(An Autonomous Institution, Affiliated to Anna University, Chennai)
QUESTION BANK
NAME OF THE PROGRAMME
BE/B.Tech – CSE/IT/AIDS/CSBS/CS/AIML
YEAR / SEMESTER
II/IV
REGULATIONS
2022
COURSE CODE
MA2401
COURSE NAME
DISCRETE MATHEMATICS
FACULTY NAME
K. R. PRABAKARAN
LEVEL : UG
Contact Number
REVISED BLOOMS TAXONOMY(RBT):
L1- Remembering
L2 - Understanding
L3 - Applying
L4 - Analyzing
L5 - Evaluating
L6 - Creating
UNIT-I: ALGEBRAIC STRUCTURES
Semigroup – Monoids – Groups – Subgroups – Abelian groups – Lagrange’s theorem – Rings (examples
only) – Integral domain – Fields – Definition and examples
PART-A (2-Marks)
S.NO
QUESTIONS
CO
RBT
LEVEL
L2
1
Define integral domain and give an example
CO1
CO1
2
Find the idempotent elements of 𝐺 = {1, −1, 𝑖, −𝑖} under the binary
operation multiplication
CO1
3
If a is a generator of a cyclic group G, then show that 𝑎 −1 is also a
generator of G
4
Show that every cyclic group is abelian
CO1
CO1
5
Show that if every element in a group is its own inverse, then the group is
abelian.
6
Prove that the identity element of a group is unique
CO1
L1
7
Show that the union of two subgroups of a group need not be subgroup.
CO1
L1
8
Define isomorphism of groups
CO1
L2
1
L1
L1
L2
L1
9
State any two properties of a group
CO1
L1
10
Prove that identity element in a group is unique
CO1
L2
11
Prove that every subgroup of an abelian group is normal
CO1
L1
12
Define a ring
CO1
L1
13
Define a field in an algebraic system
CO1
L2
14
Give an example of a ring which is not a field
CO1
L1
15
Define a commutative ring
CO1
L2
PART-B (16-Marks)
CO
RBT
Level
Marks
8
L2
8
L3
8
L2
CO1
1
Show that the union of two subgroups of a group G is again a subgroup of
𝐺 if and only if one is contained in the other
2
Prove that every subgroup of a cyclic group is cyclic
CO1
CO1
3
Prove that the necessary and sufficient condition for a non-empty subset H
of a group (𝐺,∗)to be a subgroup is 𝑎, 𝑏 ∈ 𝐻 ⇒ 𝑎 ∗ 𝑏−1 ∈ H
4
State and prove the Fundamental theorem of group homomorphism
CO1
8
L2
5
Prove that every subgroup of cyclic group is cyclic
CO1
8
L3
6
Prove that the intersection of any two subgroups of a group ( G , *) is again
a subgroup of (G ,*).
CO1
8
L2
7
Prove that every finite group of order n is isomorphic to a permutation
group of order n
CO1
8
L3
8
State and prove the Lagrange’s theorem for group. Is the converse true?
CO1
8
L3
9
State and prove Lagrange’s theorem
CO1
8
L3
10
Show that the set of all non-zero real numbers is an abelian group under *
defined by a*b = ab/2
CO1
8
L2
11
State and prove Cayley’s Theorem
CO1
8
L2
CO1
12
* on R defined by x * y = x+y+2xy x,y Є R Check : (1) (R,*) is a monoid
or not. (2) Is it commutative? (3) Which elements have Inverses and what
are they?
8
L3
State and Prove Fundamental Theorem on homomorphism of
CO1
8
L2
13
2
groups.
14
CO1
If (G, *) is an abelian group,iff
8
L2
UNIT-2: COMBINATORICS
Introduction to Basic Counting Principles, Formulae behind nPr, nCr - Balls and Pins problems - PigeonHole
Principle - Recurrence relations – Generating Functions - Introduction to Proof Techniques - Mathematical
Induction
PART-A (2-Marks)
S.NO
QUESTIONS
CO
RBT
LEVEL
1
How many edges are there in a graph with 10 vertices each of degree 6?
CO2
L1
2
State Pigeon hole Principle
CO2
L1
3
How many permutations are there in the word MALAYALAM?
CO2
L2
4
Find the value of ‘n’ if
CO2
5
Define Generating Functions
CO2
L2
6
Use mathematical induction to show that 𝑛! ≥ 2𝑛-1 , 𝑛 = 1,2,3, …
CO2
L1
7
How many permutations of (a ,b , c, d, e, f, g ) end with a?
CO2
L1
8
Write the generating function for the sequence 1, a , a 2 , a 3 , a 4 ……..
