MCS-013
Discrete Mathematics
There are eight questions in this assignment, which carries
80 marks. Rest 20 marks are for viva-voce. Answer all the questions. You may
use illustrations and diagrams to enhance the explanations. For more details,
go through the guidelines regarding assignments given in the Programme Guide.
Question 1
(a)
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Explain different logical
connectives with the help of examples.
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(3 Marks)
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(b)
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Make truth table for followings:
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(3 Marks)
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i) p→(q
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∨ ~ r) ∧ (~p ∨ ~r)
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ii) p→(r
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∧ ~ q) ∧ (~p ∧ ~q)
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(c)
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Draw Venn diagram to represent
followings:
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(2 Marks)
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i) (A ∪ B) ∪ (B ∩ C ∪ D)
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ii) (A ∪ B ∩ C) ∩ (C~A)
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(d)
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Explain logical equivalence with
the help of example.
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(2 Marks)
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Question 2
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(a)
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Write down suitable mathematical
statement that can be represented by the
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(2 Marks)
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following symbolic properties.
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i)
( $ x) ( "y) ( "z) P
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ii) ( "x) ( " y) ( $
z) P
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(b)
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Write the following statements in
the symbolic form.
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(2 Marks)
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i)
Some students can pass in exam.
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ii) Everything is having life.
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(c)
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What is indirect method of proof?
Example with example.
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(2 Marks)
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(d)
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What is relation? Explain equivalence relation with the help
of an example.
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(4 Marks)
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(a) Draw logic circuit for the following Boolean
Expressions:
i)
(x y z) + (x+y+z)'
(2 Marks)
(b) Find dual
of Boolean Expression for Q,
in the figure given below.
(2 Marks)
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Figure 1: Logic Circuit
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(c)
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Explain De Morgan’s laws in
relation to Boolean Algebra.
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(2 Marks)
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(d)
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What is principle of mathematical
induction? Explain with the help of an
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(4 Marks)
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example.
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Question 4
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(a)
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How many different committees
can be
formed of 12 professionals,
each
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(3 Marks)
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containing at least 2 Professors,
at least 3 Lecturers and 3 Administrative
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Officers, from
a set of
5 Professors, 8
Lectures, and 5
Administrative
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Officers.
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(b)
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There are two mutually exclusive
events A and B with P(A) =0.7 and P(B)
=
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(3 Marks)
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0.6. Find the probability of followings:
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i)
A and B both occur
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ii) Both A and B does not occur
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iii) Either A or B does not occur
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(c)
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What is set? Explain the basic
properties of sets.
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(4 Marks)
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Question 5
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(a)
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How many words can be formed using
letter of UNIVERSITY using each
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(2 Marks)
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letter at most once?
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i)
If each letter must be used,
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ii) If some or all the letters may
be omitted.
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(b)
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Show using truth table that:
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(2 Marks)
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(
p → q) → q ⇒ p ∨ q
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(c)
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Explain
whether (p ∨ q) ® (q ®
r) is a tautology or not.
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(2 Marks)
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(d)
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Explain addition theorem in probability.
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(2 Marks)
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(e)
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Prove that the inverse of one-one
onto mapping is unique.
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(2 Marks)
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9
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(a)
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How many ways are
there to distribute
15 district objects into 5
distinct
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(2 Marks)
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boxes with:
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i)
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At least three empty box.
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ii) No empty box.
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(b)
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Explain principle of
multiplication with an example.
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(2 Marks)
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(c)
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Set A,B and C are: A = {1, 2,
3,5, 8, 11 12,13}, B = { 1,2, 3
,4, 5,6 } and
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(3 Marks)
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C= { 7,8,12, 13}. Find A Ç
B È C , A È
B È C, A È
B Ç C and (B~C)
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(d)
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In a class of 40 students; 30 have
taken science; 20 have taken mathematics
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(3 Marks)
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and 8 has neither taken mathematic
nor science. Find how many students
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have taken:
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i)
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both subjects.
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ii)
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exactly one subject
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Question 7
(a)
What is power set? Write
power set of set A={1,2,5,6,7,9}.
(b)
Draw truth table for (P®Q) N (Q®P) and
explain whether it is a tautology or not.
(c)
What is a function?
Explain domain and range in context of function, with the help of example.
(d)
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State and prove the Pigeonhole
principle.
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(3Marks)
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Question 8
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(a)
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Find inverse of the following
functions
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(2 Marks)
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f(x) =
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x 2 + 2
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x ≠ 3
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x −
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3
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(b)
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Explain circular permutation with
the help of an example.
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(2 Marks)
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(c)
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Give geometric representation for
followings:
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(3 Marks)
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i) {
3} x R
ii) {1,
2) x ( 2, 3)
(d) Show whether
√15 is rational or irrational. (3
Marks)
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