From f98aae18e0f4f0c234bcde6bded1a50664f3be6c Mon Sep 17 00:00:00 2001 From: Rushil Umaretiya <rushilwiz@gmail.com> Date: Thu, 5 Oct 2023 19:05:23 -0400 Subject: [PATCH] completed hw4 --- hw/hw4/hw4.fdb_latexmk | 9 +++-- hw/hw4/hw4.fls | 5 +++ hw/hw4/hw4.log | 83 ++++++++++++++++++++++++++--------------- hw/hw4/hw4.pdf | Bin 164388 -> 202635 bytes hw/hw4/hw4.synctex.gz | Bin 27435 -> 37823 bytes hw/hw4/hw4.tex | 66 ++++++++++++++++++++++---------- hw/hw4/images/venn.png | Bin 0 -> 42829 bytes 7 files changed, 108 insertions(+), 55 deletions(-) create mode 100644 hw/hw4/images/venn.png diff --git a/hw/hw4/hw4.fdb_latexmk b/hw/hw4/hw4.fdb_latexmk index a66f07c..6ebcf5f 100644 --- a/hw/hw4/hw4.fdb_latexmk +++ b/hw/hw4/hw4.fdb_latexmk @@ -1,7 +1,7 @@ # Fdb version 4 -["pdflatex"] 1696363890 "/home/rushil/school/math/hw/hw4/hw4.tex" "hw4.pdf" "hw4" 1696363891 0 +["pdflatex"] 1696547102 "/home/rushil/school/math/hw/hw4/hw4.tex" "hw4.pdf" "hw4" 1696547102 0 "/etc/texmf/web2c/texmf.cnf" 1693368362 475 c0e671620eb5563b2130f56340a5fde8 "" - "/home/rushil/school/math/hw/hw4/hw4.tex" 1696363890 9560 a31e0c32281544c86bb884e400a4ec65 "" + "/home/rushil/school/math/hw/hw4/hw4.tex" 1696547102 12969 6c8a8554a14b26f86dde7789b6a53aa6 "" "/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 "" @@ -150,8 +150,9 @@ "/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 "" - "hw4.aux" 1696363891 195 15cac26600106fc29820619f968893c5 "pdflatex" - "hw4.tex" 1696363890 9560 a31e0c32281544c86bb884e400a4ec65 "" + "hw4.aux" 1696547102 195 15cac26600106fc29820619f968893c5 "pdflatex" + "hw4.tex" 1696547102 12969 6c8a8554a14b26f86dde7789b6a53aa6 "" + "images//venn.png" 1696546351 42829 1ad5a73bd039d9201fc88e86ea893b87 "" (generated) "hw4.aux" "hw4.log" diff --git a/hw/hw4/hw4.fls b/hw/hw4/hw4.fls index 99c330b..3e8b5b7 100644 --- a/hw/hw4/hw4.fls +++ b/hw/hw4/hw4.fls @@ -511,6 +511,7 @@ INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryautomata.code.tex INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryautomata.code.tex INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryshapes.multipart.code.tex @@ -606,6 +607,10 @@ INPUT /usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd INPUT /usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd INPUT /usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd INPUT /usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd +INPUT ./images//venn.png +INPUT ./images//venn.png +INPUT ./images//venn.png +INPUT ./images//venn.png INPUT hw4.aux INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb diff --git a/hw/hw4/hw4.log b/hw/hw4/hw4.log index 39049e1..c24e9b4 100644 --- a/hw/hw4/hw4.log +++ b/hw/hw4/hw4.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) 3 OCT 2023 16:11 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/Debian) (preloaded format=pdflatex 2023.8.30) 5 OCT 2023 19:05 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -437,20 +437,20 @@ File: l3backend-pdftex.def 2023-01-16 L3 backend support: PDF output (pdfTeX) ) (./hw4.aux) \openout1 = `hw4.aux'. -LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 148. -LaTeX Font Info: ... okay on input line 148. -LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 148. -LaTeX Font Info: ... okay on input line 148. -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 148. -LaTeX Font Info: ... okay on input line 148. -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 148. -LaTeX Font Info: ... okay on input line 148. -LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 148. -LaTeX Font Info: ... okay on input line 148. -LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 148. -LaTeX Font Info: ... okay on input line 148. -LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 148. -LaTeX Font Info: ... okay on input line 148. +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 149. +LaTeX Font Info: ... okay on input line 149. