Concept based notes Discrete MathematicsDiscrete Mathematics 5 Content S.No. Name of Topic 1. Graph...

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Concept based notes

Discrete Mathematics (BCA Part-I)

Varsha Gupta M.Sc. (Maths)

Revised by: Mr Shiv Kishore Sharma Lecturer

Deptt. of Information Technology Biyani Girls College, Jaipur

2

Published by :

Think Tanks Biyani Group of Colleges

Concept & Copyright :

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ISBN : 978-93-81254-37-3 Edition : 2011 Price : Leaser Type Setted by : Biyani College Printing Department

While every effort is taken to avoid errors or omissions in this Publication, any

mistake or omission that may have crept in is not intentional. It may be taken note of that neither the publisher nor the author will be responsible for any damage or loss of

any kind arising to anyone in any manner on account of such errors and omissions.

Discrete Mathematics 3

Preface

I am glad to present this book, especially designed to serve the needs of the

students. The book has been written keeping in mind the general weakness in understanding the fundamental concepts of the topics. The book is self-explanatory and

adopts the “Teach Yourself” style. It is based on question-answer pattern. The language of book is quite easy and understandable based on scientific approach.

Any further improvement in the contents of the book by making corrections, omission and inclusion is keen to be achieved based on suggestions from the readers for which the author shall be obliged.

I acknowledge special thanks to Mr. Rajeev Biyani, Chairman & Dr. Sanjay Biyani, Director (Acad.) Biyani Group of Colleges, who are the backbones and main concept

provider and also have been constant source of motivation throughout this Endeavour. They played an active role in coordinating the various stages of this Endeavour and

spearheaded the publishing work.

I look forward to receiving valuable suggestions from professors of various educational institutions, other faculty members and students for improvement of the quality of the book. The reader may feel free to send in their comments and suggestions

to the under mentioned address.

Author

4

Syllabus B.C.A. Part-I

Discrete Mathematics

Number Systems : Natural Numbers, Integers, Rational Numbers, Real Numbers, Complex Numbers, Arithmetic Modulo a Positive Integer (Binary, Octal, Decimal and Hexadecimal Number Systems), Radix Representation of Integers, Representing Negative and Rational Numbers, Floating Point Notation.

Binary Arithmetic, 2‟s Complement Arithmetic, Conversion of Numbers from One of Binary / Octal / Decimal / Hexadecimal Number System to other Number System, Codes (Natural BCD, Excess-3, Gray, Octal, Hexadecimal, Alphanumeric – EBCDIC and ASCII), Error Codes.

Logic and Proofs : Proposition, Conjunction, Disjunction, Negation, Compound Proposition, Conditional Propositions (Hypothesis, Conclusion, Necessary and Sufficient Condition) and

Logical Equivalence, De Morgan‟s Laws, Quantifiers, Universally Quantified Statement, Generalized De Morgan‟s Laws for Logic, Component of Mathematical System (Axiom, Definitions, Undefined Terms, Theorem, Leema and Corollary), Proofs (Direct Proofs, Indirect Proofs, Proof by Contra-Positive), Valid Argument, Deductive Reasoning, Modus Ponens (Rules

of Inference), Universal Instantiation, Universal Generalization, Existential Instantiation, Universal Generalization Resolution, Principle of Mathematical Induction, Structural Induction.

Sets, Venn Diagrams, Ordered Pairs, Sequences and Strings, Relation (Reflexive, Symmetric, Anti-symmetric, Transitive, Partial Order), Inverse Relation (Injective, Subjective, bijective),

Coposition of Functions, Restriction and Function Overriding, Function Spaces, Lambda Notation for functions, Lambda Calculus, Equivalence Relations, Interpretation using Digraphs. Cardinals, Countable and Uncountable Sets, Infinite Cardinal Numbers, Russell‟s Paradox, Operations on Cardinals, Laws of Cardinal Arithmetic.

