From 9693b87fe7e83f51ae0dd29478e3cbde594475bc Mon Sep 17 00:00:00 2001
From: thomasabishop <tactonbishop@gmail.com>
Date: Mon, 25 Apr 2022 08:28:09 +0100
Subject: [PATCH] Last Sync: 2022-04-25 08:28:09

---
 .../Prealgebra/Add_Subtract_Fractions.md      |  92 +++++++++++++++++-
 .../Prealgebra/Least_common_denominator.md    |   0
 pdf/Algorithmic_complexity.pdf                | Bin 549310 -> 549310 bytes
 pdf/Arrays.pdf                                | Bin 83599 -> 83599 bytes
 pdf/Assembly_programs.pdf                     | Bin 58049 -> 58049 bytes
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 pdf/Axioms_of_set_theory.pdf                  | Bin 66749 -> 66749 bytes
 pdf/Basic_database_concepts.pdf               | Bin 99218 -> 99218 bytes
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 pdf/Bits,_bytes,_nibbles.pdf                  | Bin 51186 -> 51186 bytes
 pdf/Compile_from_source.pdf                   | Bin 49795 -> 49795 bytes
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 delete mode 100644 Mathematics/Prealgebra/Least_common_denominator.md

diff --git a/Mathematics/Prealgebra/Add_Subtract_Fractions.md b/Mathematics/Prealgebra/Add_Subtract_Fractions.md
index 8234a53..ce45739 100644
--- a/Mathematics/Prealgebra/Add_Subtract_Fractions.md
+++ b/Mathematics/Prealgebra/Add_Subtract_Fractions.md
@@ -36,6 +36,94 @@ $$
 * Add/subtract
 * Reduce 
 
-Demonstration: $\frac{4}{9} + \frac{1}{6}$ :
 
-The lowest common denominator is **the smallest number divisible by both of the denominators of the fractions without remainder**.
+
+### Lowest common denominator and lowest common multiple
+
+Given the symmetry between [factors and divisors](/Mathematics/Prealgebra/Factors_and_divisors.md) these properties are related. Note however that the LCM is more generic: it applies to any set of numbers not just fractions. Whereas the LCD is explicitly to do with fractions (hence 'denominator').
+
+
+* For two fractions $a, b$ (or a set), the LCD is the smallest number divisble by both the denominator of $a$ and the denominator of $b$ (or each member of the set).
+
+
+* For two fractions $a, b$ (or a set), the LCM is the smallest number that is a multiple of the denominator of $a$ and the denominator of $b$ (or each member of the set).
+
+In order to find the LCM of the set $\{12, 16\}$ we list the multiples of both:
+$$
+12, 24, 36, 48 \\
+16, 32, 48
+$$
+
+Until we identify the smallest number common to both lists. In this case it is 48. Thus the LCM of 12 and 16 is 48.
+
+The relationship between LCM and LCD is that _the least common denominator is the least common multiple of the fractions' denomintors_.  
+
+
+
+### Demonstration: addition
+We can now use this to calculate the addition of two fractions without common denominators: $\frac{4}{9} + \frac{1}{6}$.
+
+First identify the common multiples of 9 and 6:
+
+$$
+9, 18, ... \\
+6, 12, 18, ...
+$$
+
+The least common multiple is 18.
+We then think: what do we need to multiply each denominator by to get 18?
+
+In the case of the first fraction ($\frac{4}{9}$) it is 2:
+
+$$
+    \frac{4}{9 \cdot 2}  = \frac{4}{18}
+$$
+
+But what we do to the denominator, we must also do to the numerator, hence: 
+
+$$
+    \frac{4 \cdot 2}{9 \cdot 2}  = \frac{8}{18}
+$$
+
+We then do the same to the second fraction ($\frac{1}{6}$). We need to multiply its denominator by 3 to get 18 and we apply this also to the numerator.
+
+$$
+    \frac{1 \cdot 3}{6 \cdot 3}  = \frac{3}{18}
+$$
+
+We now have two fractions that share a common denominator so we can sum:
+
+$$
+    \frac{8}{18} + \frac{3}{18} = \frac{11}{18}
+$$
+
+### Demonstration: subtraction
+
+Calculate: 
+
+$$
+    \frac{3}{5} - \frac{2}{3}
+$$
+
+Once again we need to find the least common denominator for the two fractions. We start by listing the common multiples for the two denominators 5 and 3:
+
+$$
+5, 10, 15, ... \\
+3, 6, 9, 12, 15,...
+$$
+
+The lowest common multiple is 15. From the first fraction we get 15 by multiplying by 3. With the second fraction we get 15 by multiplying by 5. Thus:
+
+$$
+    \frac{3 \cdot 3}{5 \cdot 3}  = \frac{9}{15}
+$$
+
+$$
+    \frac{2 \cdot 5}{3 \cdot 5}  = \frac{10}{15}
+$$
+
+We can now carry out the subtraction:
+
+$$
+    \frac{9}{15}  -  \frac{10}{15} = -\frac{1}{15}
+$$
diff --git a/Mathematics/Prealgebra/Least_common_denominator.md b/Mathematics/Prealgebra/Least_common_denominator.md
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diff --git a/pdf/Algorithmic_complexity.pdf b/pdf/Algorithmic_complexity.pdf
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diff --git a/pdf/Bits,_bytes,_nibbles.pdf b/pdf/Bits,_bytes,_nibbles.pdf
index ccb4b97b4ebd83c929eb9226dfba1427bc066ebd..5aa1beb4cdfbe9959a9c7ebdf71d5c36b7b3a390 100644
GIT binary patch
delta 125
zcmey=&-|&MdBH(-3nL2y18oCCbpr!+O)h=k{1lhOl2io^7b_zJBO?P7BU1yAlFjM|
x53o8Lm|0jDnHU(Fo4Og98yQ=g7`hmlJ6pOrySbP<89Ex;DcBHFGFj{R2>@N8Ak_c>

