Discrete Maths12

DISCRETE MATHEMAICS NOTES OLD SYLLABUS

MSC-IT 3RD SEM

UNIT-1

1.

a. a

b. b

c. c

d. d

2. If A = {1, 3, 5} and B = {1, 3, 5, 7} then A is a ___________ of B.

a. proper subset

b. improper subset

c. superset

d. both a)and b)

3.

a. a

b. b

c. c

d. d

4.

a. a

b. b

c. c

d. d

5. If the number of elements in as set is finite, then the set is called a ____________

a. finite set

b. infinite set

c. primitive

set

d. not defined

6. In any discussion if all the sets are subsets of a fixed set, then this set is called the________

a. superset

b. main set

c. universal set

d. a) and c)

7.


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

8.

The number of elements in a set is called the __________ of that set.

a. members

b. cardinal

number

c. size

d. length

9.

a. a

b. b

c. c

d. d

1.

a. a

b. b

c. c

d. d

2.

a. a

b. b

c. c

d. d

3.


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

4.

a. a

b. b

c. c

d. d

5.

a. a

b. b

c. c

d. d

6.

A function f has an inverse if and only if f is _________

a. one-one

b. onto

c. one-one and

onto

d. one-one and

into

7.

A set is a collection of ________________

a. well defined

objects

b. undefined objects

c. objects

d. only numbers

8. Identity function on a set is __________

a. unique

b. double

c. thrible

d. zero

9.

Let A be a set and ~ be an equivalence relation on A.

Then the set of all equivalence classes forms a ______ for A.

a. intersection

b. union

c. binary


MSC-IT 3RD SEM

d. partition

10.Two sets A and B are said to be _____________, if A is a subset of B and B is a subset of A.

a. proper subsets

b. void

c. equal

d. unequal

UNIT-2

1.

a. a

b. b

c. c

d. d

2.

a. a

b. b

c. c

d. d

3.

a. a

b. b

c. c

d. d

4.


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

5. The nth term of the series 1.4 + 4.7 + 7.10 + _______ is

a. (3n-2) (3n-1)

b. (3n) (3n + 1)

c. (3n-2) (3n + 1)

d. (3n + 2) (3n +

1)

6.

a. a

b. b

c. c

d. d

7

a. a

b. b

c. c

d. d

8.

a. a

b. b

c. c

d. d

9.


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

1 "Any non-empty subset of the set of all positive integers contains a smallest element". This principle is called as

a. well ordering

principle

b. ordering principle

c. grouping principle

d. ungrouping principle

2.

a. a

b. b

c. c

d. d

3.

a. a

b. b

c. c

d. d

4.

a. a

b. b

c. c

d. d

5.


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

6.

a. a

b. b

c. c

d. d

7.

If (a, b)= 1 then a and b are said to be ___________

a. prime

b. relatively

prime

c. divisors

d. zero divisors

8.If d divides both a and b then d is called the _____________of a and b

a. greatest common

divisor

b. least common

divisor

c. divisor

d. common divisor

9. The highest power of 3 contained in 100! is

a. 48

b. 44

c. 40

d. 39

10.The number of divisors of 9504 is

a. 42

b. 43

c. 44

d. 48

1

a. a

b. b

c. c

d. d

2


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

3.

a. a

b. b

c. c

d. d

4.

a. a

b. b

c. c

d. d

5.

a. a

b. b

c. c

d. d

6.

a. a

b. b

c. c

d. d

7.

Combinatorics is the branch of discrete mathematics concerned with _______

a. counting

problems

b. abstract algebra

c. derivative

problems


MSC-IT 3RD SEM

d. integrated

problems

8.

How many ways may one right and one left shoe be selected from six pairs of shoes without obtaining a pair

a. 40

b. 35

c. 30

d. 20

9.

