08 r059210502 mathematical foundation of computer science

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Code No: R059210502 Set No. 1

II B.Tech I Semester Supplimentary Examinations, February 2008MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE

( Common to Computer Science & Engineering, Information Technologyand Computer Science & Systems Engineering)

Time: 3 hours Max Marks: 80Answer any FIVE Questions

All Questions carry equal marks

1. (a) Let p,q and r be the propositions.P: you have the fleeq: you miss the final examination.r: you pass the course.Write the following proposition into statement form.

i. P q

ii. 7p r

iii. q 7r

iv. pVqVr

v. (p 7r) V (q r)

vi. (pq) V (7qr)

(b) Define 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)denotethe statement x plays soccer. The domain is the let of all people. Writeeach of the following proposition in English.

i. x (P (x) Q (x))

ii. x (P (x) Q (x))

iii. x (P (x) V Q (x))

(b) Write the negation of each of the above propositions, both in symbols and inwords. [6+10]

3. (a) Define partial order and total order of relations.

(b) If R is a relation defined on the set Z by a R b if a-b is a non negative eveninteger. Determine if R define a partial order and total order. [4+12]

4. (a) If a, b are any two elements of a group (G, .) which commute show that

i. a - 1 and b commute,

ii. b - 1 and a commute and

iii. a - 1 and b - 1 commute.

(b) Let S be a non-empty set and o be an operation on S defined by aob=a for a,

b S. Determine whether o is commutative and associative in S. [12+4]

5. (a) Howmany different 8-digit numbers can be formed by arranged the digits1,1,1,1,2,3,3,3.

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(b) In howmany ways can 5 different messages be delivered by 3 messanger boys ifno messanger boy is left unemployed. The order in which a messanger deliverdhis message is immaterial. [16]

6. (a) Solve an - 5an - 1 + 6 an - 2 = 1.

(b) Solve an + 5an - 1 + 6 an - 2 = 42 (4)n. [8+8]

7. Derive the

(a) breadth first tree and

(b) depth first search spanning trees for the following graph. Figure 7b [8+8]

Figure 7b

8. (a) Find the chromatic numbers ofi. a bipartite graph K3,3

ii. a complete graph Kn and

iii. a wheel graph W1,n.

(b) Find the chromatic number of the following graph. Figure 8b [16]

Figure 8b

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1. (a) Let p,q and r be the propositions.P: you have the flee

q: you miss the final examination.r: you pass the course.Write the following proposition into statement form.

i. P q

ii. 7p r

iii. q 7r

iv. pVqVr

v. (p 7r) V (q r)

vi. (pq) V (7qr)


2. (a) Let P(x) denote the statement. x is a professional athlete and let Q(x)denotethe statement x plays soccer. The domain is the let of all people. Writeeach of the following proposition in English.

i. x (P (x) Q (x))

ii. x (P (x) Q (x))

iii. x (P (x) V Q (x))

(b) Write the negation of each of the above propositions, both in symbols and inwords. [6+10]



4. (a) H is a non-empty complex of a group G. Prove that the necessary and sufficientcondition for H to be a subgroup of G is a, b H ab - 1 H, where b - 1 isthe inverse of b in G

(b) Prove that the mapping f: Z Z such that f(x) = -x for x Z where Z is anadditive group of integers is an automorphism. [If f: G G is an isomorphism

from a group G to itself then f is called on automorphism of G.] [10+6]

5. (a) Howmany out comes are obtained from rolling in distinushable dice.

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(b) Find the number of distmct triples (x1,x2,x3)of non-negative integers satisfy-ing x1 + x2 + x3 < 15.

(c) Howmany integers between 1 and 1,000 have a sum of digits of integer numbersequal to 10. [16]

6. (a) Solve the recurrence relation ar = 3ar - 1 + 2, r 1, a0 = 1 using generatingfunction.

(b) Solve an - 4an - 1 + 4an - 2 = (n + 1)2 given a

0= 0, a

1= 1 . [8+8]

7. Find the minimum spanning tree for the graph shown in figure 7 below usingKruskals algorithm. Determine the cost of the minimum spanning tree. [16]

Figure 7

8. (a) How to determine whether a graph contains Hamiltonian cycle or not usingGrin berg theorem.