delta 125
zcmey=&-|&MdBH(-a}y&I18oCCbpr!+O)h=k{1lhOl2io^7b_zJBO?P7BU1yAlFjM|
w53o9$TRNILIa|1zI$D@o8o3!byI2~V85tS_Ip$_=7EX2wHiVQ+);fLy0CD>v9RL6T

diff --git a/pdf/Compile_from_source.pdf b/pdf/Compile_from_source.pdf
index 50b17b3b35eef8172db39315e0a762bfdf3729eb..8fabdd3dd039165d0988a88030a67a9764cf07d6 100644
GIT binary patch
delta 100
zcmZo}Wo~X|-VnHl#lpzkcyj0-c`##h#U3$MXG23bb5m1u6BA=gGiNt9a}x_UOQ49G
UtD~hUkhHK<upy*m^2LLj0EEXIy8r+H

delta 100
zcmZo}Wo~X|-VnHl#oWZmaB}D#c`##h#U3$MXH#c4M^|$*V<Q7IGdE)kV-o{IH*-@{
XClgCU3s+ZjXFCNOLP{oIJh%w}cvKrr

diff --git a/pdf/Conditional_Elimination.pdf b/pdf/Conditional_Elimination.pdf
index ddec612c99346ac2d3278ae5c56bf4575ed0e301..da107c33af9798bb7b01cbbac74060c08e6532dc 100644
GIT binary patch
delta 102
zcmaEGk?FxjrVU>gvRW9K8(U8PwNMVi+AO>1IIFX{p^=5DiKB^&p{tR(p`n|BnUjf=
ZqnWvZxw(ObqnnwXf(;=hlhs%31OSta9cKUl

delta 102
zcmaEGk?FxjrVU>gvYMM18Cp#KwNMVi+AO>1IIFXxtGSt}vx}jziJ`Nhsgbjjfu*CP
ZlcTA#qq&olsi~=*f(;=hlhs%31OTX49rpkL

diff --git a/pdf/Conditional_Introduction.pdf b/pdf/Conditional_Introduction.pdf
index 85a1797df9021f10b077b01b6a30b51f57f80491..0e0c2ee39f117ee376cd4e2bcd9fc3501b86e3d7 100644
GIT binary patch
delta 107
zcmeA=!qRnwrC|%>PDxe^BXbjj>H8!Z<si)M4<#Ad*`3XuOkE8vjZIBWOk9A3tD~!>
cp`oj>rGbT|shN?Jk)47KAtlqb6&ddX0RLAS#{d8T

delta 107
zcmeA=!qRnwrC|%>PDxgC6C*>*>H8!Z<si)M4<#Ad*_|yc98KI@O%0tbUCk|BjNM#J
doK0L^Tun?3T%AqboSp3yYzQftuC2&;9{?%n8}|SJ

diff --git a/pdf/Conjunction_Elimination.pdf b/pdf/Conjunction_Elimination.pdf
index 6478e4cf7058d11ab427da29a7ffc98b152e5dce..004fb4d3be6ed9795046fa55607641834e28dbe0 100644
GIT binary patch
delta 100
zcmX^6nd$6jrVXn$u~-<Hn@nD}NgmADd~p*ao3o{xvx%ddqnoLdsjH)tiL0}bnW2-T
YxsjQ(v8koAp@E%(4Iw3yMR)E10Bq|Wb^rhX

delta 100
zcmX^6nd$6jrVXn$v6!0}8BJcdNgmADd~p*ao3pWtg@v1;n}Lg~k)?s9v5}Lhg{7s7
YsfDSTiIJhBnVX%04Iw3yMR)E1085b^$^ZZW

diff --git a/pdf/Conjunction_Introduction.pdf b/pdf/Conjunction_Introduction.pdf
index d9ef8c20fbd3a227826ce7c48a45cd398a6712d2..a99ce7568bd332bf5ff8a3c2b182be02dee61236 100644
GIT binary patch
delta 102
zcmbR6jA_C%rVY+(SS^gqO$;Y{tdWDTHfOI{#p>*4<_g3nj;03YhDI)i&W5h82BvO?
XM&`!GCg!dtPId}5gp^E<*t7!x5Y8NC

delta 102
zcmbR6jA_C%rVY+(Sj|n0j0`4wtdWDTHfOI{#p-Ni>f~nRXzc9f>Sk_i>gMKbX6R;M
YY-HhV?q+IiU}0#dU_(gB<cLi>02N>yi~s-t

diff --git a/pdf/Defining_a_computer.pdf b/pdf/Defining_a_computer.pdf
index f02299f43ac0f90937a8df77f02da403e07e70e8..9ce2122764c2f609bd154be19425950437cafe56 100644
GIT binary patch
delta 128
zcmbPtS!dp5orV_17N!>F7M2#)7Pc1l7LFFq7OpMaA~LKNMivGJ(<NoN<si)MHZt5_
wxt$%IEKJOtUCj)QOifK)oE%LoTujZ3U7ZZf-JF4xnVo_SAtl>OG`O`G0lCB=P5=M^

