Punjab Technical University 2010-3rd Sem B.Sc BioInformatics .(BI) (rd) DISCRETE AND NUMERICAL MATHEMATICS - Question Paper
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Total No. of Questions : 13] [Total No. of Pages : 03
J-3462[S-1318] [2037] B.Sc.(BI) (Semester - 3rd)
DISCRETE AND NUMERICAL MATHEMATICS (B.Sc.(BI)-302)
Time : 03 Hours Maximum Marks : 75
Instruction to Candidates:
1) Section - A is compulsory.
2) Attempt any Nine questions from Section - B.
Section - A
Q1)
a) The Inclusion/Exclusion Theorem for three sets says that if there are three sets, A, B and C, then |A u B u C|=|A| + |B|+|C|
. Note that the converse of this theorem is not true. Given a set of 29 students, where 8 need housing and financial aid; 12 need housing, financial aid, and an e-mail account; 17 need an e-mail account and financial aid- 2 ,+2l(Aiefe!d.rfnaincial aid,
19 need e-mail accounts, and 4 students dont need anything.
(i) How many students need both housing and e-mail?
(ii) Change the numbers to 57, 8, 3, 21, 21, 32, 31, 8. What is your answer now?
(iii) Which answer is bogus and why? (2 x 3)
b) A = {1,2,4,5}, B = {a,b,c,f} and C = {a,5} are three given sets. (2) Compute (A C) - (A C) x B
Where , -, ? and x are well-known set-theoretic operations.
c) Construct truth table for the following formula: (2) ~ (PV~Q) <=> (P = > Q)
d) Obtain the equivalent disjunctive normal form of the formula :~ R V(PVR)
(2)
e) Define the following concepts giving one example of each. (3)
(i) 1-1 Onto function.
(ii) Anti-symmetric relation on a set.
(iii) Partial Order Relation.
f) (i) min-term in a Boolean Algebra. (2) (ii) 2s complement of a binary number.
g) Determine the validity of the conclusion (represented by C:) from the given set of premises.
P -> ~Q, P V R, ~R V ~S, S with conclusion C: ~Q. (2)
h) Find equivalent form in Predicate/Propositional Calculus of the following statements :
(i) Nine plus ten equal nineteen.
(ii) Some patients like all doctors.
(iii) Any integer is either positive or negative.
(iv) Every integer is also a real number. (4)
i) Write down formula for
(i) Picards method of successive approximation.
(ii) Modified Eulers method.
(iii) Newtons Raphson method. (3 + 2 + 2)
(9 x 5 = 45)
a
Q2) Which of the following are statements? Give reasons for your answer.
(i) What a lovely day !
(ii) 2+3 = 6
(iii) 2+x = 7
(iv) What is the time now?
Q3) Give an example of a finite linearly ordered set. You have to verify all the properties of a linearly ordered set for your example.
Q4) Let U = {1, 2, 3, 4, 5, 6, 7}, A = {1, 2, 7}, B = {1, 2, 3, 4, 5}, C = {4, 6, 7}.
(i) Find A-B and B-C.
(ii) Check that (A B) C = (A C) (B C).
(iii) State and verify any one of the De Morgans laws for A and B.
Q5) Let A = {1, 2, 3} and Rl and R2 be relations on A given by R = {(1, 1), (1, 2), (2, 3)}
R2 = {(1, 3), (2, 1), (1, 1), (3, 1)}
(i) Write the relation matrices of R1 and R2.
(ii) Find the relation matrix of R1R2.
(iii) Check whether RXR2 is an equivalence relation.
Q6) Check that (P VQ) V (~ PV~Q) is a tautology.
Q7) Draw the switching circuit corresponding to the function ((x+y).(yz))+(x+z).
Q8) In a survey, 80 Computer Science teachers were asked which of the three programming languages, Pascal, Fortran and C, have they used to introduce programming to students- 40 teachers said they have used Pascal, 35 said they have used Fortran, 25 said they have used C, 20 said they have used both Pascal and Fortran, 17 said they have used Pascal and C, 5 said they have used Fortran and C, and 3 said they have used all the three languages.
(i) How many teachers have used Fortran alone?
(ii) How many teachers have used both Pascal and C?
(iii) How many teachers have used Fortran, but not C?
(iv) How many used none of the three languages?
Q9) Approximate y and z at x = 0.1, using Picards method for the solution of the equations dy/dx = z, dz/dx = x3(y+z) given that y(0) = 1 and z(0) = 0.5
Q10)Solve dy/dx = y-(2x/y), y(0) = 1 in the range 0< = x< = 0.2 using Eulers method.
Q11)Use Runge Kutta method of find y when x = 1.2 in steps of 0.1 given that dy/dx = x2+y2 and y(1) = 1.5.
Q12)Given Sin45 = 0.7071, Sin 50 = 0.766, Sin 55 = 0.8192, Sin 60 = 0.866, Find Sin 52, using Newtons forward Interpolation formula.
Q13)Write short notes on the following
(a) Gauss elimination method.
(b) Bisection Method.
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