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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 $2 = 2, :1: 7E 4 and .3 7E 1. ( Incorrect ) Pmo 1. Let .CL‘ and 2: be arbitrary real numbers. Assume the hypothesis, :52 : 2.
3. Suppose the conclusion is false.
Then a: = 4 and 2 =1.
But41247é2...
. . .Which contradicts the assumption that xy = 2.
Therefore the conclusion must be true.
We assumed the hypothesis and proved the conclusion, so the implication must
be true.
Since :1: and 3 were arbitrary, the theorem is true. 9051939“? F0 (a) Give the smallest line number with an incorrect assertion. 4/ b What strategy is the author of the roof attempting to use at that point?
p [/Orm/ év may ﬂack/EM» (c) What speciﬁcally is he trying to do in that line? ﬂP/éa/é % 4%) Céj/‘VJL/ (d) W hat would the line be if he had done it correctly? Xe?” cé’ 2’/ (e) Either give a correct proof, or show the theorem is wrong by giving a counter—
example. 567/ AH?” M 3’24,
714'” X27 9" :2 ﬂmcéyﬁhx ’/ // . 9L9“? M :J‘I/ W , 3. (10 points)
reasons below. Use deﬁnitions and the list of equivalences to ﬁll in the missing ﬁVAﬁ((AEJ:—+x€A)V(AEQéxEA» EaAﬁﬁﬂAefameAMMAegaxeA» Géml'mét V EHA((A€.F——>$€A)V(AEQ>$€A)) EaAaﬁAEfVIEAMMﬁAEgV$€A» ﬂmﬁé Wéz: V (3%x Jaw Em%ﬁAefv@eAvbAEQVmem» AychQw EIA A Q
G
(ﬂ IN A SAE Ab
aMA 5(3AE 0035 AEFVﬂﬁAEQVxEMVxEA»
AefvbAengeAVxem» AEbeAEQVxEA» EAﬂjAefvﬁAEQVxeA) (AefAAEQVmeﬂ
(AEFDQVzeA) EfﬂQHBEA) 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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 Fall '11
 Mitchelle

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