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Matematik 1, 1968–69Børge Jessen
Forelæsningsnoter til Mat 1x
§0. Indledning mm.
§1. Matematisk logik.
§2. Mængdelære.
§3. Relationer. Afbildninger eller funktioner.
§4. Kompositioner.
§5. Grupper. Transformationer. Permutationer.
§6. Ringe og legemer.
§7. Reelle tal. Komplekse tal. Kvaternioner.
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== (f~cr1-~ v J z1 -[x., Y t -r~ r,]:;; fx; y1-x.2 ~2J z.z. +r~ ;z +X;.3't] z:,)
(x1)x2.)[{tor2)(zox.2il = 0)x2)(r,Zy-'/.2.~) ~~~" +Y.t~J =: ( x1 [.f, ~-~z~J -~[~L; -t-J$ z,] ) x,IJt~ r~z,] -t~ [y,-z,-Jt,;;J)
4f~~. ~.e_A,~~~~Icd~·jd-t:~~ .i4 dAÅf~~h ~ ~ 1:1 a.hlil~-~. ~~~ia-7Rt.~dt~~-4tt~~~~dtr~~
7ctbt (!,o)~cd~~~1J<.d~~~; dv71~G, ~:J~ 1/t t:d"~~.;cdde .. z. lid -4;-~4 ~ tt1>~L) =f=-(tJ,~) }i~ d ~~fJ~ atbt d~ (X1;X~); ,U,.~ J (tl0te~)(K11 ~)=(jtJ).J
t{. 11 .. s. t:t1x" - tt.:t K~ = 1 4.2 X.1 -t et1 x2. -= tJ;
,/w {et1~~~)+{o,a)~~ ~~ ~ 4t.-4st-ut;e: ~
~·~~~iU~~·14 ~ af ~ fl =f(l(,o) /xe:Ae] ttf ~~ ~· {t:~
f{ 1o) + {~~o) -= (x t-J) o)~ (x,tJ)· (J';~) = (X f> o). ~
~~,d~ R .t4 ~ ~ ~~·he~-'h.e.J ~ J afl-tt~ x M (x., o)~~ .htJ~ ~~ 4f (IR,+,•) -;.;.o {k+,·)1 -t.tlueu d{i,+,·)-k.,t4 a~~~~~~~~~ ~-%~ ~ 1. ~d (0,1);~~
~ ~t~ (-t1o) =--{t; o);~~ /øv d~~ -de-
~ Cx1,~)
0oxt)-= (K">~)+ 1 (X;.., O) ..
Jev,; ~- {x,;x3.)= ('j.0 tJ)-ri(j'2(0)) 4~ ~-:tt)~:~e;~'J.~ 1dd" ?A;~~~~~Ivvv~~:
JJ(!A,,f;~ -d~ct.t:U ~(C_,-rJ•)1 4~ ,trk~·~ ~ ~~ {!R) 'ti,)~~ ~ ta-4 ~ 4.~~~::~ d~· t~ ~~=-1_,~a;t~~6(
~~~~~~~et/~ ~ = ':<1 -r i z4 .) ~t-> ..<L G 7/(.
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- ~ -~ 'z. -r;;;- -= 2:.. ..,.. w,; ~w- = ~. /Q".
~~ x,___,. F ~ ctibs:. ~ ~~ æ(ft!:t4~. ~ CCJ-+J" J~~~ ~~.,; (/1()-l)·)~fl~~ ~~~;~-
4 ~ ,t~t:tJ/tu~~) fr~ ~~ ~ (C;+J-) t6 ~~)~h~~~at··~~,v~, 4du; ~ ~ ~~ af!U~ '7~ ~ ,___, ~~ 1kv /l~ ~ ~. Jiu G f 4~ 4t!~-~~/ ~ ~ (Cftt))z.-= 'f(t1J= rrf-1)= -~~ :1~ ~M~d~)4L~~ L~-i~~ ~ t:t)lwu~·,u-1,~~~~ re~)-= z ~ Cf{l)= -t. 1·cld fip~~~ p; /oY ~ 'K==-~ -t t Zz ~ d 'f(:Z) -= '?~)'t' f:P{C) tf{~) =:.