CO2
L2
9
Use mathematical induction to show that 1 + 2 + 3 + ... + n = n( n + 1)/2
CO2
L2
CO2
10
Prove that if n and k are positive integers with n=2k, then n!/2 k is an
integer.
PART-B (16-Marks or 8-Marks)
L1
L2
CO
RBT
Level
Marks
L1
8
CO2
1
Let m be any odd positive integer. Then prove that there exist a positive
integer n such that m divides 2 𝑛 − 1
2
Solve the recurrence relation 𝑎𝑛+1 − 𝑎𝑛 = 3𝑛 2 − 𝑛, 𝑛 ≥ 0, 𝑎0 =3
CO2
L1
8
CO2
L1
8
Use the method of generating function to solve the recurrence relation
3
𝑎𝑛 = 4𝑎𝑛−1 − 4𝑎𝑛−2 + 4 ; 𝑛 ≥ 2 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝑎0 = 2 𝑎𝑛𝑑 𝑎1 = 8.
3
4
Find the number of integers between 1 to 250 that are not divisible by any
of the integers 2,3,5 and 7
CO2
Solve the recurrence relation an = -3an-1 – 3an-2 – an-3 given that
CO2
L1
8
L3
8
L2
8
L1
8
L2
8
L1
8
L3
8
L2
16
L2
16
L1
8
L1
8
L1
8
L2
8
L2
8
5
a0 = 5 ,a1 = 9 and a2 = 15.
6
Find the number of positive integers ≤ 1000 and not divisible by any of
3,5,7 and 22.
CO2
7
Solve the recurrence relation an = 3an-1 + 1, n ≥ 1,with a0 = 1, by the method
of generating function .
CO2
8
Prove that number of subsets of set having n elements is 2 n
CO2
9
Use mathematical induction to prove the inequality n< 2n for all positive
integer n
CO2
10
Solve the recurrence relation an – 7an-1 + 6an-2 = 0, for n ≥ 2 with initial
conditions ao= 8 and a1 = 6. Use generating function
CO2
(i)Prove that in any group of six people there must be atleast 3mutual
friends or 3 mutual enemies.
CO2
11
12
(ii)Prove by mathematical induction that 6n+2 +72n+1 is divisible by 43 for
each positive integer n
(i)There are six men and five women in a room. Find the number of ways
four persons can be drawn From the room if (a) they can be male or female
(b)two must be men and two women (c)they must all are of the same sex
CO2
(ii)Find the number of distinct permutations that can be formed from all the
letters of each word (a) RADAR (b) UNUSUAL
Solve an = 6an-1-11an-2+6an-3 with initial conditions a0 = 2, a1 = 5 and
CO2
13
a2 = 15
Prove by principal of mathematical induction, for ‘n’ positive integer
14
CO2
1 2 + 22 + 32 +………………….+ n2 = n(n+1)(2n+1) /6
15
How many bit strings of length 10 contain (a)exactly four 1’s (b) atleast
four 1’s (c) atmost four 1’s (d) an equal number of 0’s and 1’s?.
CO2
16
How many positive integers n can be formed using the digits 3,4,4,5,5,6,7,
if n has to exceed 5000000?
CO2
17
Solve the recurrence relation an = 3an-1 +1, n ≥ 1, with a0 = 1, by the
Method of generating function.
CO2
4
a) Find the generating function of Fibonacci sequence
18
CO2
b) Solve an = 6an-1-11an-2+6an-3 with initial conditions a0 = 2, a1 = 5 and
a2 = 15
L3
16
UNIT-3: BASIC GRAPH THEORY
Graphs and digraphs, complement, isomorphism, connectedness and reachability, adjacency matrix, Eulerian
paths and circuits in graphs and digraphs, Hamiltonian paths and circuits in graphs and tournaments
PART-A (2-Marks)
S.NO
QUESTIONS
CO
RBT
LEVEL
1
How many edges are there in a graph with 10 vertices each of degree 6?
CO3
L2
2
Define complete bipartite graph with example
CO3
L1
3
Define Complete graph
CO3
L2
4
Define the isomorphism between the graphs
CO3
L3
5
State handshaking theorem
CO3
L1
6
Define a pseudo graph graph with an example
CO3
L1
7
How many edges are there in a graph with 10 vertices each of degree 5?
CO3
L2
8
Define Graph and Subgraph with example
CO3
L1
9
State handshaking theorem
CO3
L2
10
Prove that every cyclic group is Abelian
CO3
L1
11
Define isomorphism of two graphs
CO3
L2
12
Draw a complete bipartite graph of K2,3 and K3,3 .