In how many ways can a lady wear five rings on the fingers (not the thumb) of her right hand?

a. 6620

b. 6720

c. 6520

d. 6700

10.The next permutation to 4123 in the reverse Lexicographic order is

a. 3412

b. 3421

c. 2413

d. 4312

UNIT-3

1.

a. a

b. b

c. c

d. d

2.

a. a

b. b

c. c

d. d

Q3

a. a

b. b

c. c

d. d

Q4


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

Q5

a. a

b. b

c. c

d. d

Q6

a. a

b. b

c. c

d. d

Q7Combinatorics is the branch of discrete mathematics concerned with _______

a. counting

problems

b. abstract algebra

c. derivative

problems

d. integrated

problems

Q8 How many ways may one right and one left shoe be selected from six pairs of shoes without obtaining a pair.

a. 40

b. 35

c. 30

d. 20

Q9 In how many ways can a lady wear five rings on the fingers (not the thumb) of her right hand?

a. 6620

b. 6720

c. 6520

d. 6700

Q10 The next permutation to 4123 in the reverse Lexicographic order is


MSC-IT 3RD SEM

a. 3412

b. 3421

c. 2413

d. 4312

UNIT-4

a. a

b. b

c. c

d. d

Q2

a. a

b. b

c. c

d. d

Q3

a. a

b. b

c. c

d. d

4.

a. a

b. b

c. c

d. d

5. A poset S is said to be totally ordered set if for a, b in S

a. a > b

b. a = b

c. b > a

d. all a), b)and

c)holds

6.

Boolean Algebra is an algebra of ___________

a. logic


MSC-IT 3RD SEM

b. sets

c. rings

d. groups

7.

If S is a poset and a, b are in S such that a > b and there is no c in S such that a > c and c > b then we say that

a. b covers a

b. a covers b

c. a uncovers b

d. b uncovers a

8.

In a distributive lattice, if an element has a complement, then it is ____________

a. unique

b. 1

c. 0

d. double

9. The set L = {1, 2, 3, 4, 6, 12} the factors of 12, forms a lattice under the relation

a. multiplication

b. addition

c. divisibility

d. subtraction

10.

The set of natural numbers with 'a divides b' is _____________

a. not a totally ordered

set

b. ordered set

c. totally ordered set

d. unordered set

UNIT-5

a. a

b. b

c. c

d. d

2.

a. a

b. b

c. c

d. d

3.


MSC-IT 3RD SEM

. a. a

b. b

c. c

d. d

4.

A Poset S is said to be ___________ Set if for a, b in S exactly

one of the conditions, a > b, a = b or b > a holds

a. totally ordered

b. ordered

c. not ordered

d. completely

ordered

5.

A relation means ______________ on a set S.

a. dual relation

b. binary relation

c. reflexive relation

d. symmetric relation

6.

A ______________ is a set S with a relation R on it which is reflexive,

anti-symmetric and transitive

a. equivalent set

b. ordered set

c. implicit set

d. Partially ordered

set

Q7 If L is a finite lattice then L is

a. supremum

b. infimum

c. bounded

d. unbounded

Q8 If S is a poset and a, b are in S such that a > b and there is no c in S such that a > c and c > b, then we say that_________

a. b covers b

b. a covers a

c. a covers b

d. b covers a

9. The elements in level-1 are called ___________

a. electrons

b. atoms

c. neutrons

d. molecules

10. The least upper bound of a set A is called ______________

a. Infimum

b. Supremum


MSC-IT 3RD SEM

c. greatest element

d. least element

1.

a. a

b. b

c. c

d. d

2.

a. a

b. b

c. c

d. d

3.

a. a

b. b

c. c

d. d

Q4

a. a

b. b

c. c

d. d

Q5

a. a

b. b

c. c

d. d

Q6 A finite machine M consists of a finite set Q of _____________

a. input symbols

b. output symbols

c. states

d. functions


MSC-IT 3RD SEM

Q7 A finite-state machine is an abstract model of a machine with a primitive ______________

a. circuit

b. internal memory

c. structure

d. external memory

Q8 A _____________ is associated with every finite automation

a. digraph

b. machine

c. cantor-machine

d. state-machine

Q9 Each automaton M with input alphabet A defines a ______________ over A.

a. Turing machine

b. cantor

c. language

d. machine

Q10 The ___________ of M is a labeled directed graph whose vertices are the elements of Q.

a. finite diagram

b. diagram

c. infinite diagram

d. state diagram

UNIT-61

.

a. a

b. b

c. c

d. d

Q2

a. a

b. b

c. c

d. d

Q3

.