7., r r~;. 7 d d Cf ~ ~ t~ a.fkl~, 7 ttU:t ~~/t~~~·~ z=zt-rtJ~~-2) ~ rC-r.)= ptz,)-rtpltJrt:;J= ~-t~; 4-l~f-4~ ~~x:
~d~~~i;d :z-=~-tC4~-ui :'{~=== {Z:.,+t~)(i;1-~~2)·= :z./-r~~- Øvt~~ ~ /;-~ Ji:/~J"Z-z =rl~/--+~2. ~~~~~~ -~af x, ~~df,H ~+o -4; =,:r~-_. :;~~~;~4~7'4~~
~~~/ ~ 7/(2 ~ ~ tC ~Ø-t~ tå~~ ~~ ~= X1 -rt~2 ~~ ";~ (~>~). ~ ~ ~ ~/tAl~ ~~bt!t~/ ~ ~-~~ ~-~~~n;~t~~~; ft~ tal ~~~~et~~.;;~.?~ ~~4!i.l~~~n~ ~~t-r:t 4~.;~~~~-~~~~~~~ ~~~~.~~~~4~t-d
4~ ltt ~hk,~~ ~4~/at-c&./l.- ffl ~4 ~it:d X~·w-i~
J~-tw--/ ~ lz/-t-/ttrf., z ~~ ~ ~!~ ~
~~?U~~~~)~~~ ~~ }':Z./etn@+ tl~/~~9 = /~J {~e+tske) ..
~~e ~.e.t-.~1~~ ~x=o ~8~ ~lrt:,t ~ ?;,. X=f=() el/v fJ ~ ~ ~ iTr.; 4t ~~S .. !w-W ~-VI.d"~4~~~1k4 !J+tJO:l7i) ~eF, ~'i~~d~kt <: ~ ~.~\; ~ ~ ~:.: ~~~ /rN ~a~~~~ ~1=0/l~~~~ %~t~·~ [-n:-J 7t [; ø{d-~ ~~ 1 ~ ;4'1~· (~~~ ~ ~t:t4l~'rd'~;U,t~~ ~~i~ [4111[ .. )
%r ~ ::f=IJ ~ 6dtd- fir=- &IJ -t ts~e ~~~:z., ~ ~Jtl.i:l/4 ~~ ~~~~ V=~~/lz.J~1J, trd, l.t·tt4 4 ~ "; ~~~ ~~~ ~~~~'11h ~ ~ 141Å~ ~ ~
z ~ l ~1 (~$-r ts~rg) ..~ w- =/tu-l (~(f+ t.sU:. tf )Jl
f.t~~~ 'Zw- = J<z.IJ~q/ [ ~e~tp-s~~s~tp + t(&>~S~<f>+Sil.&tm.p)]
= /<//W/ [ ~(9+f) +l s14(8+tr)] ..
~~~~ fJct ~~~~le:~ af dl.~ icd ~dd·~ ai'sMrp-> ~FUt 4 ~ ~ ~~ ~ 4~ Jl?'>2-fo/ th:eld~ ~~d~- ~~~~(!Pli)~ q;f~ ~ff~ Pr ~='1" ~ ~~~~ ~ti~~~ ~l?h'-fdw ~~ ~~ ~-t~ ~~tnt~~~for ~ "; ~; fkt ~ ~ 'kVr4 (~ ~ 1878)$, d 4 'U.d~ ~ rr-ed4 f~ { 1t = u 4 ~ i4.L (>J. =-!J "1 ~~~ (>t='l) ~ /lvtu ~ "-1~~~/VVt: .r~~~~~ }:H t~~ 4f ~t~~~~~~af~
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rx,+x"t+~J +~~; +- c ro -ry.,t-r:~.:f +fJ~) = c~_,.~) -r (x,+~) ;, + ( ~ -t-r~.)J +r "1 + ~$) 1.. •
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211
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~t~ it=- 12.= -Æ2 = -1 e
t, ;
4> -1 l . -~ ~J
~ • l
• )
l< --f
--t
~ •
-J • t
-f
~~ ~ ,t = -lfi lt -=- i = - l-Æ tf = ~ = -ji.
k s~ h:~, d~~ ~ttf-1ii ~~ ~-> ~
(Ko +X1l-t-X:;.j +Xz4)·(~-t r., -t+ f~.i-t!Jf<) = ~t~ i t ~j-t?f:) ~ :<:o -:: ~ ~ - )(1 ~1 - XL j l. - .xj ~8
~1 = ~ V1 +x, ~tJ -t~~s -xal2..
