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Proofs

Part of maths and physics class notes

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Aleck Maphanga
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0% found this document useful (0 votes)
49 views14 pages

Proofs

Part of maths and physics class notes

Uploaded by

Aleck Maphanga
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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LctC<..

t T: (RI'\ ----'71R"" -" /;nee re +rciMS-fufW1Cl_';;;;e_ Wees


Let:: be- 0... r,·neq, trcw,Sfu(">'Vl0.1::"lOVJ .
/-t -o/ � � T(x) =O hef- s/�-:s oli'e.
--chV\ ,s T
�e"" i'o /--1:-o-/ - - hctS o"'(j t"1e.
+ri vio/e. op loss·1".j /-fri via.( solution .
Be-JH� ; P-oo +:
(:___ 7): Nee.-v-, aa,,., T i<S f-{-o/ . OfY\do.i T /i'vieir ,�
. Assv.v"V)e__ , s 1-ia- / 01...-..ce T ,·.s; /iv1eiQ.r
vol flc oht 1,, = 0 ( KJ \::. Q. /)
Orr\cl.a-& T
1-I: -i,;0Uo0s- {hat T c9-J See.. . s,·Vlce___ T
f-1-ol (s.__., is 0 di'e. en1qste.. o?lo�siv.9 l}r, T(�) -ol
,·s 1-r:o-/ ) o is f'f-e anTj So/uti'ov1 o-r TCJ() =�
dws T(x ') ::::co het �le3� die. -fn·v,a.!€__ of I o.ss;;d)
i.e. TC'!,_) -=O h.a.s oY\� +l.,e. "1Y<·✓1'ctl solutiov1
l� ) : Nee,,.,___ o.av-i --( ) _ "1et- ci 8l s;- J/e -fr,'v,'a/e. op(.
1 x. 0 - has on"!J -8,e_ ---f-ri'vra.( So{c..vcJVJ.
A §v..Me.. (Kontrapositief)
Teens�d,qhei'dsb_ew�S : G�te/ T ,:s nc'1c J-/-o1 vile.
Prrof � 'ior'\iro.ol.c c-hOV\ : �f PCJ\;-e. T , s; not- f - &-o - I .
Dc:;1. n. .b e .si:z::<.Q"' daa r rv'I; nst eV\� e eV\ b vJa -t" o/c' e___ b ee (J
(Contrapositive)

----rhe.,...,_ -/::l-.ere ex1's;ts at /eQ.st one. -b -/),10.f i's f'he_


_is vo.v"\ rn;',)ttevis twee. vcl-&or-e __, sl- LA eV\ y_ ·,V\ fR "\J
1 �e., 0 t at /-ea.sf: --rwo uec-60: S'Clj -LA a""ol v /1--\ fR_ �
clil w,\ s-e T ( l,() == b e,,_ 1(v-)
= )?-.. fv!Ct£i ; T ,:s J
L·e .
\'\ l \r\ee.r
T( l,() = b a....._d T( -v) = i?_ . B u i T , :S
'\ J ol u� ls
IV\ eq,) so T/IJ. -u \ = T(.y_)- T(v-) = b -b = O
(__ )
D'1 e. ve k:.:to, u -y is n.i'e v)l,{ r Y'lie ) wO..vt t u 4 V
the. ue.cf:v, � -v is v\O t rc'<O s;:i...-. Ce Ll =f:. v-
Uus het die. v3I.
So ,bt-ie ect,,uC?.:bovi
T(x. - \ -== o nie n�t- o/i'e +rivi'cile cpi._ n;e__
hu S: MO,e. +ha,. ;\.{,� -lie -fnv. sc{.
Teer1sfydi_g. Tiwf is:. T /-I-of. /Co,--,frQdrchovi. thus. T'-' ,·r /-to - /
== 0. V, +O.V.,_ + _ _ _ . -1-0, Vp
w:l\eleu,/1 vJees, o{.r.Jr
,
b e... ar b,'frctrtt.J/ t.e.
,/ ct�o( E-t

ru ::: 'f' (c, '!. I +. - - - _.L. (_f V'I')


-::- ('f"C,) �' +- - - ;t. ( rCp) '!..P
-E �q .... ��, YrJ � W
w
., - -;

de.e(t"4;""- f e vo.�
I �n
, '
'
(� D.. S'<.� losF-c e o /i

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