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# 210502 Mathematical Foundation Of Computer Science

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210502 Mathematical Foundation Of Computer Science

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### 210502 Mathematical Foundation Of Computer Science

1. 1. www.studentyogi.com www.studentyogi.com Code No: R05210502 Set No. 1 II B.Tech I Semester Supplimentary Examinations, November 2008 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks 1. (a) Let p,q and r be the propositions. P: you have the ee q: you miss the nal examination. r: you pass the course. Write the following proposition into statement form. i. ii. 7 iii. 7 iv. pVqVr v. ( 7 ) ( ) vi. ( ) (7 ) (b) De ne converse, contrapositive and inverse of an implication. [12+4] 2. (a) Let P(x) denote the statement. “x is a professional athlete” and let Q(x)denote the statement” “x plays soccer”. The domain is the let of all people. Write each of the following proposition in English. i. ( ( ) ( )) ii. ( ( ) ( )) iii. ( ( ) ( )) (b) Write the negation of each of the above propositions, both in symbols and in words. [6+10] 3. (a) De ne a bijective function. Explain with reasons whether the following func- tions are bijiective or not. Find also the inverse of each of the functions. i. f(x) = 4x+2, A=set of real numbers ii. f(x) = 3+ 1/x, A=set of non zero real numbers iii. f(x) = (2x+3) mo d7, A=N7. (b) Let f and g be functions from the positive real numbers to positive real numbers de ned by ()=2 ()=2 Calculate f o g and g o f. [10+6] 4. Show that the set G = x/x = 2a 3b and a, b Z is a group under multiplica- tion. [16] 1 of 2 www.studentyogi.com www.studentyogi.com
2. 2. www.studentyogi.com www.studentyogi.com Code No: R05210502 Set No. 1 5. (a) In howmany ways can we place 4 red balls, 4 white balls and 4 blue balls in 6 numbered boxes. (b) Howmany integers between 1 and 1,00,000 have the sum of digits equal to 18. [16] 6. (a) Solve an - 7 an - 1 + 10 an - 2 = 6 + 8n given a0 = 1 and a1 = 2. (b) Solve an + an - 1 = 3n 2n . [8+8] 7. (a) What are the steps involved in Kruskal’s algorithm for nding a minimum spanning tree. (b) Describe the procedure to obtain all possible spanning trees in a given graph. [8+8] 8. (a) Write a brief note about the basic rules for constructing Hamiltonian cycles. (b) Using Grinberg theorem nd the Hamiltonian cycle in the following graph. Figure 8b [16] Figure 8b 2 of 2 www.studentyogi.com www.studentyogi.com
3. 3. www.studentyogi.com www.studentyogi.com Code No: R05210502 Set No. 2 II B.Tech I Semester Supplimentary Examinations, November 2008 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks 1. (a) Let p,q and r be the propositions. P: you have the ee q: you miss the nal examination. r: you pass the course. Write the following proposition into statement form. i. ii. 7 iii. 7 iv. pVqVr v. ( 7 ) ( ) vi. ( ) (7 ) (b) De ne converse, contrapositive and inverse of an implication. [12+4] 2. Prove using rules of inference or disprove. (a) Duke is a Labrador retriever All Labrador retriever like to swin Therefore Duke likes to swin. (b) All ever numbers that are also greater than 2 are not prime 2 is an even number 2 is prime Therefore some even numbers are prime. UNIVERSE = numbers. (c) If it is hot today or raining to day then it is no fun to snow ski to day It is no fun to snow ski today Therefore it is hot today UNIVERSE = DAYS. [5+6+5] 3. (a) Let A,B,C 2 where A = (x,y) / y = 2x + 1 , B = (x,y) / y = 3x and C = (x,y) / x - y = 7 . Determine each of the following: i. ii. iii. ¯ ¯ iv. ¯ ¯ 1 of 3 www.studentyogi.com www.studentyogi.com
4. 4. www.studentyogi.com www.studentyogi.com Code No: R05210502 Set No. 2 (b) State and explain the applications of the pigon hole principle. [12+4] 4. (a) If f is a homomorphism from a group (G,.) into (G’,.) then prove that (f(G),.) is a subgroup of G’. (OR) Prove that the homomorphic image of a group is a group. (b) The set, S, of all ordered pairs (a, b) of real numbers for which a = 0 w.r.t. the operation de ned by (a, b) (c, d) = (ac, bc+d) is a group. Find i. the identity of (G, o) and ii. inverse of each element of G. [10+6] 5. A mother distributes 5 di erent apples among 8 children. (a) How many ways can this be done if each child receives at most one apple (b) How many ways can this be done if there is no restriction on the number of apples a child can receive. [16] 6. (a) Solve an + 5an - 1 + 6an - 2 = 5, n 2, given a0 = 1, a1 = 2 using generating functions. (b) Solve the recurrence relation an = an - 1 + n(n + 1) 2 , n 1. [8+8] 7. (a) Derive the directed spanning tree from the graph shown Figure 7a Figure 7a (b) Explain the steps involved in deriving a spanning tree from the given undi- rected graph using breadth rst search algorithm. [8+8] 8. (a) Distinguish between Hamiltonian cycle and Euler cycle. Give examples. (b) Determine whether Hamiltonian cycle present in the graph shown in Figure 8b [16] 2 of 3 www.studentyogi.com www.studentyogi.com
5. 5. www.studentyogi.com www.studentyogi.com Code No: R05210502 Set No. 2 Figure 8b 3 of 3 www.studentyogi.com www.studentyogi.com