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

Q4

a. a

b. b

c. c

d. d

Q5

a. a

b. b

c. c

d. d

Q6

a. a

b. b

c. c

d. d

Q7 If an algebraic system A satisfies closure, associative and identity then A is a

a. group

b. monoid

c. vector

d. ring

Q8 Square roots of unity is an abelian group with respect to

a. division

b. addition

c. multiplication

d. subtraction

Q9 The semigroup S/R is called the _____________

a. totally ordered

b. quotient

semigroup

c. not ordered

d. completely

ordered

Q10 The set Z with the binary operation 'subtraction' is __________ a subgroup

a. not


MSC-IT 3RD SEM

b. subset of

c. always

d. superset of

1

a. a

b. b

c. c

d. d

Q2

a. a

b. b

c. c

d. d

Q3

a. a

b. b

c. c

d. d

Q4

a. a

b. b

c. c

d. d

Q5

a. a


MSC-IT 3RD SEM

b. b

c. c

d. d

Q6

a. a

b. b

c. c

d. d

Q7

a. a

b. b

c. c

d. d

Q8 If the code is to be error-correction, then D must be _____________

a. onto

b. into

c. one-one

d. one-one and

onto

Q9 The word c = 1010110 is transmitted through a binary symmetric. If p = 0.02 is the probability of receipt of a signal, then the probability that c is received as r = 1011111 is

a. 0.0036

b. 0.000036

c. 0.00036

d. 0.000306

Q10 The word c = 1010110 is transmitted through a ___________

a. channel

b. binary channel

c. binary symmetric

channel

d. symmetric channel

UNIT-7

1

a. a

b. b


MSC-IT 3RD SEM

c. c

d. d

Q2

a. a

b. b

c. c

d. d

Q3

a. a

b. b

c. c

d. d

Q4

a. a

b. b

c. c

d. d

Q5

a. a

b. b

c. c

d. d

Q6

a. a

b. b

c. c

d. d

Q7


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

Q8 A commutative ring is said to be an integral domain if it has no ____________

a. zero divisors

b. inverse

c. multiples

d. identity

Q9 A finite integral domain is a _________

a. subfield

b. vector

c. field

d. ring

Q10 If R is a Boolean ring then R is a ___________

a. commutative

ring

b. subring

c. integral ring

d. integer

UNIT-81

a. a

b. b

c. c

d. d

Q2

a. a

b. b

c. c

d. d

Q3

a. a

b. b

c. c

d. d


MSC-IT 3RD SEM

Q4

a. a

b. b

c. c

d. d

Q5

a. a

b. b

c. c

d. d

Q6 A ____________ is a statement which is either true or false, but not both.

a. argument

b. conclusion

c. bi-conditional

d. proposition

Q7 Reasoning is a special kind of thinking called as _________

a. inferring

b. logics

c. bijective

d. contradictive

Q8 The basic unit of our objective language is called a ____________

a. prime divisor

b. prime statement

c. bijective statement

d. statement

Q9 The class of variables which are quantified stand for only those objects that are members of a particular set and is called the ____________

a. universe

b. discourse

c. universe of discourse

d. injective of discourse

Q10 The validity of an argument doesnot guarantee the truth of the _____________

a. permutation

b. commutative

value

c. conclusion

d. identity value

Q2 A finite alternating sequence of vertices and edges is called as a ___________

a. tree

b. vertex

c. walk


MSC-IT 3RD SEM

d. forest

Q3 A graph is also called a __________

a. 2 - complex

b. 1 - complex

c. 3 - complex

d. 4 - complex

Q4 A simple graph in which there exists an edge between every pair of vertices is called a __________

a. complete graph

b. void graph

c. incomplete

graph

d. subgraph

Q5 An edge having the same vertex as both its end vertices is called a __________

a. self-edge

b. self-loop

c. self-line

d. self-curve

Q6 An edge that has not acquired any takes is a _________

a. forest

b. tree

c. bridge

d. line

Q7 State true or false i) A graph G is said to be connected if there are exactly two paths between every pair of vertex in G. ii) A component is not a graph

a. i) F ii)T

b. i) T ii)F

c. i) T ii)T

d. i) F ii)F

Q8 State whether true or false i) Every tree is a bipartite graph ii) A bipartite graph has a self-loop

a. i)T ii)T

b. i)T ii)F

c. i)F ii)F

d. i)F ii)T

Q9 State whether true(T) or false(F) (i) A tree is a connected graph without any circuits (ii) A single vertex in a graph G is not a sub-graph of G

a. (i) F (ii) T

b. (i) T (ii) T

c. (i) F (ii) F

d. (i) T (ii) F

Q10 State whether true(T) or false(F) (i) Every graph is its own subgraph. (ii) A single edge in G, together with its end vertices, is also a sub-graph of G.