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 149. +LaTeX Font Info: ... okay on input line 149. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 149. +LaTeX Font Info: ... okay on input line 149. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 149. +LaTeX Font Info: ... okay on input line 149. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 149. +LaTeX Font Info: ... okay on input line 149. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 149. +LaTeX Font Info: ... okay on input line 149. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 149. +LaTeX Font Info: ... okay on input line 149. (/usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] \scratchcounter=\count310 @@ -470,51 +470,72 @@ Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 48 (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live )) -LaTeX Font Info: Trying to load font information for U+msa on input line 150. +LaTeX Font Info: Trying to load font information for U+msa on input line 151. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd File: umsa.fd 2013/01/14 v3.01 AMS symbols A ) -LaTeX Font Info: Trying to load font information for U+msb on input line 150. +LaTeX Font Info: Trying to load font information for U+msb on input line 151. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd File: umsb.fd 2013/01/14 v3.01 AMS symbols B ) [1 {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] -Underfull \hbox (badness 10000) in paragraph at lines 160--162 +Underfull \hbox (badness 10000) in paragraph at lines 161--163 [] [2] -Underfull \hbox (badness 10000) in paragraph at lines 195--198 +Underfull \hbox (badness 10000) in paragraph at lines 197--200 [] [3] [4] -LaTeX Font Info: Trying to load font information for OMS+cmr on input line 249. +LaTeX Font Info: Trying to load font information for OMS+cmr on input line 246. (/usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd File: omscmr.fd 2022/07/10 v2.5l Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10> not available -(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 249. +(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 246. -Underfull \hbox (badness 10000) in paragraph at lines 249--250 +Underfull \hbox (badness 10000) in paragraph at lines 246--247 [] -[5] [6] [7] [8] [9] [10] [11] (./hw4.aux) ) +[5] +Underfull \hbox (badness 10000) in paragraph at lines 262--263 + + [] + +[6] [7] +<./images//venn.png, id=42, 385.44pt x 238.9728pt> +File: ./images//venn.png Graphic file (type png) +<use ./images//venn.png> +Package pdftex.def Info: ./images//venn.png used on input line 298. +(pdftex.def) Requested size: 227.62204pt x 141.124pt. + [8 <./images//venn.png>] [9] +Underfull \hbox (badness 10000) in paragraph at lines 316--318 + + [] + +[10] +Underfull \hbox (badness 10000) in paragraph at lines 322--323 + + [] + +[11] (./hw4.aux) ) Here is how much of TeX's memory you used: - 15661 strings out of 476091 - 318629 string characters out of 5794081 + 15676 strings out of 476091 + 318873 string characters out of 5794081 1867330 words of memory out of 5000000 - 35766 multiletter control sequences out of 15000+600000 + 35778 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,11n,104p,445b,326s stack positions out of 10000i,1000n,20000p,200000b,200000s + 102i,11n,104p,555b,326s stack positions out of 10000i,1000n,20000p,200000b,200000s </usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx9.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr9.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsl10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb> -Output written on hw4.pdf (11 pages, 164388 bytes). +Output written on hw4.pdf (11 pages, 202635 bytes). 