(b) Prove or disprove that there is an Hamiltonian cycle in the following graph.Figure 8b [16]

Figure 8b

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1. (a) Construed a truth table for each of there (easy) compound statements

i. (p q)(7p q)

ii. p (7qV r)

(b) Write the negation of the following statements.

i. Jan will take a job in industry or go to graduate school.

ii. James will bicycle or run tomorrow.

iii. If the processor is fast then the printer is slow.

(c) What is the minimal set of connectives required for a well formed formula.[8+6+2]

2. Prove using rules of inference or disprove.

(a) Duke is a Labrador retrieverAll Labrador retriever like to swinTherefore Duke likes to swin.

(b) All ever numbers that are also greater than2 are not prime2 is an even number2 is primeTherefore some even numbers are prime.UNIVERSE = numbers.

(c) If it is hot today or raining today then it is no fun to snow ski todayIt is no fun to snow ski todayTherefore it is hot todayUNIVERSE = DAYS. [5+6+5]



4. Show that the set G = { x/x = 2a 3b and a, b Z} is a group under multiplica-tion. [16]

5. (a) How many integers between 1 and 104 contain exactly one 8 and one 9

(b) How many integers between 105 and 106

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i. have no digit other than 2,5 or 8

ii. have no digit other than 0, 2, 5 or 8. [8+6+6]6. (a) Solve the recurrence relation an = an - 1 +

n(n + 1)2

, n 1

(b) Solve the recurrence relation ar = ar - 1 + ar - 2 using generating function. [8+8]

7. (a) Define spanning tree. What are its characteristics.

(b) Derive all possible spanning trees for the graph shown in Figure 7. [6+10]

Figure 7

8. (a) Determine whether the following two graphs are isomorphic or not. Figure 8a,8a

Figure 8a

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Figure 8a(b) Show that two simple graphs are isomorphic if and only if their Complements

are isomorphic. [16]

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1. (a) Let p,q and r be the propositions.P: you have the flee

q: you miss the final examination.r: you pass the course.Write the following proposition into statement form.

i. P q

ii. 7p r

iii. q 7r

iv. pVqVr

v. (p 7r) V (q r)

vi. (pq) V (7qr)


2. Prove using rules of inference or disprove.

(a) Duke is a Labrador retrieverAll Labrador retriever like to swinTherefore Duke likes to swin.

(b) All ever numbers that are also greater than2 are not prime2 is an even number2 is primeTherefore some even numbers are prime.UNIVERSE = numbers.

(c) If it is hot today or raining today then it is no fun to snow ski todayIt is no fun to snow ski todayTherefore it is hot todayUNIVERSE = DAYS. [5+6+5]

3. (a) Define 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) mod7, A=N7.

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(b) Let f and g be functions from the positive real numbers to positive real numbersdefined byf(x) = 2xg(x) = x2

Calculate f o g and g o f. [10+6]

4. (a) If G is the set of even integers i.e., G = { ... - 4, - 2, 0,2,4.....}, then prove thatG is an abelian group with usual addition as the operation.

(b) Let G = { - 1, 0, 1} . Verify whether G forms a group under usual addition.[12+4]

5. (a) A palindrome is a word that reads the same forward or backward. How many

9-letter palindromes are possible using English alphabets?(b) There are five different roads form city A to city B, three different roads form

city B to city C, and three different roads that go directly form A to C.

i. How many different ways are there form A to C altogether?

ii. How many different trips are there form A to C and back to A that visitB at least once. [16]

6. (a) Solve an - 8an - 1 + 16 an - 2 = 0 given a2 = 6 and a3 = 80.

(b) Solve an + 5an - 1 + 6 an - 2 = 3n2 - 2n + 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 first search algorithm. [8+8]

8. (a) Write a brief note about the basic rules for constructing Hamiltonian cycles.

(b) Using Grinberg theorem find the Hamiltonian cycle in the following graph.Figure 8b [16]

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Figure 8b

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08 r059210502 mathematical foundation of computer science

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