# exam1-answers - NAME A “5 WC’J’S Sets and Logic...

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Unformatted text preview: NAME A “5 WC’J’S Sets and Logic: MHF3202 (4628) First Hour Exam WY. Mitchell January 27, 2010 1 Start your answers on the worksheet; if you need more space, ( ) continue them on the back. Be sure to show your work to receive 2 l ) full credit. 3. (10) No calculators or notes are allowed on this test. 4 ( ) f ) Be sure to write out your proofs neatly in the proper class style. T ot. 1. (10 points) (a) Compute the following power sets: o IfA:{{®}7{1}}then73(A)= €525) i - PM): 5% {W} W. (b) Give the definition of H F, using setbuilder notation. {X’f V361: X52); (0) Give one strategy for using, as a given, a formula of the form Harrow). 32;? gas/4:7 4a) Zr 4 1e mfg/«ML m4 7‘4/2503)’ 73m nae ﬁj/M @09) I": fﬁly/Wﬂ // (d) Explain why the following is an incorrect theorem: VmEZEyEZ(r+2y=O) 2. (10 points) Consider the following incorrect proof: Theorem. For any real numbers :1: and 2 such that \$2 = 2, :1: 7E 4 and .3 7E 1. ( Incorrect ) Pmo 1. Let .CL‘ and 2: be arbitrary real numbers. Assume the hypothesis, :52 : 2. 3. Suppose the conclusion is false. Then a: = 4 and 2 =1. But4-1247é2... . . .Which contradicts the assumption that xy = 2. Therefore the conclusion must be true. We assumed the hypothesis and proved the conclusion, so the implication must be true. Since :1: and 3 were arbitrary, the theorem is true. 9051939“? F0 (a) Give the smallest line number with an incorrect assertion. 4/ b What strategy is the author of the roof attempting to use at that point? p [/Orm/ év may ﬂack/EM» (c) What speciﬁcally is he trying to do in that line? ﬂP/éa/é % 4%) Céj/‘VJL/ (d) W hat would the line be if he had done it correctly? Xe?” cé’ 2’/ (e) Either give a correct proof, or show the theorem is wrong by giving a counter— example. 567/ AH?” M 3’24, 714'” X27 9" :2 ﬂmcéyﬁhx ’/ // . 9L9“? M :J‘I/ W , 3. (10 points) reasons below. Use deﬁnitions and the list of equivalences to ﬁll in the missing ﬁVAﬁ((AEJ:—+x€A)V(AEQ-éxEA» EaAﬁﬁﬂAefameAMMAegaxeA» Géml'mét V EHA((A€.F——>\$€A)V(AEQ->\$€A)) EaAaﬁAEfVIEAMMﬁAEgV\$€A» ﬂmﬁé Wéz: V (3%x Jaw Em%ﬁ[email protected]» AychQw EIA A Q G (ﬂ IN A SAE Ab aMA 5(3AE 0035 AEFVﬂﬁAEQVxEMVxEA» AefvbAengeAVxem» AEbeAEQVxEA» EAﬂjAefvﬁAEQVxeA) (AefAAEQVmeﬂ (AEFDQVzeA) EfﬂQ-HBEA) WWW/1) VA (A627 & MW ? :3 ﬁWW.;' lﬂﬂéfﬁﬁ l/ (M (M 1/ 7%, ’77/ 052%} ii Maia/MHI/ ) 6“” W4). 3x65” é 4. (10 points) Use the class style to prove the following theorem: Theorem. Suppose B is a set and J3 is a family of sets, and .7: g 73(8). Then U}— Q B. ﬁ/0ﬂf Aﬁmﬂ f/ﬂ/ém J 5/05). if x Ae/ an: gréf/my memhw yﬁi M ‘2 ygj )Mfﬂ/XéﬂJﬂ/p éw/ﬂ/ﬂ’ ...
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exam1-answers - NAME A “5 WC’J’S Sets and Logic...

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