delta 128
zcmbPtS!dp5orV_17N!>F7M2#)7Pc1l7LFFq7OpMaA~LMzCPpTP(<NoN<si)MHZt5_
yxt-n2O$-d3jEzjp%*-qe-7L)voGc6-UENG94P2cpoGk1VYzQgYUZTOR#RveUx*y^I

diff --git a/pdf/Design_features_of_language.pdf b/pdf/Design_features_of_language.pdf
index 0fdaea323bcdd1ff029dee9a74c18e650753072a..af4884ebc1c7da5e6bb30d5f9871c4fe4a04d400 100644
GIT binary patch
delta 125
zcmaEImHEk4<_!)H)Gdt6jZL%-4Al(`)HS*Eee+XX5=&AQG+eBV42+BnOpHtoKuR{J
yKZxdZGIKI?F>tgnG%zzXH#9Xeaj|r^bT)7@GcYxGaW!<YQ?PL*pky-3s|NtIj3KK4

delta 125
zcmaEImHEk4<_!)H)Xhzd3{A8R4Al(`)HS*Eee+XX5=&AQG+eBV42+BnOpHtoKuR{J
yKZxdZGPiU#H8nPOa&dBTbaOFub#gOxadb5|aCI>>cQH4%Q?PL*pky-3s|Ntt;3A^{

diff --git a/pdf/Disjunction_Elimination.pdf b/pdf/Disjunction_Elimination.pdf
index 52a34f63773b524e08b76c7a824a1a1126390731..827cf95d0b011e79815e75d4840c2bb4c20aa22c 100644
GIT binary patch
delta 107
zcmex=jP?I9)`l&Nd!kt_jLc1pryq=Fl!Gw0KZ#~cXLB}lbTf1`bTl?LaWOG5voJF>
dHFk1wbagawHF35yax<_~upy*mdS4RbM*!9Y9t!{f

delta 107
zcmex=jP?I9)`l&Nd!kv*O^l3;rXP%El!Gw0KZ#~cXLEKjFflbYb#XIxadR~@a56V{
db#-zyGBYr9G&3?aGBdJMupy*mdS4RbM*!5J9qj-B

diff --git a/pdf/Disjunction_Introduction.pdf b/pdf/Disjunction_Introduction.pdf
index e1a50d2551047869744e90eea15267a328d2df3e..26d10c7f119cbfbf2f94f04bf3198de299ca25d1 100644
GIT binary patch
delta 100
zcmX@IiuuSY<_$;Auvi$Gn@m1&Mjp)A{Q8U?yR)H@k%^&&p|PW@qlt@?qk*NPi<y&!
YnX!S3k(-gTtD&8O4Iw3ybFSO~09HpGz5oCK

delta 100
zcmX@IiuuSY<_$;Au$Y?|8BIQMMjp)A{Q8U?yR(a<le3wnft#DDqnWXRp_!4Hg_((=
Yg^8KDiHot3rG=e>4Iw3ybFSO~08w@wU;qFB

diff --git a/pdf/HTTP_request_types.pdf b/pdf/HTTP_request_types.pdf
index 62f81e8c4b0288900a4e1e8cf72e0e1cc0ba983c..49c680c3a9b74db68853aaf2de68a589a1083947 100644
GIT binary patch
delta 102
zcmZ4cmTB!<rVS_8v050J8yihNw@wbi+Wc|dJT_-1Hw!~^OCt*-GZO<t6AL3ZCj&<(
ZS4Tr9LrWJ+6Gsa>1sg(2CSTfm8vv?+9>)Lx

delta 102
zcmZ4cmTB!<rVS_8v6`D085&MLw@wbi+Wc|dJT_-H6E{OgOD6*Z7YlPI7eh-IV@p>f
ZQzHXcLqk(%BPTaI1sg(2CSTfm8vwDl9_s)A

diff --git a/pdf/Indeterminacy.pdf b/pdf/Indeterminacy.pdf
index 6082265cbbdad548e74c06f11240094f82ed2ce5..ea127ab316382485ac33d981436718294f45a282 100644
GIT binary patch
delta 106
zcmbRCo^9HDwuUW?7He26jLc1@+pJ-f2eY=vuVH+{>1<?Z;bv&)YHDU;>}c+6VrFJ$
dZsKHN=4|Zd;%sVaVP>abLrBT?zHN*gi~w`299;ka

delta 106
zcmbRCo^9HDwuUW?7He3{O^l4D+pJ-f2eY=vuVH+{>Fj9kXk_f>>SEw*Zt7%iX>4ZV
dXyWQ@YG7&MYG`0!;Ap2{LrBT?zHN*gi~xPK9C-i$

diff --git a/pdf/Law_of_the_Excluded_Middle.pdf b/pdf/Law_of_the_Excluded_Middle.pdf
index 6075aba93e58fbf1e2d35bfb14335616d4dcbb0c..9cdc28357f0f5fff7b732b60e6711a909c63bd8d 100644
GIT binary patch
delta 100
zcmZ2BjcM65rVXYoEEY!QCX+2&<iU*1F)gQ9oz30MEDQ~uEX>Tzjhx&J3@y#g%nb~T
Xj17z&O^uw5jqDU`2q~GI*|iq{ZWJ0j

delta 100
zcmZ2BjcM65rVXYoEaoOgMw2aD<iU*1F)gQ9oy{yQjg5>9jh&rc+|1k@EzC?^3@j{N
XU7ZXZUEN$Qoa_{A2q~GI*|iq{koX&W

diff --git a/pdf/Logic_circuits.pdf b/pdf/Logic_circuits.pdf
index 316c97d81ee8e9d2a94d5edf1d76cf2d62f8d6e5..e9a19e6eaeae24bde3f6b3b8c89b2b3eb7d83d37 100644
GIT binary patch
delta 127
zcmex<p7Z;8&W0AoElg*lSuBh!45nX<W|9Z9wttCcs^$tbb}?`?b9AvZbT)D{wlp`g
xFmrWub~QA2bv3ngHgR;cQ?MbZB$i9xCp9m<BtyZ-*l;?d8<XbtmuXCVi~yK)BX9r!

