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Unformatted text preview: PROBLEM 1 a. (2 pts) Evaluate —17 mod 5. 3 b. (2 pts) What is the prime factorization of 12! (factorial of 12)? ll)‘ u x :oxaxsx’lxaxx xHxS x 7’ 2x35 1x1} S x zxsx '7 x ulex 3x3x ’2:x.lfxn x
.. . I .. I .. . Zﬂdg 0
z‘xsyx yawn c. (3 pts) What is the LCM (least common multiple) of the following integers 22 3  5 and
23  7? 23K 3x§2< 7 ~8x i575? 3 DOA? ~ 9(4) (:1. (3 pts) What is the GOD (greatest common divisor) of the following integers 23325 13
and 2233711? 212‘}? :Hxﬁrgc PROBLEM 2 Give the big—O estimate for each of the following functions. Provide a simple function 9(22)
of the smallest order a. (2 pts) f(£L') = 3:210g(:1:3 —— 1) + 561'5.
x1 H x b. (2 Pts) f(93) = W2 + 3)/2l V /‘ c. (2 pts) f(:r) = 2”” + x6. (1‘. ('4 pts) Show that f = 3x2+4x is 0(362) by providing constants C and $0 as evidence. PROBLEM 3
a. (5 pts) Compute the following: M V) fic— .2" [>1 (.4 h
22? r” 1;. ) 7/
,/z “82.1" J) = "" b. (2 + 3 pts) Determine if the following functions are bijections from R to R. i. f(ac) = 2:2: +1. LI¢> ii. = 23:2 — 1. PROBLEM 4 a. (6 pts) Show using set identities that = (50:? )m‘r. b. (4 pts) What is the Cartesian product A x B for sets A = {sunny, rainy} and B 2
{nights, days}. A X B: {Cgkhm}, nigh—rs ), (gunmé’ O‘OUjS)/
C m:‘«% ,nfakrs), (mm?) OMS) PROBLEM 5 a. (4 pts) Use a direct proof to show that the sum of two odd integers is even.
S “PPQ We have, 2 bold {Ixre 0d any 0, and 02 t
The“ 0,: ‘ "FDY' gomc ln‘feﬂer m 02: 2n+l ‘FDV Cvme I(h'fv€?er r1 Thus. O,+0z: 2m+ +2nPl : 2Cm+n3+ 2
/ uh‘ch is e (fen. b. (6 pts) Show using rules of inference that the premise “Everyone who exercises has a
sore muscle” and “Jimmy exercises” imply that “Jimmy has a sore muscle.” Use E
to denote “ac exercises” and S(:c) to denote that “a: has a sore muscle.” “71% C500» 300)] $[E£J’immgt)~> Scamng ECTFMW3>
A ECTMMQ) 9 SCTI‘mmg) MM t‘ . S Cj“MM4 >_ PROBLEM 6 Let 5(23) denote “a: is a. student,” F(x) denote “a: is on the faculty,” and A(x,y) denote “x
asked y a question.” Let a: be drawn from the universe of all people in the world (i.e., ac need
not necessarily be a student or faculty). Translate the following into logical statements. a. (4 pts) Every student has asked Professor Smith a question. .61“ < S‘CX) __; A (1%] Smith)) b. (6 pts) Some student has not asked any faculty member a question. 3%, tn} (fem Ample» we”) ), PROBLEM 7
(5 pts) Show that [(p —> q) /\ (q —> 7")] —> (p —> 7") is a tautology. Lcm>«<%—»r>1—>cw>r> 7W , [CwPVQﬂA (1%vr)j —> (qpvr) why/QT) vtCtqfvrlv (ﬁpvr) .2 (“w1:8) foonvr) AT ] (VA—106%, Cay/hr) V CﬂPvr) C‘1PV4%)V cka‘ 4‘) V6%V1/)VY
1V\/ T Vr 1:: (5 pts) Show using identities that —(p V (79 /\ q)) and (—np /\ ~1q) are logically equivalent. v
u.— .’ /\
’1 (PV («we») "P #5 EPA?” 91 T: /\ W
W W .1 i ‘1? M F My 1 E bwﬂv («VA«w * F “‘73”?
a 7 v («3: m g) .—
in ...
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