diff --git a/hw/hw5/hw5.fdb_latexmk b/hw/hw5/hw5.fdb_latexmk index f29065a..bdcae2d 100644 --- a/hw/hw5/hw5.fdb_latexmk +++ b/hw/hw5/hw5.fdb_latexmk @@ -1,7 +1,7 @@ # 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 "" "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -152,8 +152,8 @@ "/usr/share/texmf/web2c/texmf.cnf" 1681034085 39561 34c98e380bf7c7201ee6a7909aff625a "" "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1693368379 4623455 fa0568a71dd9a288d6c226ee477506c6 "" "/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 "" (generated) "hw5.aux" "hw5.log" diff --git a/hw/hw5/hw5.log b/hw/hw5/hw5.log index d9280d1..471aca3 100644 --- a/hw/hw5/hw5.log +++ b/hw/hw5/hw5.log @@ -1,4 +1,4 @@ -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 -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 diff --git a/hw/hw5/hw5.pdf b/hw/hw5/hw5.pdf index 2b141cf..3139a0b 100644 Binary files a/hw/hw5/hw5.pdf and b/hw/hw5/hw5.pdf differ diff --git a/hw/hw5/hw5.synctex.gz b/hw/hw5/hw5.synctex.gz index 16acb4f..6b17d0c 100644 Binary files a/hw/hw5/hw5.synctex.gz and b/hw/hw5/hw5.synctex.gz differ diff --git a/hw/hw5/hw5.tex b/hw/hw5/hw5.tex index df873a6..391ff2d 100644 --- a/hw/hw5/hw5.tex +++ b/hw/hw5/hw5.tex @@ -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}