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Posted By: RAFI Member Level: Gold Posted Date: 18 Jun 2008
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2005 Indira Gandhi National Open University (IGNOU) M.C.A Computer Aplications MCS-013 : DISCRETE MATHEMATICS, December, 2005 Question paper
MCS-013
MCA (Revised)
Term-End Examination
December, 2005
MCS-013 : DISCRETE MATHEMATICS
Time : 2 hours " Maximum Marks : 50
Note : Question number 1 is compulsory. Attempt any three questions from the rest,
1. (a) Write the negation of the following statements : 2+2
o (i) For all x, x < x.
(ii) There exists x such that x = 2.
(b) Construct the circuit that produces the following output :
x' A (y V z')'- 4
(c) Let A = (1, 2, 3, 4, 5}. Construct a relation R from A to A such that R is reflexive and symmetric but not transitive. . . 4
(d) Prove by induction that h — n is divisible by 3 for
all positive integers. 4
(e) Determine all the integer solutions to
x + x + xg + x4 = 9, where x. > 1, i = 1, 2, 3, 4. 4
2. (a) A sequence of ten bits (O's and I's) is randomly
generated. What is the probability that at least one
of the bits is 0 ? 5
(b) Find the number of permutations of the word ATTEMDANT.
3
(c) Write the contrapositive of the statement
'If x is a positive real number, there is a number y such that y = x.'
3. (a) Given five points inside a square whose side has
length 2, prove that two are within a distance of 42 of each other, 5
(b) Prove, that ((p V q -> r) A (~p)) -> (q -* v) is a
tautology. 5
4. (a) A committee of three individuals decides a proposal.
Each individual votes either yes or no. The proposal is passed if it receives at least two yes votes. Design a circuit that determines whether the proposal passes. 5
rr+x"
(b} Find the domain and range of the function J-------,
where x takes real values.
(c) State whether the following statement is true or false. Give reasons for your answer.
"For any 3 sets A, B and C, and functions
f : A —> B, g : B -> C such that g o f is surjective,
then f and g must be surjective."
25. {a} How many boolean functions of n variables are
there ? Give reasons for your answer. 4
(b) Check whether the following argument is valid, using
a truth table. 6
"If Shalini leaves home before 9.00 AM or if she takes a taxi, she will reach office in time. She did leave after 10.00 AM and she did reach office in time. Therefore, Shalini must have taken a taxi."
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