2015 Question paper of year 2015 of BE second year IT of subject Discrete Mathematical Structure New CGPA patturn. Question paper
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1. Write the correct answer from the options given below : [14 marks] 1) Determine whether or not each of the following is a partition of the set N of positive integers a) [{nn > 5}, {nn < 5}] b) [{nn > 6}, {1, 3, 5}, {2, 4}] c) [{nn2 > 11}, {nn2 < 11}] d) None 2) Which of the following statements is false ? a) (P? Q) ? (~P ? Q) ? (P ? ~Q) is equal to ~Q? ~P b) (P? Q) ? (~P ? Q) ? (P ? ~Q) is equal to Q? P c) (P? Q) ? (~P ? Q) ? (P ? ~Q) is equal to Q? (P ? ~Q) d) (P? Q) ? (~P ? Q) ? (P ? ~Q) is equal to [(P? ~P) ? Q]? (P ? ~Q) 3) To show that the circuit corresponding to the Boolean expression (P ? Q)? (~P ? Q)? (~P ? ~Q) can be represented using two logical gates, one shows that this Boolean expression is equal to ~P ? Q. The circuit corresponding to (P ?Q? R) ? (~P ?Q? R) ? (~P ? ~Q? ~R) computes the same function as the circuit corresponding to a) (P? Q)? ~R b) P? (Q ? R) c) ~P? (Q ? R) d) (P? ~Q)? R 4) A relation R on a set A is called a partial order iff it is a) Reflexive b) Antisymmetric c) Transitive d) All above 5) A function f : A ?B is bijective iff it is a) injective b) surjective c) one to one and onto d) none 6) Which one is the contrapositive of q?p ? a) p?q b) ¬p? ¬q. c) ¬q? ¬p d) none of these 7) Each of the following defines a relation on the positive integers N : (1) "x is greater than y". (3) x + y = 10 (2) "xy is the square of an integer". (4) x + 4y = 10. Determine which of the relations are : reflexive a) 1 b) 2 and 3 c) 4 d) none 8) Let A = Z + the set of positive integers. Define the relation R on A by aRb if and only if ab. R is a) transitive b) asymmetric c) both d) none 9) Which of the following two sets are equal ? a) A = {1, 2} and B = {1} b) A = {1, 2} and B = {1, 2, 3} c) A = {1, 2, 3} and B = {2, 1, 3} d) A = {1, 2, 4} and B = {1, 2, 3} 10) A relation R on a set A is called an equivalence relation iff it is a) Reflexive and symmetric b) Transitive c) Both d) None 11) If B is a Boolean Algebra, then which of the following is true. a) B is a finite but not complemented lattice b) B is a finite, complemented and distributive lattice c) B is a finite, distributive but not complemented lattice d) B is not distributive lattice 12) Let L be a lattice. Then for every a and b in L which one of the following is correct ? a) a? b = a ?b b) a? (b? c) = (a? b) ? c c) a? (b ? c) = a d) a? (b? c) = b 13) Abelian group satisfies additional ___________ property than group. a) transitive b) inverse c) identity d) commutative 14) Integral domain in _____________ have property with no zero deviser. a) ring b) field c) chain d) none
SECTION – I 2. Attempt any five questions : [20 marks] 1) Write the PCNF of P ? (P? Q). 2) Obtain the CNF of (P ? (P?Q))?Q. 3) Define properties of binary relations in a set. 4) Explain equality of set and proper set. 5) Show implication (P?Q)?Q?P? Q and of (P ? Q) ?(P?Q). 6) Show that (p?q) ? (q?p) is logically equivalent to p?q with truth table.
3. What is POSET ? Give procedure to draw Hasse diagram for powerset of S = {a, b, c}. [8 marks]
SECTION – II 4. Attempt any 5 questions : [20 marks] 1) Definition lattice with GLB and LUB example. 2) P1 = 123 P2 = 123 P3 = 123 P4 = 123 P5 = 123 P6 = 123 123 213 321 132 231 312 Draw permutation and prepare table for (P, *). 3) Write note on Group code. 4) Let A = B = {a, b, c}. Consider the relation g = {(a, b), (b, c), (c, c)}. Is g onetoone ? Is g onto ? Why ? With example explain. 5) What are the different type of functions. 6) Which of the partially ordered sets in figures (a), (b) and (c) are lattices ? Justify your answer. 5. Write algorithm to convert infix expression to polish Notation with example. [ 8 marks]
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