:S = ~ )'1. -X, V3 -+X! ~o+ I<.J't1 2'~ = ~rl + x1'U~ -~11 +~ro .
~ ~~ ld"._"t~ ~ ,_ ~ e.s~t~~~ ~dd ~ ~~;ut~~~ ~ ~ 1N.d t~~ af ck ~ -bd~ tt,; t~'H/t ~ l/(~ t;~"/[;;~ ~ fo,o-ts~
(~) xt)x~) '<3) +(1"'~1,;~)~3)~{X,tYo,X1+:Jnx2.t-1f~>~+rs) (Xo, J(, >X2J X3) • (1jfJ>d1'J~ll>~& )= {>zo>Z,>~'~-3)/ ~ z.,,~,z.2,~ 4dt#~1k.:l~~rylf.~ et {!R~J+) IJI)~ ~ ~~~-/Y~' ~ {fJ1 tJ,b~o)? *!4.å ~ ~ 1t1 ~ (~,xflx2;X.S) ~ (-~)-x1J-~)-x3)~ ~~~~d~f~~~~~~d~ ~td~~dUtr-l~~ ~ ~ tUU.v~~ ,A.thtt 4 (W~+)"')~~.~
l :. ( o, fl ()l o) i =; ( ø,tJ, f, (J)
~:::; (",6,tJ,1),
•t •a ;2 ) (, =J -= "1e.- = .:-(l,() ,l) l o
~~ rtdA ~~-== ,t=-~~ Jt; = i =- -.ti< - . i; := ~ ~ -;t::;
tJ/dA ~ /tN ~ ~ (><a,X1;X,~,x3) 0øJXf; X~ JX3)
= (Xt~ 1 D,IJ,O)+(X11 01 b,_{))G + (x~1 ~,1J,O)/ +(~)fltOtb) ~. ~ ~ (4lK-1;K:lJX3 ) ==- (!fo>Ot(),t;) f (f!11 1J1D;O) t' + ( 12.1b1l) e O)/ T (1~ l!J1 (Jt O) ~) ~ 4/.hl Xc '=#tJ; x, -~1) K.2 ::::.1~ >kJ= fr.r ~
~~~.~:
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1'1u 1 x) 1'1~8-6,? z 1.3
f7-e~ t~~~~ 7 dv" ~~~ ~ f~v·æ:.~ ~~ ~Yulu .ttt·~ KG/l( ~~""f~~~~-~~~
X-=- Ko -t X1 t -t X2..j -t Xj /e_; ~ )X'11~) ~ E /l( i
~..u!~~a4~A4~~ t/æ ~~ ~ (7/(~-J-)·); 4~~ ~4-~~~~~~~~~~~~
JJer·~ f4L~ ~ ~ d~Pv~ ~ ~t -rx2-j +~~ ~~ af:~ x= \-t X.1 Z -t }{zj + Ksfl. J~ x=~- X11.1-~~-x1~ ~ ~ .J .__HA A•L ,_ _,· 1A . • ~ r ~f", , )/- ~ - i 1 l. 2. ~ ~ M X. v~~ )(X::.~ +-xt "1-K.z+~ ø
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(;"'- = rf =- ~ -= 'i~ == -1..) o:;;~~~x~~.~~~ ~-~
>< "=' x~ t x1 b + ~ j + x.,~ ) ~) x1 J ,(.~J ~ 6 ~ 4a~atb-d-~~~
1_, Yl~ ~ J~ s~ ; /K ~ 4~ x~-1.
