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Posted Date: 26 Jan 2009 Posted By: A santhosh Member Level: Gold
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2008 Jawaharlal Nehru Technological University, Hyderabad Information Technology Code No: R05 II B.Tech I Semester Regular Examinations, November 2008 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE Question paper
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 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)
(b) Define 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 today then it is no fun to snow ski today
It is no fun to snow ski today
Therefore it is hot today
UNIVERSE = DAYS. [5+6+5]
3. (a) Consider f; Z+ ! Z+ define 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) Define a lattice. Explain its properties.
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 8 x G, Log10 x e G‘
and (x, log10 x ) 2 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 fixed 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]
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
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