Jawaharlal Nehru Technological University Kakinada 2010 B.Tech Information Technology 2-1 ENGG.R07 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE SUPPLY SET-2 - Question Paper

Code No: X0522/R07 Set No. 2

II B.Tech I Semester Supplementary Examinations, May 2010

MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE

( Common to Computer Science & Engineering and info
Technology)

Time: three hours Max Marks: 80

ans any 5 ques.

All ques. carry equal marks


1. (a) Let p,q and r be the propositions.
P: you have the ?ee
q: you miss the ?nal exam.
r: you pass the course.
Write the subsequent proposition into statement form.
i. P ? q
ii. 7p ? r
iii. q ? 7r
iv. pVqVr
v. (p ? 7r) V (q ?~ r)
vi. (p?q) V (7q?r)

(b) De?ne converse, contrapositive and inverse of an implication. [Marks 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 a few 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. [Marks 5+6+5]

3. (a) Determine whether the subsequent relations are injective and/or subjective function. obtain universe of the functions if they exist.
i. A = {v,w, x, y, z}
B = {1, 2, 3, 4, 5}
R = {(v,z),(w,1),(x,3),(y,5)}
ii. A = {1, 2, 3, 4, 5}
B = {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). [Marks 8+8]

4. (a) If G is the set of all positive rational numbers, then prove that G is an abelian
group under the composition de?ned by o such that aob = (ab)/3 for a, b ?
G with usual addition as the operation.

(b) Let P(S) be the power set of a non ?empty set S. Let n be an operation in
P(S). Prove that associate legal regulations and commutative legal regulations are actual for the operation
n in P(S). [Marks 12+4]

5. (a) How many out comes are found from rolling in distinushable dice.

(b) obtain the number of distmcttriples (x1,x2,x3)of non-negative integers satisfying x1 + x2 + x3 < 15.

(c) How many integers ranging from one and 1,000 have a sum of digits of integer numbers
equal to 10. [Marks 16]

6. (a) Solve an - 5an - one + 6an - two = 2n + n

(b) Solve an - 4an - one + 4an - two = (n + 1)
2
provided a0 = 0, a1 = one . [Marks 8+8]

7. (a) De?ne spanning tree. elaborate its characteristics.

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

8. A mathematics department plans to o?er 7 graduate courses next semester,
namely combinatorics (C), group theory (G), Field theory (F), numerical analysis
(N), topology (T), applied mathematics (A), and real analysis (R). The mathematics graduate students and the courses they plan to take are as follows: [Marks 16]
learner Courses
Abe C,F,T
Bob C,G,R
Carol G,N
Dewitt C,F
Elaine F,N
Fred C,G
learner Courses
George A,N
Herman F,G
Ingrid C,T
Jim C,R,T
Ken A,R
Linda A,T

Earning:   Approval pending.