a. (i) T (ii) F

b. (i) F (ii) T

c. (i) T (ii) T

d. (i) F (ii) F

1


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

Q2 A finite alternating sequence of vertices and edges is called as a ___________

a. tree

b. vertex

c. walk

d. forest

Q3 A graph is also called a __________

a. 2 - complex

b. 1 - complex

c. 3 - complex

d. 4 - complex

Q4 A simple graph in which there exists an edge between every pair of vertices is called a __________

a. complete

graph

b. void graph

c. incomplete

graph

d. subgraph

Q5 An edge having the same vertex as both its end vertices is called a __________

a. self-edge

b. self-loop

c. self-line

d. self-curve

Q6 An edge that has not acquired any takes is a _________

a. forest

b. tree

c. bridge

d. line

Q7 State true or false i) A graph G is said to be connected if there are exactly two paths between every pair of vertex in G. ii) A component is not a graph

a. i) F

ii)T

b. i) T

ii)F

c. i) T

ii)T

d. i) F

ii)F

Q8 State whether true or false i) Every tree is a bipartite graph ii) A bipartite graph has a self loop

a. i)T

ii)T

b. i)T

ii)F

c. i)F ii)F

d. i)F


MSC-IT 3RD SEM

ii)T

Q9 State whether true(T) or false(F) (i) A tree is a connected graph without any circuits (ii) A single vertex in a graph G is not a sub-graph of G

a. (i) F (ii)

T

b. (i) T (ii)

T

c. (i) F (ii)

F

d. (i) T (ii)

F

Q10 State whether true(T) or false(F) (i) Every graph is its own subgraph. (ii) A single edge in G, together with its end vertices, is also a sub-graph of G.

a. (i) T (ii)

F

b. (i) F (ii)

T

c. (i) T (ii)

T

d. (i) F (ii)

F

UNIT-9

a. a

b. b

c. c

d. d

Q2

a. a

b. b

c. c

d. d

Q3

a. a

b. b

c. c

d. d


MSC-IT 3RD SEM

Q4

a. a

b. b

c. c

d. d

Q5 A connected graph without circuits is called a _____________

a. unique

tree

b. vertex

c. decision

tree

d. mail tree

Q6 A graph G is a tree if and only if it is ______________

a. disconnected

b. minimally

connected

c. maximally

connected

d. connected

Q7 A tree G with n vertices has _____________ edges

a. n - 1

b. n - 2

c. n - 3

d. 2n

Q8 A tree in which there is exactly one vertex of degree 2, and all other remaining vertices are of degree one or three, is called a ___________

a. Rooted tree

b. Binary tree

c. Connected

tree

d. complete

tree

Q9 A tree without any edge is called as a ______________

a. group

tree

b. self-tree

c. branch

d. null tree

Q10 State T or F For a given graph G, the following conditions are equivalent i) G is connected and is circuitless ii) G is a minimally connected graph.

a. i) F ii)T

b. i) T ii)F

c. i) T ii)T

d. i) F ii)F

Unit-10


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

Q2

a. a

b. b

c. c

d. d

Q3

a. a

b. b

c. c

d. d

Q4

a. a

b. b

c. c

d. d

Q5

a. a

b. b

c. c

d. d

Q6 A connected graph G is an Euler graph if and only if it can be decomposed into __________

a. roots

b. trees


MSC-IT 3RD SEM

c. forests

d. circuits

Q7 If some closed walk in a graph contains all the edges of the graph, then the walk is called an ____________

a. rooted line

b. Hamiltonian

line

c. Connected line

d. Euler line

Q8 If we remove any one edge from a Hamiltonian circuit, the path left is called a ___________

a. closed circuit

b. Hamiltonian path

c. connected line

d. open circuit

Q9 State T or F i) An Euler graph G is arbitrarily traceable from vertex v in G if and only if every circuit in G contains v ii) If a graph has exactly has exactly two vertices of odd degree, then there exists a path joining these two vertices.

a. i) F ii)T

b. i) T ii)F

c. i) T ii)T

d. i) F ii)F

Q10 State T or F i) An Euler graph G is arbitrarily traceable from vertex V in G if and only if every circuit in G contains v. ii) If a graph has exactly two vertices of odd degree, then there does not exist a path joining these two vertices.