delta 127
zcmex<p7Z;8&W0AoElg*lS<Fq0Or~FqW|9Z9wttCcs^$tbb8~fZHFPpIF)?(sFf}zW
xcXc+iG%$2EGB7bWaWgcsQ?MbZB$i9xCp9m<BtyZ-*l;?d8<XbtmuXCVi~xEwBFF#$

diff --git a/pdf/Logic_gates.pdf b/pdf/Logic_gates.pdf
index fc7c736a79e19dc70a27c09df46a45d9d7cc2576..2b98a1671a9548d2fe63f992fb1e055c0390a6ac 100644
GIT binary patch
delta 118
zcmaFyM&iXAiG~)&7N!>F7M2#)Eo{XatQJNV21e5>G}z=I%<Z!^*qphY4P6Z_j9gtU
oO&u*<44j;doJ@=zjf@SP3@seZ+>Fd!?G$VXDcP=Q!sgBd0Lxk)MgRZ+

delta 118
zcmaFyM&iXAiG~)&7N!>F7M2#)Eo{XatmY<0CdSh%G}z=I%<Z!^*qphYot!N#+*}>a
ooh%$J94$>v94$>OEes3|4P1;ZoDE$K>=bMWDcP=Q!sgBd0MvCJQ~&?~

diff --git a/pdf/Logical_possibility_and_necessity.pdf b/pdf/Logical_possibility_and_necessity.pdf
index 20b6abeaeda716f9d1c52a17e7a3a93d6ce518d5..dbed976fe24f936166b84b2035c85035ac2f9a60 100644
GIT binary patch
delta 107
zcmdnJpMCd!_J%Etfmc{9jLc2Ur-xl(l!Gw0S6*R!&*@}jYHDC=<Yeq>>S*rb<ZNl`
eYUpZc;%s4JZ02g@W^7`oU_(I3_Eq;71sMU(cpb0+

delta 107
zcmdnJpMCd!_J%Etfmc|~O^l4priWc&l!Gw0S6*R!&*@}tWaeV&;^gFLX5nV)Xk=(%
e>}F}?Z0Ty@W^C%}WMN{bU_(I3_Eq;71sMU|Rvs7t

diff --git a/pdf/Lovelace_quote.pdf b/pdf/Lovelace_quote.pdf
index d2647ddf44658c9b28dd124aef60239e47766d87..d81361c159a54149b80f4554e78ca54ca0e0caa3 100644
GIT binary patch
delta 100
zcmbRBgK^dm#tlYgEEYx<29wRo<iU*1k!5#ToSn^_EFI0woeUifEZkfzE!@nUjE&74
XjZF;PEDT)TOzaeF2q~FdS-lVd#zGsV

delta 100
zcmbRBgK^dm#tlYgEaoOgCX>y|<iU*1k!5#ToJ}kZ%uQX4++3X8oZVd9%p8r)-CQk<
XTr6FUO-&75jO-L_2q~FdS-lVd(48Ca

diff --git a/pdf/Morphology.pdf b/pdf/Morphology.pdf
index a1b1759f114537bc65fbb346aead7a48d70c38b4..442a666a571425285b2ca8917bea9fcd3c677a78 100644
GIT binary patch
delta 107
zcmex=o8|v)mWC~ivsGCwjLeNqr!P=tl!Gw0A68{lXLokCbT)T#GBGi+Ff(;Ba&dAs
dv2-yob#ieuGH`TuG%>bQupy*mx~?wcB>;uP9DV=*

delta 107
zcmex=o8|v)mWC~ivsGEmO^ggprY}%sl!Gw0A68{lXLmMmHgt4!Hghv`HZwLic62i}
daI$c9adt7ZurM$-vox?%upy*mx~?wcB>;QJ96kU5

diff --git a/pdf/Negation_Elimination.pdf b/pdf/Negation_Elimination.pdf
index 6731a89e898c867f1672445763d572902ff28114..45f6fa46aabf9478cbea1512320350c8c76eab7e 100644
GIT binary patch
delta 102
zcmdmUn`zH&rVU@0v050Jn^;W#wM-7e+AO<#EvvJUqp_otft#tTvxS+tiMg4DqmzlV
ZlbgA*rGcZVxrvdTf(;=hlX=&!2LO_A9eDr%

delta 102
zcmdmUn`zH&rVU@0v6`D08JSQ1wM-7e+AO<#EvvJErMbD2nWK@VnW341iJ_B$v4xAN
Zfu*T|n~9^jshg#pf(;=hlX=&!2LN&H9Mu2-

