MTE-13 ASSIGNMENT BOOKLET Bachelor s Degree Programme Discrete Mathematics (Valid from 1st January, 2013 to 31st December, 2013.) It is compulsory to submit the assignment before filling in the exam form. School of Sciences Indira Gandhi National Open University Maidan Garhi, New Delhi (For January, 2013 cycle)
Dear Student, Please read the section on assignments in the Programme Guide for elective Courses that we sent you after your enrolment. As you may know already from the programme guide, the continuous evaluation component has 30% weightage. This assignment is for the continuous evaluation component of the course. Instructions for Formating Your Assignments Before attempting the assignment please read the following instructions carefully. 1) On top of the first page of your answer sheet, please write the details exactly in the following format: ROLL NO :............................ NAME :............................ ADDRESS :.................................................................................... COURSE CODE :......................................................... COURSE TITLE :............................. STUDY CENTRE :............................ DATE............................ PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. 2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. 3) Leave a 4 cm margin on the left, top and bottom of your answer sheet. 4) Your answers should be precise. 5) While solving problems, clearly indicate which part of which question is being solved. 6) This assignment is valid only up to December, 2013. If you fail in this assignment or fail to submit it by December, 2013, then you need to get the assignment for the January 2014 cycle and submit it as per the instructions given in the Programme Guide. We strongly suggest that you retain a copy of your answer sheets. Wish you good luck. 2
Assignment (To be done after studying the Blocks) Course Code: MTE - 13 Assignment Code: MTE-13/TMA/2013 Maximum Marks: 100 1) Which of the following statements are true and which are false? Give reasons for your answer. i) The statement If x 2 + y 2 = 0 for any two positive integers x and y, then xy is a prime. is false. i iv) The sentence There is no smoke without fire. is a statement. The Boolean expression (x 1 x 2 ) (x 1 x 3 ) is in conjunctive normal form. If three students are to be allotted hostel rooms and there are four vacant rooms, this can be done in 20 ways. v) If 5 points are chosen at random inside a square of side 2 cm, then the distance between at least two of these points will be atmost 2 cm. vi) a n = a n 2 + n, a 1 = 0 where n is a power of 2 is a linear recurrence relation. v The generating function of the sequence {1,2,3,4,...,n...} is (1 z) 2. vi If g(x) is the generating function for {a n }, the (1 x)g(x) is the generating function for the sequence {b n } where b n = a n 1 n. ix) Any graph is isomorphic to its complement. x) Any bipartite graph is planar. (20) 2) a) Consider the propositions 2+3=5 and The Sun rises in the West. i) Write the disjunction of the statements and give its truth value. Write the conjunction of the statements and give its truth value. i Write the exclusive disjunction of the statements and give its truth value. (3) b) Rewrite the following statements using logical quantifiers. Also write down the negation of the statements, both using logical quantifiers and as mathematical statements. (3) i) The minimum temperature in Delhi was less than 17 in December this year. Every odd natural number is the sum of three odd primes. c) Write down the converse of each of the following statements: i) If p is a prime number and a and b are any two natural numbers and if p divides a or b, then p divides ab. In a triangle ABC if AB 2 + AC 2 = BC 2, then BAC = 90. (2) 3
d) Apply precedence rules to find the truth value of the statement p q r q if p is the statement 2+3=5, q is the statement The Sun rises in the East and r is the statement If x 2 +y 2 = 0 for x, y R, then x = 0, y = 0. (2) 3) a) If a second order linear recurrence with constant coefficients 4u n + au n 1 + bu n 2 = f(n),(n 2) has the solution u n = 1 2n + 3.2 n. Determine a, b and f(n). b) A bank pays you 4.5% interest per year. In addition, you receive Rs.100 as bonus at the end of the year(after the interest is paid). Find a recurrence for the amount of money after n years if you invest Rs.2000. (10) 4) a) Consider the number of words of length n made by using letters a and b that do not contain 2 consecutive a s. Denote this number by a n. i) What are the values of a 1, a 2, a 3 and a 4? Derive a recurrence relation for a n and solve it. b) Consider the graph given below: i) Write the degree sequence of the graph. i Exhibit a cycle of length 9 in this graph. Does this have any complete graph as its subgraph? If yes, exhibit one. iv) Find δ(g) and (G) for this graph. (10) 5) a) Write the expression x 1 x 2 x 3 x 4 in conjunctive normal form and disjunctive normal form. (6) b) Find the Boolean expressions that represent the output of the logical circuits in Fig. 1. (4) 6) a) Use( generating )( ) function ( to prove the identity r s r + s = k n k n n k=0 ). (5) b) Solve the recurrence relation (5) b n = b n 1 + n 2 + n(n + 1) 2 7) a) Show that a total of 83160 11-letter words can be formed with the letters of the word ABRA- CADABRA. 4
Figure 1: Figure for exercise 5(b) b) From a survey of 120 people, the following data was obtained: 90 owned a car, 35 owned a computer, 40 owned a house, 32 owned a car and a house, 21 owned a house and a computer, 26 owned a car and a computer, 17 owned all the three. i) How many people owned neither of the three? How many people owned only a car? i How many people owned only a computer? (10) 8) a) Consider the graph given below: i) Draw its complement. 5
Show that each of the graphs in Fig. 3 is isomorphic to one of the subgraphs of the graph in Fig. 2. For example, you can show that the graph in Fig. 4 is isomorphic to a subgraph of the graph in Fig. 2 by producing the subgraph H with V(H) = {f,b,d,c}, E(H) = {(f,b),(b,c),(c,d),(d,f)} and the isomorphism g(f) = a,g(b) = B,g(c) = D,g(d) = C. b) Is it possible to draw a 6 regular graph on 11 vertices? If yes, draw such a graph. If no, give reasons for your answer. (10) 9) a) Draw the Ferrar graph of the following partitions: i) 21 = 8 + 7 + 3 + 2 + 1. 30 = 9 + 8 + 6 + 3 + 2 + 1 + 1. Also, write down the conjugate partitions of each of these partitions. (4) b) Using the recurrence relation S m n+1 = Sm 1 n + ms m n, calculate the values of S m n for 1 m,n 6. (6) 6