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fixed 2.23.26

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Rushil Umaretiya 2023-10-10 23:32:10 -04:00
parent 0198c59c2d
commit cef0c8a333
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5 changed files with 14 additions and 13 deletions
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hw/hw5/hw5.fdb_latexmk
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# Fdb version 4
["pdflatex"] 1696985184 "/home/rushil/school/math/hw/hw5/hw5.tex" "hw5.pdf" "hw5" 1696985184 0
["pdflatex"] 1696995106 "/home/rushil/school/math/hw/hw5/hw5.tex" "hw5.pdf" "hw5" 1696995107 0
"/etc/texmf/web2c/texmf.cnf" 1693368362 475 c0e671620eb5563b2130f56340a5fde8 ""
"/home/rushil/school/math/hw/hw5/hw5.tex" 1696985183 10795 c3ecd31e4c963c0a6444d89b0c163c0e ""
"/home/rushil/school/math/hw/hw5/hw5.tex" 1696995106 11312 e84734c8a338ea4d42bec57e0b42e2d7 ""
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"/usr/share/texmf/web2c/texmf.cnf" 1681034085 39561 34c98e380bf7c7201ee6a7909aff625a ""
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"/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1693368408 7880218 5644a8bb2704ddc1060e0c139ce5389a ""
"hw5.aux" 1696985184 195 5f40eaf408ff79ba46902478f1cd62f4 "pdflatex"
"hw5.tex" 1696985183 10795 c3ecd31e4c963c0a6444d89b0c163c0e ""
"hw5.aux" 1696995107 195 5f40eaf408ff79ba46902478f1cd62f4 "pdflatex"
"hw5.tex" 1696995106 11312 e84734c8a338ea4d42bec57e0b42e2d7 ""
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"hw5.aux"
"hw5.log"

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hw/hw5/hw5.log
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This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/Debian) (preloaded format=pdflatex 2023.8.30) 10 OCT 2023 20:46
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/Debian) (preloaded format=pdflatex 2023.8.30) 10 OCT 2023 23:31
entering extended mode
restricted \write18 enabled.
file:line:error style messages enabled.
@ -502,9 +502,9 @@ Here is how much of TeX's memory you used:
35748 multiletter control sequences out of 15000+600000
522110 words of font info for 69 fonts, out of 8000000 for 9000
1141 hyphenation exceptions out of 8191
102i,14n,104p,445b,318s stack positions out of 10000i,1000n,20000p,200000b,200000s
102i,14n,104p,445b,324s stack positions out of 10000i,1000n,20000p,200000b,200000s
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Output written on hw5.pdf (10 pages, 190913 bytes).
Output written on hw5.pdf (10 pages, 191641 bytes).
PDF statistics:
125 PDF objects out of 1000 (max. 8388607)
78 compressed objects within 1 object stream

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hw/hw5/hw5.tex
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@ -298,12 +298,13 @@ Consider these functions from the set of students in a discrete mathematics clas
\begin{enumerate}[a)]
\item Prove that a strictly increasing function from \(\AllReals\) to itself is one-to-one.
\begin{proof}
Lets take the strictly increasing function \(f(x) = x + 2)\). We can prove that this function is one-to-one by proving that for all \(x, y \in \AllReals\), \(f(x) = f(y) \implies x = y\).
\begin{align*}
f(x) &= f(y)\\
x + 2 &= y + 2\\
x &= y
\end{align*}
Since the function is strictly increasing, we can assume that \(\forall a,b \in \AllReals, a > b \implies f(a) > f(b)\). We can prove that this function is one-to-one by proving that for all \(x, y \in \AllReals\), \(f(x) = f(y) \implies x = y\). In order to use this, we'll use cases,
\begin{enumerate}
\item \(x > y\). Since \(f\) is strictly increasing, \(f(x) > f(y)\), which contradicts \(f(x) = f(y)\). Thus, \(x > y\) is impossible.
\item \(x < y\). Since \(f\) is strictly increasing, \(f(x) < f(y)\), which contradicts \(f(x) = f(y)\). Thus, \(x < y\) is impossible.
\item \(x = y\). This is trivially true.
\end{enumerate}
Since \(x > y\) and \(x < y\) are impossible, we can conclude that given \(f(x) = f(y)\), \(x = y\), and thus \(f\) is one-to-one.
\end{proof}
\item Give an example of an increasing function from \(\AllReals\) to itself that is not one-to-one.\\
The piecewise function, \(f(x) = \begin{cases}