diff --git a/pdf/Negation_Introduction.pdf b/pdf/Negation_Introduction.pdf
index ffaece5cefd727905d959ecc7ec498ede8e70ad1..ff3b905151841ccf313ada9fe4a9b23e9948998e 100644
GIT binary patch
delta 99
zcmX?pkLmb5rVYF+SuBjqO(qMjlm|05o2*o2b#ilYG&40eadvbzHZgHAH+FS&b}}(H
Xa&mKXG;(!yGPYB&A)sXP>~(7Zq&6Gq

delta 99
zcmX?pkLmb5rVYF+S<Fq0j3x`Nlm|05o2*o2bux9eFgA6za5FP7b#-wvH8VGJF>*FB
Wu`n_*aW(?V*(ul%P%?S;x-|fYY8w;)

diff --git a/pdf/Object_language_and_meta-language.pdf b/pdf/Object_language_and_meta-language.pdf
index f4d0c3acb046dc4399a8fb906c8c69e1bf560604..7e96509fdaf3282ba46b2634a83562763d561239 100644
GIT binary patch
delta 104
zcmeBZ=IChVXxPGN`;EoI$lPSQ<2OcmFl&3tH^yt6PR?$oCdMX?j!s4vMurxShUO;5
c#%_)lrlwA=j+RE|b_zCb1e9#8Vd7x~04&5CasU7T

delta 104
zcmeBZ=IChVXxPGN`;Eoi#K>s6<2OcmFl&3tH^yt6PL7rqKxAxUYG!QYV(4OIU}<h(
a<Y?;XY-ZtTVQyrnVB<zW$<`Vs9!3BiN*fvg

diff --git a/pdf/Package_management_in_Arch.pdf b/pdf/Package_management_in_Arch.pdf
index cb6c27a48431f767f06357bf9d43c741c11c3fbf..56b0cbbe71f5b38b52ec1c9f157af9f8a2da4f7e 100644
GIT binary patch
delta 131
zcmbQbnq}&0mWC~ieY_eLM&`z5+6IQ|1_tVyT>8HGDK3d6sR|k{Rz?O!Mg}HErUqao
z+qdyDo@H}3ayB=yFfnp<Gc|QKF*CGu1<E^_x>*`InY&sV85r0p*bq`O{g?>jc>uK>
BAn*VH

delta 131
zcmbQbnq}&0mWC~ieY_gxCPs#)+6IQ|1_tVyT>8HGDK3d6sR|k{Rz?O!Mg}HErUqao
z+qdyDo@H}301BD7npwI!89BN-S(rPzxH=lTI2pS-S(-SRo0;1w*bq`O{g?>jc>u*}
BA&vk5

diff --git a/pdf/Primary_key.pdf b/pdf/Primary_key.pdf
index f88ba843d1910b8d0a46dbe8b7a8bf82229bf445..b8d561fb353522c5b31c923adc3ce2f42bb7ff08 100644
GIT binary patch
delta 106
zcmZ4Tm1W6SmWC~itBqJKjLeOVr*AM~l!Gw0Up8Xg&*tRpW@hSaXl!m^Y-Vg=Ze(QO
dY+-I<;b>`YU}WU#WNB=tU_(I3^edK(7XVxR9D4u&

delta 106
zcmZ4Tm1W6SmWC~itBqLAO^ghUrf)D}l!Gw0Up8Xg&*tP}U})fM?r7*}>T2ffX6)=_
dV&G!nXkqATY;I=kWNd7wU_(I3^edK(7XWRj9QgnM

diff --git a/pdf/Queue.pdf b/pdf/Queue.pdf
index ae979cc5bbd619361fb130240a8f6d3aada75cb0..94608b6e02181704d42ffc2f5b0d2707e63b4ec4 100644
GIT binary patch
delta 2588
zcmV+%3gh*Gw+Vr_39xDg4`E?2W@BSDW->QpH#uZAgLwvrc?JQuc?JS`NCh}DH#0Do
z_f7&H1voM{GccDLPXhQq+j84D5`EWK;BD=ll5imifKyXbWyf)n$;C;Wd*TO1qGcws
z=#W$<nXjMI07;Ru<;y<oKJ1jsAP96fx(^PQQBVRQe-L8Y#0-DJjg(+UMGVbW#!Nvk
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zfyKVwO3{Agxf?ub{E4U~f0X2fqOTd!)ClFeHPrl<u9ff{2_;Dk`n+_gQ{Qd>RnLaK
zRBG$zG_+t9TR}{*%C~w_8%JveCo7WhgeJMj5EBX6q|qjv5^0+Lrp@a%*29WwR-3m^
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delta 2588
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diff --git a/pdf/RESTful_APIs.pdf b/pdf/RESTful_APIs.pdf
index bbf94cff2a4200c472bc085c53a2171acc24c469..5cedef829a950a8081346dd8465a1d95393227c4 100644
GIT binary patch
delta 108
zcmbQRi*3R#wuUW?k0!8M7?~RzPk%mvQ4Ye~&NY#79hb9{i;0=Bv!k1%p|OdXi>tAP
fg^{s~nW3SDg|U%|o1vMVf(;=h+keky6lDYevZ@^y

