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24 March, 2014 Exam 2 Math 2200 UID: Name: Instructions: • Answer each question in the space provided. Scratch paper will be pro vided and should be attached to your test when turned in, but your final answer should be presented as neatly as possible in the space provided. • No calculators are allowed. • You may refer to a note card no larger than 3 x 5 inches. • Answers should be justified as appropriate. Question Points 1 10 2 10 3 10 4 10 5 10 6 5 7 5 8 0 Total: 60 Score Math 2200 Exam 2, Page 2 of 8 1. (10 points) Let A= {1,2,3} B = {3,4,5} C= {1,4} List all elements of the following sets: (a) AU(BflC) L1 A (b) (A—C)uB 7g5f 3L, C) (c) A—(CUB) 1, Cui3 A— .s: 1 I31 1z (d) A x C ( i), (i,L Czi), (?,i) (3,1), (vi) (e) P(C) (the power set of C) I , qj j, q 24 March, 2014 Math 2200 Exam 2, Page 3 of 8 2. (10 points) Consider the function (a) Show that f f Z x Z L) 1 ç( f Z given by f(x, y) = x — y. is not one-to-one. 5tv1c_ (b) Show that —+ 24 March, 2014 is onto. LeéZ. 3 i- _—-€. 3. (10 points) Consider the relation a c(mod2) and 6 d(mod3). ‘-‘-i on the set Z x Z given by (a, b) r (c, d) if is reflexive. (a) Show that f- 24 March, 2014 Exam 2, Page 4 of 8 Math 2200 ) ( k 1 EZl. ci’ b 7) ct (‘ 3) ()‘4) t4 is symmetric. (b) Show that (c,d (c) Show that 54A 1> r’ of tS . yV1k’L(1. -ad 0 (3) (3)-’ (oo) 6 CO,o) #(?,3) H3y is transitive. (cd,) (d)? (e) Zx 5C’ ‘—(g. (d 2) (-i ) O dI,(d is not antisymmetric. [oo)d (d) Show that (-odi). cc )’- (k,) Cc d 1 S d Iiec • p 1 d(d 3) [y Lb () )E (, “) ‘---(rE) cj A Ce(’ i), d€(vd 3%), t’’ (Q)’-(c,d) Exam 2, Page 5 of 8 Math 2200 24 March, 2014 011 the set Z x Z given by 4. (10 points) Consider the relation d (mod 3) as in the previous problem. Problem 3 (a. b) r (c. d) if a c (mod 2) and b shows that this is an equivalence relation. (a) List four elements of Z x Z in the same equivaleice class as (0, 0). (0,) (-;-3) (, ) 1 L (L (b) List all equivalence classes of the relation. L(oo) ) J iJ] 1 C (o [Ri) Math 2200 Exam 2, Page 6 of 8 24 March, 2014 5. (10 points) Prove or disprove each of the following statements: (a) If ac and bc, then (ab)c. Fcthe 21’, qjq 44iLj (b) If xy and xz, then xyz. ‘ cLwLe yz S”-Ex svzL.< iv1frL1 x\yz 3 .id Math 2200 Exam 2, Page 7 of 8 24 March, 2014 6. (5 points) Find the greatest common divisor of 222 and 629. (LC1 2222-’- tS zz J8 V6-37 537f cd(3ü) ,o 3_i 7. (5 points) Find integers s and t such that 17s + 12t I7 L2 2s 5:: 2(I22.s) L 5.5-2 s(7 n) — I ‘) & -7 2• 1 = 1. 24 March, 2014 Exam 2, Page 8 of 8 Math 2200 8. (5 points (bomis)) Let a, n (i) a e Z and ii > o. Show that the following are eqiivalent: 0(modn) (ii) a mod ri — 0 (iii) nla (Ij) rc ( ‘) vck SO- f ‘ (‘; ) c nc .i. O 0. — TP c ‘t sc = SoVr-C +O_ &;)— () 1Q h- (-o). The€ n).