Hello I don't think the answers I got were correct for the paper. For instance, the second question obviously have an answer of A by the distributive and implication law. But the tutor gave an answer of E.

Could someone with prior exp in discrete mathematics help me to solve these questions so I can check my answers. (answers not given by prof, not found in sch website or any other website). TQSM

2. Which of the following statements is logically equivalent to ~p V (q Ar)? A. (p - q) A (p-r). B. ~(PA -q) A (r - p). C. (-PAq) V (pA~r). D. (PAq) - (~qv~r). E. None of the above. 3. Which of the following statements is the negation of Vx By Vz ((P(x) A Q())) - R(z))? ((2)a - ((nov(x)d) ) ~ZAKEXA V (()anov (x)d)~ ZEMAXE 8 (nov ( x)d)~ + (2)y~) ZE AXE ) D. 3x Vy 3z (P(x) A Q(V) A ~R(z)). E. None of the above. 4. Let A = {-2, -1,0,1,2), B = {0,1,4) and C = {-4, -3, -2, -1,0,1,2,3,4). Which of the following statements are true? (i) VXEC((x EA) +> (x2 EB)). ((x = z*) + (83 Xx a= (A)) J3 xA (!!) (iii) 3x E A Vy E A ((x # 0) A (xy E B)). A. Only (i). B. Only (ii). C. Only (i) and (ii). D. Only (i) and (iii). E. None of the above.C512315 C512315 E. LetA and B be any sets. A U B be the universal set. and let 'P(X) denote the power set ofX. 9. Given a set A with two elements. how many relations onA are not transitive? Which of the followmg statements are true? A. 1_ [ii lfA n B = o.then?(A) nP(B] = i3. B. 3. (ii) (PM. n B] = 3'01) n ?(B]. C. 5. [iii] (AUE]n(uB):(AnE]u(riB). D 13. A. Only (ii). E None of the above. E. Only (iii). 5- OHIY N am \"0- 1|]. How many possible partitions are there on a set With 3 elements? D. Only (ii) and (iii). A. 1_ E. None of the above B. 3_ C. 5. D. 7. E. None of the above. W 11. Given A : {12.3.4.5} and the partial order R on A as follows: W R = (rm) :2: e A] u {(4.1). (5,2). (4,2). (1.2), (4,5)}. E I F How many distinct linearizations of}? are there? I ' A. 2 7. Given the following statements on any set A, B 7 (i) If}? is a reflexive relation on A. then lAl 5 IRI. : :0 iii) If}? is a symmetric relation on A. then |A| S lRl. E None of the above. [iii] If}? is a transitive relation on A, then lAI S IRI. Which of the statements above are true? 12. Given the partial order relation R on A in question 11 above. which of the followiing statements A. None ofli]. (iilorliiilistrue. aretrue? Only m is true. (i) There is no largest element. Only (in and (iii are true. (ii) The smallest element is 4. Only (ii) and (iii) are true. All of (i), [ii] and (iii) are true. manta (iii) The maximal elements are 2 and 3. Only (i). Only In and [iii]. Only [ii] and (iii). All of (i). (ii) and (iii). None of the above. 8. Consider the following relations R and S on 222. where alb means a divides 1). IR)! 4: Elk E E32 (klx A My) 1532 :5 Elk E 322(1th Aylk) Which of the folloWing statements is true? wens\"? A. Neither R nor Sis an equivalence relation. E R is an equivalence relation butS is not. C. 5 is an equivalence relation but R is not. D Both R and 5 are equivalence relations. -3txf9- -4of9- CS12315 CS12315 13. When flipped, a biased coin lands on its head with a probability of p, and lands on its tails with a 17. Which of the following are countable sets? probability of (1 - p). Dueet flips the coin ten times. What is the probability of getting exactly 3 (i) The set of relations on heads, with all the 3 heads in a row, that is, the 3 heads appearing consecutively? (ii) The set of rational numbers between 0 and 1 exclusive. A. 3p3(1 - p)'. (iii) The set Z* of all strings over Z. B. (3 x 7)p3(1 - p)'. A. Only (i). C. ('3)p3(1 -P)'. B. Only (ii). D. 8p3 (1 - p). Only (i) and (ii). E. None of the above. Only (ii) and (iii). E. All of (i), (ii) and (iii). 14. Define a set S recursively as follows. (1) 1 ES and 2 ES. (base clause) 18. Which of the following statements are true? (2) If x, y ES, then = E S. (recursion clause) (i) Ks - {e} is planar where e is one of the edges in K5. (3) Membership for S can always be demonstrated by (finitely many) (ii) Ks is both Eulerian and Hamiltonian. successive applications of clauses above. (minimality clause) (iii) There is a simple planar graph with 6 vertices, 8 edges, and 7 faces. Which of the following statements is true? A. Only (i). A. Vx ES (x E Z = x is even). B. Only (i) and (ii). (1> X =03 x)SaxA C. Only (i) and (iii) (IZA+x)Sa AXA D. All of (i), (ii) and (iii). D. ISI = 121. E. None of the above. E. None of the above. 19. Which of the following statements are true? 15. Given the recurrence relation an = 5n + an-1 with initial value a, = 4, what is the value of azo? (i) Let G be a simple, undirected graph with 6 vertices, 5 edges and no cycles. A. 1054 Then it is not possible to have only one vertex in G with degree 1. 1050 (ii) Let G be a simple, undirected graph with n vertices and e edges. 1049 If G is not connected, then e