delta 108
zcmbQRi*3R#wuUW?k0!91n;01yO@BUtQ4Ye~&NY#79hbACqp6XrrHiGri@B+plevq9
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diff --git a/pdf/Recursion.pdf b/pdf/Recursion.pdf
index 2dc0cba5994d3e95c4b03899736042d550f9067f..e84d0b7cea23cc775fdab5fc35992d140617fc78 100644
GIT binary patch
delta 110
zcmZ4RfNQ}6u7(!IEld>)SuKprjSZ&PE@YB}Ft;yQ$aI>=+0f0|$k@`<*udD#$<)x;
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delta 110
zcmZ4RfNQ}6u7(!IEld>)S<Ow13@oSDE@YB}Ft;yQ$aI>=*}&P{*~!Go)!fa|#L(5r
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diff --git a/pdf/Reiteration.pdf b/pdf/Reiteration.pdf
index 000bf73cda6559fedcd3f9b9a4b6261ad490616e..e13810cf4e0103c9c673eecd0bc1ec88be169c87 100644
GIT binary patch
delta 100
zcmeBMz}&Zhc?0K377HVDlgYd%<-v^21}C4eI-41qyE;3XxEdQ9TbP;~I$1cnIT;w3
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delta 100
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diff --git a/pdf/SQL_syntax.pdf b/pdf/SQL_syntax.pdf
index ce550da12c351abdd97ae75b983f2f88792e63e0..3ac26b97e62872494ed7a76a5aba916531cbc45c 100644
GIT binary patch
delta 122
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delta 122
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diff --git a/pdf/Semantic_versioning.pdf b/pdf/Semantic_versioning.pdf
index ce6a23994b100e72a8f83cbfd194b5c7d326eca9..bc15df630027fc34ce56f4055dd88aba77685364 100644
GIT binary patch
delta 106
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delta 106
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diff --git a/pdf/Soundness.pdf b/pdf/Soundness.pdf
index 72f70527aedb6a5dd6aa1e7d45b73820d4915fc3..d2e6e8324f3ce3f52c5e371b9cfdca435c12c6d9 100644
GIT binary patch
delta 105
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delta 105
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diff --git a/pdf/Stacks.pdf b/pdf/Stacks.pdf
index 939f032cacccb2f366c4ec464221b45ee4907e65..4f9e172d41673f19c26db73602c242eea6e002da 100644
GIT binary patch
delta 2619
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zk<QX`4d6=ZcHM{t+&7fzD4q`D#RArVAG_$|r&(TmzMKQ*JX=m4`uJIwuWY{&=5nnL
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z>*E(}Ys0rS9ipO!5*DrtGv<P+&h49!r)Y<e1pVEX{rDMp*EE~-LL{lbs9ME`#*IBy
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dMnpn6L`5@0FgQatK{7@`I5symw>VG&P62HY3*`U+

