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quiz1solnmat25 - SDlU'l'lth Quiz 1 Thursday August 7 1. (5...

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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 ix-9\<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. (Do-M 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 xeA-MB. 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 cchEosi-five 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) BX-S ‘= 3V "‘5 ...
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quiz1solnmat25 - SDlU'l'lth Quiz 1 Thursday August 7 1. (5...

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