a^2\) is divisible by 4. This means that \(4 \mid a^2\). This means that \(4 \mid a^2 + 2\) and \(4 \mid a^2\). We can then use identites to show that \(4 \mid a^2 - (a^2 + 2) \equiv 4 \mid 2\). This is a contradiction because \(4 \nmid 2\). Therefore, \(4 \nmid a^2 + 2\).
|
||||
\end{proof}
|
||||
|
||||
\pagebreak
|
||||
|
||||
\prob{30}
|
||||
Find the integer a such that\\
|
||||
\begin{enumerate}[a)]
|
||||
\item \(a \equiv 43\pmod{23}\) and \(-22 \leq a \leq 0\)
|
||||
\begin{align*}
|
||||
a = -3
|
||||
\end{align*}
|
||||
\item \(a \equiv 17\pmod{29}\) and \(-14 \leq a \leq 14\)
|
||||
\begin{align*}
|
||||
a = -12
|
||||
\end{align*}
|
||||
\item \(a \equiv -11\pmod{21}\) and \(90 \leq a \leq 110\)
|
||||
\begin{align*}
|
||||
a = 94
|
||||
\end{align*}
|
||||
\end{enumerate}
|
||||
\pagebreak
|
||||
|
||||
\prob{34}
|
||||
Decide whether each of these integers is congruent to 3 modulo 7.\\
|
||||
\textbf{Note: } \(a \equiv b \pmod n\) means that \(n \mid (a - b)\).
|
||||
\begin{enumerate}[a)]
|
||||
\item 37
|
||||
\begin{align*}
|
||||
37 \not \equiv 3 \pmod 7 \text{ because } 7 \nmid (37 - 3) = 34
|
||||
\end{align*}
|
||||
\item 66
|
||||
\begin{align*}
|
||||
66 \equiv 3 \pmod 7 \text{ because } 7 \mid (66 - 3) = 63
|
||||
\end{align*}
|
||||
\item -17
|
||||
\begin{align*}
|
||||
-17 \not \equiv 3 \pmod 7 \text{ because } 7 \nmid (-17 - 3) = -20
|
||||
\end{align*}
|
||||
\item -67
|
||||
\begin{align*}
|
||||
-67 \equiv 3 \pmod 7 \text{ because } 7 \mid (-67 - 3) = -70
|
||||
\end{align*}
|
||||
\end{enumerate}
|
||||
\pagebreak
|
||||
|
||||
\prob{44}
|
||||
Show that if \(n\) is an integer then \(n^2 \equiv 0\) or 1\(\pmod 4\).
|
||||
\begin{proof}
|
||||
We will conduct a proof by cases on \(n\).
|
||||
\begin{enumerate}
|
||||
\item [\(n\) is even]
|
||||
\begin{align*}
|
||||
n = 2k \text{ for some } k \in \AllIntegers\\
|
||||
n^2 = 4k^2 = 4(k^2)\\
|
||||
\exists \bar{k} \in \AllIntegers, k^2 = \bar{k}\\
|
||||
n^2 = 4\bar{k}\\
|
||||
n^2 \equiv 0 \pmod 4
|
||||
\end{align*}
|
||||
\item [\(n\) is odd]
|
||||
\begin{align*}
|
||||
n = 2k + 1 \text{ for some } k \in \AllIntegers\\
|
||||
n^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1\\
|
||||
\exists \bar{k} \in \AllIntegers, k^2 + k = \bar{k}\\
|
||||
n^2 = 4\bar{k} + 1\\
|
||||
n^2 \equiv 1 \pmod 4
|
||||
\end{align*}
|
||||
\end{enumerate}
|
||||
Therefore, \(n^2 \equiv 0\) or 1\(\pmod 4\).
|
||||
\end{proof}
|
||||
\pagebreak
|
||||
|
||||
\prob{46}
|
||||
Prove that if \(n\) is an odd positive integer, then \(n^2 \equiv 1 \pmod 8\).
|
||||
\begin{proof}
|
||||
We will conduct a direct proof. Let \(n\) be an odd positive integer.
|
||||
\begin{align*}
|
||||
\exists k \in \AllIntegers, n = 2k + 1\\
|
||||
n^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1 = 4(k)(k+1) + 1\\
|
||||
\end{align*}
|
||||
Note, either \(k\) or \(k+1\) must be even, therefore
|
||||
\begin{align*}
|
||||
\exists \bar{k} \in \AllIntegers, k(k+1) = 2\bar{k}\\
|
||||
n^2 = 4(2\bar{k}) + 1 = 8\bar{k} + 1\\
|
||||
n^2 \equiv 1 \pmod 8
|
||||
\end{align*}
|
||||
\end{proof}
|
||||
\pagebreak
|
||||
|
||||
|
||||
\end{document}
|
||||
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Reference in New Issue
Block a user