a. i) F ii)T

b. i) T ii)F

c. i) T ii)T

d. i) F ii)F

UNIT-111

a. a

b. b

c. c

d. d

Q2

a. a

b. b

c. c

d. d

Q3


MSC-IT 3RD SEM

a. a

b. b

c. c

d. d

Q4 A connected planar graph with n vertices and e edges has ____________ regions

a. e - n

b. e + n

c. e - n + 2

d. e - n - 2

Q5 A drawing of a geometric representation of a graph on any surface such that no edges intersect is called an _______________

a. isomorphism

b. homomorphism

c. embedding

d. epimorphism

Q6 A graph G is said to be a _________________ if there exists some geometric representation of G which can be drawn on a plane such that no two of its edges intersect.

a. non-planar graph

b. planar graph

c. line graph

d. null graph

Q7 An embedding of a planar graph G on a plane is called a _____________ of G.

a. representation

b. plane

c. line

representation

d. plane

representation

Q8 State T or F i) A plane graph G divided the plane into a number of regions. ii) A region is defined in a non-planar graph.

a. i) F ii)T

b. i) T ii)F

c. i) T ii)T

d. i) F ii)F

Q9 State whether true(T) or false(F) (i) A graph consisting of only isolated vertices is 1-chromatic. (ii) For coloring problems, we consider only simple connected graphs

a. (i) T (ii) F

b. (i) F (ii) T

c. (i) T (ii) T

d. (i) F (ii) F

Q10 State whether true(T) or false(F) (i) Every tree is not a Bipartite graph (ii) A Bipartite graph contains self-loops

a. (i) T (ii) F

b. (i) F (ii) T

c. (i) T (ii) T


MSC-IT 3RD SEM

d. (i) F (ii) F

UNIT-12

1

a. a

b. b

c. c

d. d

Q2

a. a

b. b

c. c

d. d

Q3

a. a

b. b

c. c

d. d

Q4

a. a

b. b

c. c


MSC-IT 3RD SEM

d. d

Q5 A matrix whose entries are either 0 or 1 is called a ___________

a. asymmetric matrix

b. binary matrix

c. diagnol matrix

d. symmetric matrix

Q6 If I(G) is an incidence matrix of a connected graph G with n vertices, then the rank of I(G) is ____________

a. n

b. n + 1

c. n

d. 2n

Q7 On an incidence matrix I of a graph G, if two edges a, b are parallel edges then the corresponding columns are ___________

a. different

b. multiples of 2

c. identical

d. multiples of 5

Q8 State T or F i) The vertex corresponding to the deleted row is called the reference vertex. ii) Vertices of a connected graph cannot be made the reference vertex.

a. i) F ii)T

b. i) T ii)F

c. i) T ii)T

d. i) F ii)F

Q9 State T or F i)The reduced incidence matrix of a tree is singular ii) Each largest non-separable subgraph is called a block.

a. i) F ii)T

b. i) T ii)F

c. i) T ii)T

d. i) F ii)F

Q10 The entries along the principal diagnol of the adjacency matrix are all _____________

a. unity

b. non-zeroes

c. real numbers

d. zeroes

UNIT-131

a. a

b. b

c. c

d. d


MSC-IT 3RD SEM

Q2

a. a

b. b

c. c

d. d

Q3

a. a

b. b

c. c

d. d

Q4

a. a

b. b

c. c

d. d

Q5 A directed graph is referred to as a ___________

a. oriented graph

b. dis-oriented

graph

c. line graph

d. null graph

Q6 A vertex v in a digraph D is said to be a ______________ if it is of degree 1

a. leaf vertex

b. singleton

c. cord vertex

d. pendent vertex

Q7 State T or F. A digraph D is said to be an arborescence if it satisfy the following two conditions: i) D contains no circuit (neither a directed circuit nor a semi-circuit). ii) There exists exactly one vertex v of zero in-degree (this vertex v is called the root of the arborescence).


MSC-IT 3RD SEM

a. i) F ii)T

b. i) T ii)F

c. i) T ii)T

d. i) F ii)F

Q8 State T or F. Let D be a directed graph (i) A closed directed walk which traverses every edge of D exactly once, is called a closed directed walk. (ii) D is said to be an Euler digraph if it does not contain a directed Euler line

a. i) F ii)T

b. i) T ii)F

c. i) T ii)T

d. i) F ii)F

Q9 The number of edges incident out of a vertex v is called the ___________ of v.

a. in valence

b. branch valence

c. out valence

d. point valence

Q10 Two directed edges are said to be __________ if they are mapped onto the same ordered pair of vertices.

a. coincident

b. parallel

c. directive

d. edges

Discrete Maths12

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