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-189,6 +190,7 @@ We can find the following values for \(P(x)\) and generalize a formula for \(P(n &= \frac{(n+1)^2}{(n+1)(n+2)}\\ &= \frac{n+1}{n+2} \end{align*} + Thus, \(P(n) \implies P(n+1)\) and \(P(n)\) is true. \end{proof} \end{enumerate} \pagebreak @@ -233,15 +235,10 @@ Use mathematical induction to prove that if \(p\) is a prime and \(p | a_1a_2 \c \textbf{Inductive Step:} Given P(n), we will prove P(n+1). \begin{align*} P(n+1) &= p|a_1a_2\cdots a_na_{n+1} \implies p|a_i \text{ for some } i \in \AllIntegers .\\ + a_k &= a_1a_2\cdots a_n\\ + &= p|a_ka_{n+1} \implies p|a_i \end{align*} -Since we know that \(p\) is prime, -\begin{align*} - p|a_1a_2\cdots a_na_{n+1} &\implies p|a_1a_2\cdots a_n \text{ or } p|a_{n+1}\\ - &\implies P(n) \text{ or } p|a_{n+1}\\ - &\implies p|a_i \text{ for some } i \in \AllIntegers \text{ or } p|a_{n+1}\\ - &\implies P(n+1) -\end{align*} -We have shown that \(P(n) \implies P(n+1)\), thus \(P(n+1)\) is true. +Since we assume \(P(n)\), we know that \(p|k \implies p|a_i\). Given that \(\forall a,b,c \in \AllIntegers, a|b \implies a|bc\), we can say that \(p|a_ka_n+1 \implies p|a_i\) and shown that \(P(n) \implies P(n+1)\), thus \(P(n+1)\) is true. \pagebreak \unit{2.1} \prob{2} Use set builder notation to give a description of each of these sets. @@ -256,45 +253,74 @@ We have shown that \(P(n) \implies P(n+1)\), thus \(P(n+1)\) is true. \end{align*} \item \(\{m,n,o,p\}\) \begin{align*} - C = \{x | x \text{ is a lowercase letter in the alphabet from m to p}\} + C = \{x | x \text{ is a lowercase letter in the English alphabet, alphabetically from m to p}\} \end{align*} \end{enumerate} \pagebreak \prob{6} For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other. \begin{enumerate}[a)] - \item the set of people who speak English, the set of people who speak English with an Australian accent + \item the set of people who speak English, the set of people who speak English with an Australian accent\\ + \begin{align*} + \{\text{English speakers with an Australian accent}\} \subset \{\text{English speakers}\} + \end{align*} + The set of people who speak English with a Australian accent is a subset of all people who speak English, as every member of the subset of individuals with an Australian accent are members of the superset of all English speakers. \item the set of fruits, the set of citrus fruits + The set of all citrus + \begin{align*} + \{\text{all citrus fruits}\} \subset \{\text{all fruits}\} + \end{align*} + The set of all citrus fruits is a subset of all fruits, as every member of the subset of citrus fruits are members of the superset of all fruits. \item the set of students studying discrete mathematics, the set of students studying data structures + \begin{align*} + \{\text{Students studying discrete mathematics}\} \text{ is not related to } \{\text{Students studying data structures}\} + \end{align*} + The set of students studying discrete mathematics is not related to the set of students studying data structures, as there is no overlap between the two sets. Neither is a subset of the other. \end{enumerate} \pagebreak \prob{12} Determine whether these statements are true or false. \begin{enumerate}[a)] \item \(\emptyset \in \{\emptyset\}\) + True. The empty set is a member of the set containing the empty set. \item \(\emptyset \in \{\emptyset, \{\emptyset\}\}\) + True. The empty set is a member of the set containing the empty set and the set containing the empty set. \item \(\{\emptyset\} \in \{\emptyset\}\) - \item \(\{\emptyset\} \in \{\{\emptyset\}\}\) - \item \(\{\emptyset\} \subset \{\emptyset, \{\emptyset\}\}\) - \item \(\{\{\emptyset\}\} \subset \{\emptyset, \{\emptyset\}\}\) - \item \(\{\{\emptyset\}\} \subset \{\{\emptyset\}, \{\emptyset\}\}\) + False. The set containing the empty set is not a member of the set containing the empty set. + \item \(\{\emptyset\} \in \{\{\emptyset\}\}\) True. The set containing the empty set is a member of the set containing the set containing the empty set. + \item \(\{\emptyset\} \subset \{\emptyset, \{\emptyset\}\}\) True. The set containing the empty set is a subset of