delta 2619
zcmV-B3dHq>qX~wi39y<15H&Y4Wiw_mV`MosV>4tiVY8|N{sIs+H!@{2W-()AIW=Q5
zWHDibxCMu}1p&9X1p-JtC^s}RF)%AIF()uECn*XqO>bmGVRU66C`39kFfuYQG%__X
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zJ;|RXx$W(9M8VfR5kk)sT(PlSu!E`ZidNinIaILx50YgZWW_<UTJj-A)PrPsG?Wk}
z50Mo@4k08TCd=87!3MI!1DcucdH@7-v8u-a?y5Wn>u8ntG4Q`|0IAm<K+6Bj0i-&7
z?Wrx~Gd5IzF!mv`%Gsf_JC{^LN|!{~-K5<@Vu$n=(cLAukKE2Ka8$rbg_!lvA@5_)
zqmQsh!oU5<7(2$~BU22o1tujXF(xBS{%8xNwbZI6ZWpGt%p8-X7RcLD<GQ-s(9j_8
zU6wVBvM%Pe>QyJ-_R!V3YRfRWt5qkphA(xoW6uSDCg+%(Ve-1an@zFF=FJASQ4MR}
zuD+|yNo(T9J9&L{UhZqNyuGsxP4kVKyiR=a+a5#ZrlF^`p|g6_`IZT6CM0d-_6(&q
zhq|h%ns1-<#%?b)@eO;%(3@jib^(vNsGB*QZqT@>lj#=ub;~_N{oV&H^sP_H-~Uhk
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zkP!z{WaRaQ32C(k147=X-~diVAZw!t`Z+F!_?X0-DIFZjEwhk!K%l_KB#0GGh1%is
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ziK<2!oMv=nj6`_l$ct1GLzzJttPRqCRc(Z6sEMSlBpTV!u5{#Mk<T|$zAZ2m@;(Iy
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zBELPfYD{2)nJTDPIo<TEYP`=X!~B0!3aw3DC^xl2qYaLGKk}%8TuYxoKUB<r2L|aS
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zY&r&kX*?;23X&)Os5(&Akb!V`a<dj3O~Pc+<6?wIR^veSS-kiqNhc3g5eLM!0gs?`
zUdGdObUdF-6T*7*x=dzo!O^3C9(`AJf>jGT9^xF2f=Be2p3*aVPA}+ddP%S74UOp?
z9nmpO=slg#2b$79=#<XroMv=Eb6U`nF6kEv2LO-zEQPU*ZXc!}qvO@d^)K%)YC729
zDw!;&alV5QI13L%#PCTmG9p@xur?p8VT0|U!nB!?yvxK1eeSuqqSy3)4gDRMi1R%A
znxiz3atDX?qy%n8*)*F&I-A86Pxom>)u$F&**pQl{L{s0a$kmAP_{_&G#k^;^z$++
z6W{7{p*3(^KIBP4%deSd`z!r(jdA-F#*g2;y1Be=F&^02LAZ7m;nEo-T(?~?>bhXG
z&jklut(ezl5A$*5aVu|sL;6~tM*r$q9zDhK_cv$no?n9HFSB`eQG)4v)}B^mAQ|qT
z3g2TyExTFA)qJtGTX4I>-<;fOHQJoe<nuC~(g{8#PI7NQTEDXWYuj$ipJMm3%U^EZ
zzNqYebctyWhI5Fs_@;<)(-^L!&V>77W4I7cUue%^c|nlL<5BQ`e4ncp8D#%R%DNiz
zp`k_9T1a|)l_8l$GE2j<?Jg{EE&5lIXC0p9Q+WRM*|YPvZyTQXF081c*?IqJ(|lkc
zYb~g~q3nXb0+vHC-$UwVB%8E+@!{LzA(?Ms+sCRRMMT50;DO<3Ts&+ETb-a~w`>#7
z=c8;4y1?f2{yxEfcQHvpJ_rX<748Y!fpOa`!-iHFP{FNcJ>bE)^`H@uhX%KSmD#mV
zz@0~T9l~~o+W|y3*d1X{0GFWWa92Yw;9d}zg{_h}5%vkVf222XCq#dTJMuQyaIe;(
zn?$A+?l*VYPn~`Cb1VM&*bk0cFBt+MxwUL=4<l5!wzHal*k%?X+M8;RyWMnt@Vez)
z>5Eti?sgIviGO6h^X{gzX7H7&e=L8N=0%CV3M}{NMO+oQ0G&7KxO^yxsh;4kEWy{S
zzT!>x+Z^U>44$-O``d?e&uzyVRJRUQxLr`i<51nxq3+(VyGSj2gjCTWRgWW$o{n_q
z{<;QKYyjndZJ<1=f$Hc{ppHEq=<W@14XIj3s#u3o-B8LON6Md$bmx|`26SUV;jR5v
zFV*gGpyK;3sE(UT({Rexq3R8&@;h=WqZUxv@>26SPPK^>-(D8@`f>@V;O6kxbX=_7
zx>v7W!wbW;82ox!PSZJd_0aHx4?h}0gHviK1g*M%?wP_ygrCIt#$)_Q#^<URpsoUZ
znI~WH!;&`E(37uvs%UplRk_kDe4VJ}rW)mHHD-?*)mF0wjncImxkn9esZn-|8d0lZ
zTh)Y~ovFh+J(#7V(`+=jF5|rHlkn>GC>|6gZ1ID%7{mjdpg13l;}Wa$<)~cd$)J3Q
z%RzR3S0wqDqyU-wtdHMX`R?Onmd!!be<ISyPpW+N0-OZjy}DLW^zk#SrY!qJn&2q(
zBAun>8o-s*?Ya>QxNj)aQ9K>Qiv_F!KX%c_PqV!Cd^rcqdA6K9^zpMUU)g>m%;j1g
zLQ`vX08iMpI@QPT!`2}0g46D{VMF`XoUEpQ;%hP(#q+`4GM$bGt9iN4v)KS@;m#-i
z*2gc{)`o9uIz&YcB`jPQX3Paso!d7dPtgt`3HrM&`|&gIu4y*ug-B9=QMHN>jT?Ka
zqT0tN$xDby^tU2uLdokzGC%hAlX{!_H|;egFbZXEWOH<KWnpa!Wo~3|VrmLGGaxXR
zpHBi-LqbMEFhfN(LNzuyGBr6yF*7+uLqs?@MnyP6H84RiHa;LcLPkO`Lq#=0H8wdi
dH91BxGdV>=L^wD`ML0q=FhMXjw>VG&P64Mx4VVA`

diff --git a/pdf/Syllogism.pdf b/pdf/Syllogism.pdf
index 0c63c18bb9636d9c613b3ce2bd2c08013e73048f..e2fd8f852a9c48f33d0e6d060b4889585bca8042 100644
GIT binary patch
delta 105
zcmZ4Wfo0VPmWC~iC-hk?jLc1^pV4QO2eY=n*Jo5?cQ$l1G_!PcHgtAzGIDcrb#Zbs
cG%<B`baFK_wRAIbF|$*!A*5t_ggN6S0A!^c&;S4c

delta 105
zcmZ4Wfo0VPmWC~iC-hm&O^l4EpV4QO2eY=n*Jo5?cXl;5GIVn>baFFrvvhW`G`BD}
cFmW|8H8gfLHF0w@cd=8jA*5t_ggN6S08;}TU;qFB

diff --git a/pdf/The_Pragmatic_Programmer_1999.pdf b/pdf/The_Pragmatic_Programmer_1999.pdf
index 0e1002092b73fbaa77fa3808225409de7e5ac4d5..26a9c5ed4d38f2aa923d7c6bac6e23422c0d21e7 100644
GIT binary patch
delta 108
zcmeypitG0(u7(!IEli7>SS*apO{Xt!Vv+~5wx4QZ`p55VX<}+(VdiS#WbEW*;N)m%
f?q*<OXlP_?YG&@_VruGWr(i=!$#(9^Oa)8;81o&z

delta 108
zcmeypitG0(u7(!IEli7>Sj<g~jHfSeVv+~5wx4QZ`p55VX5`}JXkz4Q<mzl`;^JiD
fWa(yTX>MrYYGh(&YHZ|cr(i=!$#(9^Oa)8;CR!d0

