fix formatting

hw/hw6/hw6.fdb_latexmk

@@ -1,7 +1,7 @@
 # Fdb version 4
-["pdflatex"] 1698093234 "/home/rushil/school/math/hw/hw6/hw6.tex" "hw6.pdf" "hw6" 1698093235 0
+["pdflatex"] 1698093499 "/home/rushil/school/math/hw/hw6/hw6.tex" "hw6.pdf" "hw6" 1698093500 0
   "/etc/texmf/web2c/texmf.cnf" 1693368362 475 c0e671620eb5563b2130f56340a5fde8 ""
-  "/home/rushil/school/math/hw/hw6/hw6.tex" 1698093234 10597 99b46c2c3b3f88a118fc653eb5889b88 ""
+  "/home/rushil/school/math/hw/hw6/hw6.tex" 1698093499 10607 17031e54ea748adbb730bc9254f89a7c ""
   "/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 +150,8 @@
   "/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 ""
-  "hw6.aux" 1698093235 113 e525f25688a88382d92e4852bbc94338 "pdflatex"
-  "hw6.tex" 1698093234 10597 99b46c2c3b3f88a118fc653eb5889b88 ""
+  "hw6.aux" 1698093500 113 e525f25688a88382d92e4852bbc94338 "pdflatex"
+  "hw6.tex" 1698093499 10607 17031e54ea748adbb730bc9254f89a7c ""
   (generated)
   "hw6.aux"
   "hw6.log"

hw/hw6/hw6.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)  23 OCT 2023 16:33
+This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/Debian) (preloaded format=pdflatex 2023.8.30)  23 OCT 2023 16:38
 entering extended mode
  restricted \write18 enabled.
  file:line:error style messages enabled.
@@ -479,17 +479,7 @@ LaTeX Font Info:    Trying to load font information for U+msb on input line 134.
 File: umsb.fd 2013/01/14 v3.01 AMS symbols B
 ) [1
 
-{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [4] [5] [6]
-Underfull \hbox (badness 10000) in paragraph at lines 252--254
-
- []
-
-[7] [8]
-Underfull \hbox (badness 10000) in paragraph at lines 272--274
-
- []
-
-[9] (./hw6.aux) ) 
+{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [4] [5] [6] [7] [8] [9] (./hw6.aux) ) 
 Here is how much of TeX's memory you used:
  15641 strings out of 476091
  318265 string characters out of 5794081
@@ -497,9 +487,9 @@ Here is how much of TeX's memory you used:
  35749 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,16n,104p,630b,420s stack positions out of 10000i,1000n,20000p,200000b,200000s
+ 102i,16n,104p,649b,420s 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/cmsy7.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 hw6.pdf (9 pages, 172285 bytes).
+Output written on hw6.pdf (9 pages, 172288 bytes).
 PDF statistics:
  112 PDF objects out of 1000 (max. 8388607)
  70 compressed objects within 1 object stream

hw/hw6/hw6.tex

@@ -249,7 +249,7 @@ from \(\AllReals\) to \(\AllReals\).
 \end{enumerate}
 \pagebreak
 
-\prob{36} If \(f\) and \(f \circ g\) are one-to-one, does it follow that \(g\) is one-to-one? Justify your answer.\\
+\prob{36} If \(f\) and \(f \circ g\) are one-to-one, does it follow that \(g\) is one-to-one? Justify your answer.
 \begin{proof}
     Yes, we will conduct a direct proof. Assuming \(g(a) = g(b)\),
     \begin{align*}
@@ -269,7 +269,9 @@ are functions from \(\AllReals\) to \(\AllReals\).
 \end{align*}
 \pagebreak
 
-\prob{74} Suppose that \(f\) is a function from \(A\) to \(B\), where \(A\) and \(B\) are finite sets with \(|A| = |B|\). Show that \(f\) is one-to-one if and only if it is onto.\\\\
+\prob{74} Suppose that \(f\) is a function from \(A\) to \(B\), where \(A\) and \(B\) are finite sets with \(|A| = |B|\). Show that \(f\) is one-to-one if and only if it is onto.
+\parskip=0.5em
+
 A function \(f: A \rightarrow B\) can map each element of \(A\) to a unique element of \(B\). Since \(|A| = |B|\), if \(f\) is one-to-one, then every element of \(A\) must be mapped to a different element of \(B\) and since \(|A| = |B|\), every element of \(B\) must be mapped to by an element of \(A\). Therefore, \(f\) is onto. In the other direction, in order for \(f\) to be onto it has to map every element of \(A\) to a distinct element in \(B\) since \(|A| = |B|\). Therefore, \(f\) is one-to-one. Since we have shown both directions, \(f\) is one-to-one if and only if it is onto.
 
 \pagebreak