the set containing the empty set and the set containing the set containing the empty set. + \item \(\{\{\emptyset\}\} \subset \{\emptyset, \{\emptyset\}\}\) True. The set containing the set containing the empty set is a subset of the set containing the empty set and the set containing the set containing the empty set. + \item \(\{\{\emptyset\}\} \subset \{\{\emptyset\}, \{\emptyset\}\}\) True. Both of these sets are equivalent, therefore the set containing the set containing the empty set is a subset of the set containing the set containing the empty set and the set containing the set containing the empty set. \end{enumerate} \pagebreak \prob{18} Use a Venn diagram to illustrate the relationships \(A \subset B\) and \(A \subset C\). - +\graphicspath{{./images/}} +\begin{figure}[H] + \includegraphics[width=8cm]{venn} + \centering +\end{figure} \pagebreak \prob{22} What is the cardinality of each of these sets? \begin{enumerate}[a)] - \item \(\emptyset\) - \item \(\{\emptyset\}\) - \item \(\{\emptyset, \{\emptyset\}\}\) - \item \(\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}\) + \item \(\emptyset\)\\ + The cardinality of the empty set is 0. + \item \(\{\emptyset\}\)\\ + The cardinality of the set containing the empty set is 1. + \item \(\{\emptyset, \{\emptyset\}\}\)\\ + The cardinality of the set containing the empty set and the set containing the empty set is 2. + \item \(\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}\)\\ + The cardinality of the set contianing the empty set, set containing the empty set, and an additional set is 3. \end{enumerate} \pagebreak -\prob{32} Suppose that \(A \times B = \emptyset\), where \(A \text{ and } B\) are sets. What can you conclude? +\prob{32} Suppose that \(A \times B = \emptyset\), where \(A \text{ and } B\) are sets. What can you conclude?\\\\ +Either set must be empty, as the cartesian product of two sets is the set of all ordered pairs of elements from the two sets. If the cartesian product is empty, then is no elements in at least one of the sets. \pagebreak \prob{44} Prove or disprove that if \(A, B, \text{and } C\) are nonempty sets and \(A \times B = A \times C\text{, then } B = C\). +The definition of the cartesian product is that \(A \times B = \{(a,b) | a \in A, b \in B\}\). If \(A \times B = A \times C\), then \(\forall a \in A, b \in B, \exists c \in C, (a,b) = (a,c)\). Since the first element of the ordered pair is the same, \(b = c\). Thus \(B = C\).\\ + +We can also conduct a proof by contradiction, given \(A \times B = A \times C\) we assume \(B \neq C\). We can assume that \(\exists x, x \in B, x \notin C\). We can also assume that since \(A\) is nonempty, there exists some value \(y\) in \(A\). According to the definition of the cartesian product, \((y,x) \in A \times B\). Since \(A \times B = A \times C\), \((y,x) \in A \times C\). However, since \(x \notin C\), \((y,x) \notin A \times C\). This is a contradiction, therefore \(B = C\). + \end{document} \ No newline at end of file diff --git a/hw/hw4/images/venn.png b/hw/hw4/images/venn.png new file mode 100644 index 0000000000000000000000000000000000000000..519caaf512702902f9c800d92023fb55dc0c2184 GIT binary patch literal 42829 zcmd?P`9IX_|35yI94Su6DV0!%I7*gmAykUUQd!2nBxDZ_*;S;nl_fh#wi(RWmoY{Z z*^>r?A=$^i%wUX}@ALVb^Ll^&i0?VKe&BgskLCWjKbGsdyuNFoef03@!!Q`^sLt)1 z_hGO@*I=*%zy3G`{KoZP%nA&~7U80)c~?hM^Xy|EFDDmwM;PqicOmc8Zns`L^VQbq z&Yj{bW;ef_(?nE6zqvAa;qTlVMaL}=GWA>B=~bc<QgP4SK7Wq?ihL1WS;kZL{Ok_f zZ}RU?^Z%S@NJ~_`TK`oiyt<;cGR}4b)hS;o&Gx9l@!@?P*%aI3FVKI8gr9r$M@M%_ zpU(Z$*;9)`>P_Qc1Kw}fYiQ#wWKR9!d(BtkY4%0ni(H0j)~335QnF+6PUh2=58Qug z_*i@I#`<>HKXcZKb$80XqAU46&}_lpR8ju7%kNa=LXz~fm8r6q_aO4<V7+*l{ihFT zN#Cs5fA5=p`Ad|1`}%?7#`fn7PzG;aA6XVxql?7Z{*!W}?C?qD6L()6BCfw+eqQzL z`Tg@##@{;Z&-h0fCl5TwUc<VKPPEsM)Y<5L&B^1GA&cfE@r9nU6RPZoYMw<emK7|q zrKmQq8Cfen)ZQ5Q`^~Li`!P)tyhp~_ClzXId({nu!zB^PO>8_o&({BPL<cph{u~{} 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