MCS-013
1. (a) What is proposition? Explain whether, x-y
>5 is a proposition or not. (2 Marks)
(b) Make truth table for followings.
(i) p→ (~q Ú ~ r) Ù (~p Ú r)
(ii) p→ (r Ú q) ⋀ (~p Ù ~q) (4 Marks)
(c) Draw a Venn diagram to represent followings:
(i) (A B) (BC) (ii) (AB) (C~A) (2 Marks)
(d) Give geometric representation for followings:
(i) R x { 4}; where R is a natural number
(ii) {2, 2) x ( 2, -4) (2 Marks)
2. (a) Write down suitable mathematical statement
that can be represented by the following symbolic properties.
(i)
(x) ( y) (z) P
(ii)
(x) ( y) (z) P (2 Marks)
(b) Write the following statements in the
symbolic form.
(i) Some birds can not fly
(ii) Nothing is correct (2 Marks)
(c) What is modus ponen and modus tollen? Write
one example of each. (2 Marks)
(d) What is relation? Explain equivalence relation
with the help of an example. (4 Marks)
3. (a) Make
logic circuit for the following Boolean expressions: (2 Marks)
(i) (x' y' z) + (xyz)
(b) Find Boolean Expression of Q in the figure
given below (2 Marks)
Figure 1: Boolean Circuit
(c) Find Boolean
Expression of Q in the figure given below. (2 Marks)
Figure 2: Boolean Circuit
(d) What
is integer partition? Write down all partitions of 8. Also find
and
. (4 Marks)
4. (a) How many different committees can be formed of
10 professionals, each containing at least 4 Professors, at least 3 General
Managers and 3 Finance Advisors from list of 10 Professors, 12 General Managers
and 5 Finance Advisors? (3 Marks)
(b) There are two
mutually exclusive events A and B with P(A) =0.5 and P(B) = 0.4. Find the
probability of followings:
(i) A does not occur
(ii) Both A and B does not occur
(iii) Either A or B does not occur (3
Marks)
(c) What is set? Explain the basic properties of
sets. (4 Marks)
5. (a) How
many words can be formed using letter of UMBRELLA using each letter at
most once?
(i) If each letter must be used,
(ii) If some or all the letters may be omitted. (2 Marks)
(b) Show using truth that: (p ® q) ® q ⇒ p ⋁ q (2 Marks)
(c) Explain whether (p ® q) ® (q ® r) is a tautology or
not. (2 Marks)
(d) Prove that: 1 + 2 + 3 + . . . + n = ½n(n + 1) using
mathematical induction. (4 Marks)
6. (a) How
many ways are there to distribute 15 district objects into 5 distinct boxes
with:
(i) At least three empty box.
(ii) No empty box. (2 Marks)
(b) Explain principle of multiplication with an
example. (2 Marks)
(c) Set A,B and C are:
A = {1, 2, 3,5, 8, 11 12,13}, B = { 1,2, 3 ,4, 5,6 } and C { 7,8,12, 13}.
Find A Ç B È C, A È B È C, A È B Ç C and (B~C)
(d) Out of 30 students
in college 15 takes art courses, 8 takes biology courses and 6 takes chemistry.
It is also known that 3 students take all the three courses. Show that 7 or
more students taken none of the course. (3 Marks)
7. (a) Explain
principle of duality with example? (2 Marks)
(b) What is power set? Write power set of set
A={1,2,3,4,5,6}. (3 Marks)
(c) What is a function? Explain domain and range
in context of function with example. (2
Marks)
(d) State and prove the Pigeonhole principle. (3Marks)
8. (a) Find
inverse of the following functions
(2 Marks)
(b) Explain circular permutation with the help of
an example. (3 Marks)
(c) What
is indirect proof? Explain with an example. (2 Marks)
(d) What
is Boolean algebra? (3 Marks)
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(Last 5 year solved question
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