CO3
L1
13
Define strongly connected graph
CO3
L2
14
Define Eulerian graph with examples
CO3
L2
5
15
Give an example of a graph which is Eulerian but not Hamiltonian
PART-B (16-Marks or 8-Marks)
CO3
CO
CO3
1
Prove that a connected graph G is Eulerian if and only if all the vertices are
even
CO3
2
Give an example of a graph which is 1. Eulerian but not Hamiltonian 2.
Hamiltonian but not Eulerian 3. Hamiltonian and Eulerian 4. Neither
Hamiltonian nor Eulerian
CO3
3
Prove that the maximum number of edges in a simple disconnected graph
G with n vertices and k components is (𝑛 − 𝑘)(𝑛 − 𝑘 +1)/ 2
4
Prove that the number of vertices of odd degree in any graph is even
CO3
5
Draw the graph with 5vertices,𝑨, 𝑩,𝑪, 𝑫, 𝑬 such that 𝑑𝑒𝑔(𝐴) = 3,B is an
odd vertex, 𝑑𝑒𝑔(𝐶) = 2 and 𝑫 and𝑬 are adjacent.
CO3
6
Show that the maximum number of edges in a simple graph with n vertices
is n(n – 1)/2.
CO3
7
If G is a simple graph with n vertices with minimum degree δ(G) ≥ n/2 ,
show that G is connected
CO3
8
Prove that a simple graph with n vertices must be connected if it has more
than (𝑛−1)(𝑛−2) /2 𝑒𝑑𝑔𝑒𝑠..
CO3
L1
RBT
Level
Marks
L1
8
L2
8
L1
8
L2
8
L1
8
L2
8
L1
8
L1
8
UNIT- 4: TREES, PLANER GRAPH AND COLOURING OF A GRAPH
Trees; Planar graphs, Euler’s formula, dual of a planer graph, independence number and clique number,
chromatic number, statement of Four-color theorem
PART-A (2-Marks)
S.NO
1
QUESTIONS
Define Tree
6
CO
RBT
LEVEL
CO4
L1
2
What is Branch and Chord?
CO4
L2
3
What is proper coloring?
CO4
L2
4
What is a fundamental circuit?
CO4
L2
5
Define the Terms: Rank and Nullity
CO4
L3
6
State two properties of binary tree.
CO4
L2
CO4
L3
CO4
L2
CO4
L3
7
8
9
Define the dual of the graph?
Define chromatic polynomial
Define Prim's algorithm
10
Define Kruskal’s algorithm
CO4
L1
11
Define spanning trees in a weighted graph
CO4
L2
12
Define Independent number
CO4
L1
13
Define Clique number
CO4
L2
14
What are the applications of planar graphs
CO4
L3
15
Define planar graph
CO4
L2
16
What are the radius and diameter in a tree.
CO4
L1
17
What is proper coloring?
CO4
L2
18
Define Chromatic number
CO4
L2
19
Write the properties of chromatic numbers
CO4
L2
20
Define four color theorem.
CO4
L2
21
Define Chromatic partitioning.
CO4
L2
22
Define independent set and maximal independent set.
CO4
L2
23
Distinguish between planar and non- planar graphs.
CO4
L2
PART-B (13- Marks Or 16-Marks or 8-Marks)
1
Explain what is meant by spanning tree? Find all spanning trees
for the following graph :
7
CO
RBT
Level
Marks
L1
8
CO4
Use the algorithm of Kruskal to find a minimum weight spanning tree
2
CO4
in G
Use the algorithm of Prim’s to find a minimum weight spanning tree
in the following graph
8
L1
8
L2
8
CO4
3
4
L2
Prove that If in a graph G, there is one and only one path between every pair CO4
of vertices, G is a tree.
5
Prove that any connected graph with n vertices and n-1 edges is a tree
CO4
L1
8
6
Prove that every connected graph has atleast one spanning tree.
CO4
L2
8
7
Write an algorithm to find the shortest spanning tree in a weighted graph?
CO4
L1
8
8
Show that a graph is a tree if and only if it is minimally connected
CO4
L2
8
9
State the two algorithms to find the shortest spanning tree in a weighted
graph. Write the details of one of these algorithms?
CO4
L1
8
10
Prove that w.r.t any of its spanning trees a connected graph with nvertices and e-edges has (n-1) tree branches and (e-n+1) chords?
CO4
L3
8
8
11
Prove that graphs K5 and K3,3 are non planar.
