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Unformatted text preview: SDlU'l'lth Quiz 1
Thursday August 7 1. (5 pts) A function f is uniformly continuous if the following property holds:
V6 > 0, 36 > 0 so that if — y < 6, then — < 6. Write the negation of the above property. (There should be no ~ in your ﬁnal
answer.) m(’dg>o 33>o Sal'hcd \x~9\<8 ~—> \?Cx)'*¥(g‘)l<€l v; 390 Mgasm €33 M \wjus ——é \‘?(x)~?(g)l<€)
:3 3g70 V3>o ~(ix—9\<'8>) \HX)~F(~3)l<E) E 32>0 ix9\<g 952/; oﬁgiml Siakmeni implicitly has Vx,3 in me‘ 019
'HM \X~9\<8 so had/imam” \jn m ngﬁon ‘H/UU‘G ghould 2. (5 pts) Prove: \Oe 0k 3X19 beﬁﬁ We \x~3\<8) ‘
\
AgBifandonlyifAﬂBz/l. purl 1
AéB—> AnBrA.
PM 1, Age a ADBS‘A «Lelv xeAnB. Then xeA cmd xes. (DoM need xes). 1—530 PmBSA.
“*3. Aéev—‘eASAns. Le+ xeA. Then XGB blc A915. 50 ‘xeA omd xeB,
Hera“we xeAnB‘ WMJFom A; ADE. Sing; ACB~§Ang§A and AQBﬁAS: AngJ AQB‘QAzAan
Vari— a L AnBIA —) ASS. 1 Lel— xeA. Since A: A03; in particular AgADE. There$02
xeAMB. go xeB. 50 ASE (a) (5 pts) Suppose you want to prove that p —+ q. Explain what a proof by
contrapositive is, and Show via truth table it is equivalent. The cchEosiﬁve 0(— gs Nabawf)
W5 Ovvf equivaievﬁ WC “May have m Scmua
'hui’h Jrawe 2 (b) (5 pts) Let f be the function : 3m — 5. Use the contrapositive to
prove: if :I: # y, then f(:c) 7Q f(y). (This says that f is injective). \
M Con‘i‘anosi‘I'ive 0‘; Xt‘j "3 Quota”)
is Hx3=FCxp *3 x23. Vvosf 1 Ho: fem)
3) BXS ‘= 3V "‘5 ...
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 Winter '11
 Wiley
 Differential Equations, Equations

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