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(/./;) PART-A C I . ," \) /,,/// K') a. For any t hree s~t~;J),13and C, prove that 101owmg ~/':::.::" (06 Marks) An (BilC) = (APr,B) ~ = (AnB) il (AnC). ((~\ ,,) b. In a survey of 60 \~~0ple)t was found that 25 read weekly ~ag~ziiles, 26 read fortnightly magazines, 26 read morltJ\1y magazines, 9 read both week~ ~;?dmonthly magazines, II read both weekly arl'd' j9'~ nightly magazines , 8 rea~~;'p th fortnightly and monthly ~agazines and 3 read aWt?rb~Inagazines. find . );~:, '( . 1) the number ofpeople whq,'!"ead at least one of the t I£ihnagazmes and ii) the ~umber ofpeopl~ wHd~.~~ exactly o~~ mag~~~\~ . .. (06 Mar~s) c. A certam soccer team wms (w) }'ht~probabliIty O,.~,'J.osses(L) WIth probability 0.3, and ties (T) with probability 0.1. The tea~ plays, three gamek].~er the weekend: . i) Determine the elements oftl1e-'~le t,A tt(a.t:,tqe team wins at least twice and does not v j/ lose; Also fine peA). ,\,' ), · ii) Determine the elements of the event B that the team wins, losses and ties in the same 1'> (08 Marks) order; Also find PCB). I,
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Prove the following: i) P ~ (q ~ r)
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(q A r) V (P A r)
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i) -, P ii) P A q iii) P~v ,("t~) P ~ q q. (06 Marks) c. Check the validity ofthe foIF~~in'g argument: \.:-:;:-~") . If the band could not plaY'J9PK ~usic or the refreshment.S::w~~ not delivered on time, then the New Year's party w6urcrJl,ave been cancelled and Ali~ia~w,6uld have been angry. If the party were cancelled; 1;h~~V{funds would have had to be'-
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Write down the cdn\~'etse , inverse and contra positive of each oft1y~:)ollowing statements for which the set~8f'atl real numbers is the universe. Also indicate tl'ietr values. i) \fx,[(x>1)';>:::G~2>9)] ii) \fx,[{(x2+4X-21»0}~{(X>3)V(x~-~~}]. (06Marks) b. Check the ),(aH(1i,t/ of the following arguments : ~'/~. In triangle ..., Y~;' there is no pair of angles of equal measure. \30 c:\ If a tr~~rl~le/b:a~two sides of e~uallength , then it is isosceles. o.,f-!,;. If a trIangle, Is Isosceles, then It has two angles of equal measure. Therefore triangle XYZ has no two sides of equal length. \. :{.Qv~arks) c. ~?t1~,b3)an integer, prove that, n is odd ifand only if7n + 8 is odd. (O&M~S)
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a. the following, for all n ~ 1 using the principle of Mathematical Induction : [\~ 1 1 n \:J,r + -(06 MarKs) -, -+-+, 1.2 2.3 n(n + I) n + 1 b. Use Mathematical induction to prove that 5 divides nS - n, where n is a non - negative (06 Marks) integer. 1 of2
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lOCS34 c. _for the Fibonacci sequence Fo, F" F2, ..... (Fa = 0 ,F, = I and F, = Fn-, + Fn-2). Prove that
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i) Determine f(Ot (fI~l), f(5/3), f(-5/3). ii) Find C'(O) , ~(I))l, C'(-I), C'(3) , ["'(-3), iii) What are f-'([-5, 5]r~'nd ["'([_6, 5])? \ Prove that function f: A.•... //",.J3 is invertible ifand
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Using characteristic functionsFor any sets A, B, C containe
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IfR = {(x, y) / x > y} is a relation defined on the set, A = {I, 2, 3, 4}, write down the matrix and the digraph ofR. Also list the in - degrees and out - degrees of all vertices. <. . -" . . (06 Marks) b. Define R on A = {I, 2, 3, 4,5,6, 7, 8~:jO'IJ 1, I1} as (x, y) E R, if x - y is a multiple of 5. i) Show that R is an equivalence relation on A.] ./ ii) Determine the equivalence classes and partition of A induced by R. (08 Marks) c. For the Poset shown in the following Hasse diagram, find i) all upper bounds ii) all
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Prove that the cub~roots of Unity form a group under Prove that every ~u.bgr9up ofa cyclic group is cyclic. Let fbe a homomorphism from a group G, to a group Prove the folio "iog : i) IfH, is ,(subgroup of G, and H2 = f(H,), then H2 ii) Iff is ciiJ is morphism from G, on to G2, then r'
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'(06 ~arks) C, hen prove that d(C, r) s d(C, r) for all c E C. (Z,E£>, 0) is a ring under the binary op~rat.ions b. F~9'~) integers K and m for which (06 !\1lll'ks) 9- ¥= x + Y - K , x 0 y = x + Y - mxy. ';~ that in z., [a] is a unit if and only if gcd (a, n) = 1. C{; -.i~,/rove 'J word.
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it) Find all the units in Z'2.
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***** 20f2
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