6. 6. www.studentyogi.com www.studentyogi.com Code No: R05210502 Set No. 3 II B.Tech I Semester Supplimentary Examinations, November 2008 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks 1. (a) Let p,q and r be the propositions. P: you have the ee q: you miss the nal examination. r: you pass the course. Write the following proposition into statement form. i. ii. 7 iii. 7 iv. pVqVr v. ( 7 ) ( ~ ) vi. ( ) (7 ) (b) De ne converse, contrapositive and inverse of an implication. [12+4] 2. Prove using rules of inference or disprove. (a) Duke is a Labrador retriever All Labrador retriever like to swin Therefore Duke likes to swin. (b) All ever numbers that are also greater than 2 are not prime 2 is an even number 2 is prime Therefore some even numbers are prime. UNIVERSE = numbers. (c) If it is hot today or raining to day then it is no fun to snow ski to day It is no fun to snow ski today Therefore it is hot today UNIVERSE = DAYS. [5+6+5] 3. (a) Determine whether the following relations are injective and/or subjective func- tion. Find universe of the functions if they exist. i. = { } = {1 2 3 4 5} R = {(v,z),(w,1),(x,3),(y,5)} 1 of 3 www.studentyogi.com www.studentyogi.com
7. 7. www.studentyogi.com www.studentyogi.com Code No: R05210502 Set No. 3 ii. = {1 2 3 4 5} = {1 2 3 4 5} R = {(1,2),(2,3),(3,4),(4,5),(5,1)} (b) If a function is de ned as f(x,n) mod n. Determine the i. Domain of f ii. Range of f iii. g(g(g(g(7)))) if g (n) = f(209, n). [8+8] 4. Prove that the union of two subgroups of a group is a subgroup i one is contained in the other. [16] 5. (a) In a certain programming language, an identi es is a sequence of certain num- ber of characters where the rst character must be a letter of the English alphabet and the remaining characters may be either a letter or a digit. How many identi ers are there of length from 1 to up 8 characters. (b) How many 7-digit numbers are there with exactly one5? [16] 6. (a) Solve an - 6an - 1 + 9an - 2 = 0, = 2, given a0 = 2, a1 = 3 using generating functions. (b) Solve the di erence equation ar - 2ar - 1 = 3.2r. [8+8] 7. (a) Explain about the adjacency matrix representation of graphs. Illustrate with an example. (b) What are the advantages of adjacency matrix representation. (c) Explain the algorithm for breadth rst search traversal of a graph. [5+3+8] 8. (a) Determine whether the following two graphs are isomorphic or not. Figure 8a, 8a Figure 8a 2 of 3 www.studentyogi.com www.studentyogi.com
8. 8. www.studentyogi.com www.studentyogi.com Code No: R05210502 Set No. 3 Figure 8a (b) Show that two simple graphs are isomorphic if and only if their Complements are isomorphic. [16] 3 of 3 www.studentyogi.com www.studentyogi.com
9. 9. www.studentyogi.com www.studentyogi.com Code No: R05210502 Set No. 4 II B.Tech I Semester Supplimentary Examinations, November 2008 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks 1. (a) Let p,q and r be the propositions. P: you have the ee q: you miss the nal examination. r: you pass the course. Write the following proposition into statement form. i. ii. 7 iii. 7 iv. pVqVr v. ( 7 ) ( ~ ) vi. ( ) (7 ) (b) De ne converse, contrapositive and inverse of an implication. [12+4] 2. Prove using rules of inference or disprove. (a) Duke is a Labrador retriever All Labrador retriever like to swin Therefore Duke likes to swin. (b) All ever numbers that are also greater than 2 are not prime 2 is an even number 2 is prime Therefore some even numbers are prime. UNIVERSE = numbers. (c) If it is hot today or raining to day then it is no fun to snow ski to day It is no fun to snow ski today Therefore it is hot today UNIVERSE = DAYS. [5+6+5] 3. (a) Consider f; + + de ne by f (a)= a2. Check if f is one-to-one and / or into using suitable explanation. (b) What is a partial order relation? Let S = { x,y,z} and consider the power set P(S) with relation R given by set inclusion. ISR a partial order. (c) De ne a lattice. Explain its properties. [4+8+4] 1 of 2 www.studentyogi.com www.studentyogi.com
10. 10. www.studentyogi.com www.studentyogi.com Code No: R05210502 Set No. 4 4. (a) G is a group of positive real numbers under multiplication, G‘ is a group of all real numbers under addition.Let f G x G‘ such that x G, Log10 x G‘ and (x, log10 x ) f. Show that f is an isomorphism from G to G?. (b) If Z is the additive group of integers, then prove that the set of all multiplies of integers by a xed integer m is a subgroup of Z. [10+6] 5. (a) In howmany ways can we place 4 red balls, 4 white balls and 4 blue balls in 6 numbered boxes. (b) Howmany integers between 1 and 1,00,000 have the sum of digits equal to 18. [16] 6. (a) Solve the recurrence relation ar = 3ar - 1 + 2, r = 1, a0 = 1 using generating function. (b) Find a recurrence relation for an the number of n-digit ternary sequences without any occurrence of the subsequence ‘012’. [ A ternary sequences is a sequence composed of 0s, 1s and 2s.] [8+8] 7. Derive the minimum spanning tree from the following graph using Kruskal’s ap- proach. Shown all intermediate steps. Figure 7. [16] Figure 7 8. (a) Write a brief note about the basic rules for constructing Hamiltonian cycles. (b) Using Grinberg theorem nd the Hamiltonian cycle in the following graph. Figure 8b [16] Figure 8b 2 of 2 www.studentyogi.com www.studentyogi.com