diff --git a/pdf/The_binary_number_system.pdf b/pdf/The_binary_number_system.pdf
index 4b9130bb391ac2538a67f74bcad3711e6aa5602a..8ebcfec0fc3e0f227b634304785b2ccdeef059e0 100644
GIT binary patch
delta 134
zcmZ4Wn`6~)j)oS-ElhR#8Wu(t2FBV3hUx|e>Y7~ozWFIGi6yBD8ZK5w21Z5(CPt<P
zU?tm^=`+RhIGbBox;h#<8ksp68aca|x|tiA7@JxcxmX$-n!7kTI@>AO5K^+;&4ww3
F5deLZA*cWV

delta 134
zcmZ4Wn`6~)j)oS-ElhR#8s;WOCMMbjhUx|e>Y7~ozWFIGi6yBD8ZK5w21Z5(CPt<P
zU?tm^=`+RhIJ+1)x)~X}np(J+nmSq<8<`lIyP6spyO~&6SQxvQSlTJr5K^+;&4ww3
F5de2pA!Yyo

diff --git a/pdf/Theorems_and_empty_sets.pdf b/pdf/Theorems_and_empty_sets.pdf
index 2e3a390f78c0f8feb5f5299efa1321c5031b86d2..2ca1aeb74add7dc2fba904e57436712d0401783b 100644
GIT binary patch
delta 105
zcmex;j^+0`mWC~iS#m5EM&_o|^W+%i!L02Q<rt5%I~!V<I+;6~TNoG^JDWJ0yIES8
c8oL=dS{k~V7`vL98`&w?5K=OIwi@FD03{V0&Hw-a

delta 105
zcmex;j^+0`mWC~iS#m7qCPv27^W+%i!L02Q<rt5%I~$w2xEQ#)85meN8oOD#7+V-P
c8k?A!n;DrqyBV6fn%ODX5K=OIwi@FD04hEk^Z)<=

diff --git a/pdf/Turing_machines.pdf b/pdf/Turing_machines.pdf
index 4108017d4ea807e7a2df02d86f27e4d7d52c2ae2..7a983f2fe3f15989116fd338e8f32328ab710f03 100644
GIT binary patch
delta 131
zcmaE`nC-!0wuUW?3)<B!j4TXHv<(c^4Gh#Zx%7SWQ(O{DQWZ2@tc(ndj0{YSObtLv
zwjXb2yvpHhVCZ6I>}YK2XzJqPXzFBS=;Y>R;pXaOW@Kn-?(Aq}r(i=!$@cpb7#SG>
DfEps)

delta 131
zcmaE`nC-!0wuUW?3)<DqO^i%Tv<(c^4Gh#Zx%7SWQ(O{DQWZ2@tc(ndj0{YSObtLv
zwjXb2yvpHhY2xDMWMb-M=xpF@Wa?_*Y+!6+YHZ@{V&>#*X=!d~r(i=!$@cpb7#SG>
Dc9tTD

diff --git a/pdf/User_management.pdf b/pdf/User_management.pdf
index b65b0dd32539863309fb58fe6e8f2e98ba750ec4..08939db7425346ca67692c31c640ff39ce075352 100644
GIT binary patch
delta 131
zcmeA>%+hz5rC|$WpQMI`k-4$Cwt=C#fq}Xvm%eX)ic4Zis)B}#m63swk%5VksR3BY
z_HB}k``MfwT@1_|%`Mzq4NP53EgYSkER2mz9nD>x+$>C-T^wEQ6l@47nZ84T@frZ^
C(IOWB

delta 131
zcmeA>%+hz5rC|$WpQMJliIJh1wt=C#fq}Xvm%eX)ic4Zis)B}#m63swk%5VksR3BY
z_HB}k``Mfgjm<4h%uQTOT#b#~Tnvm|Tn)?&EL;suO-zklEgfC#6l@47nZ84T@frZd
Cupun~

diff --git a/pdf/Von_Neumann_architecture.pdf b/pdf/Von_Neumann_architecture.pdf
index 55d8128b46e9b0bc7d5551db5c61ccd38d1c61ab..4171548e6275c30292441cb1b768ef967e4a084c 100644
GIT binary patch
delta 110
zcmdmZfp6mlzJ?aY7N#xCE;Cpxj4TYMd(2>#2eY<k&tR71b2c(GvM@1qadS3yu{1Gq
hv2b=XaB(v>b~JKzGj=v}wy;yMA*5vc@ny_ei~zsd9h(3E

delta 110
zcmdmZfp6mlzJ?aY7N#xCE;CrnO^i&Yd(2>#2eY<k&tR71b9Oa%v@kI-HgR$>GBR^D
hFf=f+urzZuax=9sF>|%BFtbyzA*5vc@ny_ei~y^39QyzO

diff --git a/pdf/Why_computers_use_binary.pdf b/pdf/Why_computers_use_binary.pdf
index d5d2c09333cc6b12230e47b273eeddf1be5dba95..30435ee508a3a31d68f5e636e046f9eb26eb979e 100644
GIT binary patch
delta 110
zcmZ4Vieu3$j)oS-Elk!LtQJNV2By;;G??Td%<ah<Op)BqmZqjI<}QxTj!vfL#zvOr
eh8Bh<rq1SOZWad4jzH2*!G@5M?f;FLlo<g?FB_Nu

delta 110
zcmZ4Vieu3$j)oS-Elk!LtmY<0CT7zeG??Td%<ah<Op)BqMn;B~PNqgCmaZ1gW`-6{
gP9_F!7A~$%PR5Q#j;5}rb_zCxlx+WR%%sc+07Z%$ssI20