CO4
12
Prove that a connected graph G with n vertices and e edges has e - n+2
regions
CO4
Draw a spanning tree of the following graph given below and list all the
fundamental circuits with respect to this tree---
CO4
L2
8
L1
8
L2
8
L1
8
13
Define the chromatic number and chromatic polynomial of a graph? Find
the chromatic number and the chromatic polynomial of the following
CO4
14
graph----15
Prove that every tree with two or more vertices is 2- Chromatic
16
A graph with at least one – edge is 2-chromatic iff it has no–cycle of odd
length
(It has a cycle of even length).(or) Konig’s Theorem
17
If G is a tree with n vertices , then prove that its chromatic polynomial is
Pn( λ) = λ (λ-1)n-1
18
Prove that a graph of n vertices is a complete graph iff its chromatic
polynomial is Pn( λ) = λ (λ-1(λ-2)………………..( λ- n+1). (N/D-2016)
19
State and prove five- color theorem
20
Write a note on chromatic polynomials and their applications.(A/M-2017)
9
UNIT-5: LOGIC AND BOOLEAN ALGEBRA
Propositional calculus - propositions and connectives, syntax; Semantics - truth assignments and truth tables,
validity and satisfiability, tautology; Adequate set of connectives; Equivalence and normal forms; Compactness
and resolution; Formal reducibility - natural deduction system and axiom system; Soundness and completeness.
Introduction of Boolean algebra, truth table, basic logic gate, basic postulates of Boolean algebra, principle of
duality, canonical form, Karnaugh map.
PART-A (2-Marks)
S.NO
QUESTIONS
CO
RBT
LEVEL
1
Define Tautology with an example.
CO5
L1
2
Without using the truth table, prove thatp qrqpr
CO5
L2
3
Prove that PQ  RQ  PRQ
CO5
4
Show that p (q r) and ( p q)  ( p r) are logically equivalent
CO5
5
Show thatP(QR)(QR)(PR)R , without using truth table
CO5
6
Show that (𝑝 → (𝑞 → 𝑟)) → ((𝑝 → 𝑞) → (𝑝 → 𝑟)) is a tautology.
CO5
7
What are the contra positive, the converse and the inverse of the
conditional statement “If you work hard then you will be rewarded”.
CO5
8
Using truth table show that P  ( P  Q )  P .
CO5
L1
9
Construct the truth table for P Q
CO5
L1
10
Express the statement “Good food is not Cheap” in symbolic form.
CO5
L2
11
Using truth table show that
CO5
L2
12
Define Boolean algebra
CO5
L2
CO5
13
Give the contra positive statement of the statement, “If there is rain, then I
buy an umbrella”
14
Without using truth table show that P   Q  P  P   P  Q .
CO5
15
Prove that X + XY = X
CO5
L1
16
What is principle of duality? Give Example
CO5
L2
17
Define Minterm and Maxterm
CO5
L1
10
L1
L3
L2
L1
L2
L1
L2
17
Draw a general K-map for four variables A, B, C and D
CO5
18
Express A  B in terms of the connectives   ,  .
CO5
PART-B (13- Marks Or 16-Marks or 8-Marks)
Without using truth table find the PCNF and PDNF of
CO
P→(QP) (P→ (QR))
CO5
2
Obtain the principal disjunctive normal form and principal conjunction normal
Form of (P->R) (Q↔P) using equivalences.
Without using truth table find PCNF and PDNF of
CO5
[𝑃 ⟶ (𝑄 ∧ R)] ∧[¬P→(¬Q∧¬R)]
In any Boolean algebra, prove that the following statements are equivalent:
4
1) 𝑎 + 𝑏 = 𝑏
2) 𝑎. 𝑏 = 𝑎
3) 𝑎′ + 𝑏 = 1
L2
RBT
Level
Marks
L3
8
L3
8
L2
8
L3
8
CO5
1
3
L2
CO5
4) 𝑎. 𝑏′ = 0
5
In a Boolean algebra, prove that i) (a + b)’ = a’.b’ ii) (a . b)’ = a’+b’
CO5
L2
8
6
State and prove the De Morgan’s laws in a Boolean algebra
CO5
L3
8
7
Explain about basic logic gates
CO5
L2
8
8
Define Karnaugh map with examples
CO5
L2
8
9
Give the Boolean function F (W, X, Y, Z) = ∑ (0, 4, 8, 9, 10, 11, 12, 13, 15).
Reduce it byusing K-Map.
CO5
L2
8
10
Reduce F (A, B, C, D) = ∑ (1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 14, 15) using K-Map
CO5
L3
8
11
Using K-Map, Simplify the following expression in four variables
L2
8
L3
8
CO5
F (A, B, C, D) = m1 + m2 + m4 + m5 + m9 + m11 + m12 + m13.
12
Convert the Boolean function F(X, Y,Z)= (X+ Y)(Y+ Z) into canonical form
__________
11
CO5