Graph Theory, Undirected Graph, Digraph, Weighted Graph, Similarity Graphs, Paths and

Cycles, Hamiltonian Cycles, Shortest Path Algorithm, Isomorphism of Graphs, Planar Graphs.

Trees, Characterization of Trees, Spanning Trees, Breadth First Search and Death First Search Method, Minimal Spanning Trees, Binary Trees, Tree Traversals, Decision Trees and the Minimum Time for Sorting, Isomorphism of Trees.

□ □ □


Content

S.No. Name of Topic

1. Graph Theory

1.1 Simple Graph

1.2 Isomorphism 1.3 Dijekstra Algorithm 1.4 Non-Planarity 1.5 Matrix Representation 1.6 Regular Graph and Complete Graph

2. Trees

2.1 Definition and Properties of Trees 2.2 Prim‟s Methods 2.3 Tree Transversal 2.4 m-ary and Full m-ary Tree

3. Number System

3.1 Conversion from Decimal to Binary Number System

3.2 Sum of Binary Numbers 3.3 Conversion from Decimal to Octal Number

System 3.4 Conversion from Hexadecimal to Decimal Form

4. Binary Arithmetics

4.1 2‟s Complement 4.2 8-bit 2‟s Complement 4.3 BCD Code 4.4 Gray Code 4.5 EBCDIC Code

6

S.No. Name of Topic 5. Sets

5.1 Power Set 5.2 Operations on Sets 5.3 Symmetric Difference of Two Sets 5.4 De-Margan‟s Law 5.5 Russell‟s Paradox

6. Relations

7. Functions

8. Proportional Calculus

8.1 Converse, Inverse and Contraposition 8.2 De-Margan‟s Law 8.3 Quantifiers

9. Unsolved Papers 2011 to 2006

□ □ □


Chapter-1

Graph Theory

Q.1 Draw simple graphs with one, two, three and four vertices.

Ans.:

Simple graph with one vertex

•V1

Simple graph with two vertices

V1 V2

Simple graph with three vertices

V3

V1 V2

Simple graph with four vertices

V4 V3

V1 V2

8

Q.2 Show that if G = (V, E) is a complete bipartite graph with n vertices then the

total numbers of edges in G cannot exceed 2

4

n.

Ans.: Let Kp,q be a complete bipartite graph. The total no. of edges in Kp,q is p.q and

total no. of vertices will be (p+q). If we take p = q = 2

n then in complete bipartite

graph K ,2 2

n n no. of edges will be 2

.2 2 4

n n nwhich is maximum (If two numbers

are equal then their product is maximum). Hence in a complete bipartite graph of

n vertices the no. of edges cannot exceed 2

4

n.

Q.3 Show that following two graphs are not isomorphic.

V5 U6

V1 V2 V3 V4 U1 U2 U3 U4 U5

V6

G G‟

Ans.: In graph G and G‟ we find that

(i) No. of vertices in G = No. of vertices in G‟ = 6.

(ii) No. of edges in G = No. of edges in G‟ = 5.

(iii) No. of vertices of degree one in G and G‟ = 3.

No. of vertices of degree two in G and G‟ = 2

No. of vertices of degree three in G and G‟ = 1

i.e. Number of vertices of equal degree are equal. Although it satisfies all the three conditions but then also G and G‟ are not isomorphic because corresponding to vertex V4 in G there should be a vertex U3 because in both G and G‟ there is only one vertex of degree three. But two pendent


vertices V5 and V6 are incident on the vertex V4 in G whereas only one pendent vertex U6 is incident on the vertex U3 in G‟.

Hence G and G‟ are not isomorphic.

Q.4 Define the followings :-

(i) Walk (ii) Trail (iii) Path (iv) Circuit (v) Cycle

Ans.: (i) Walk : An alternating sequence of vertices and edges is called a Walk. It is denoted by ‘W’.

Example :

a d

e1 e4

e6 b e3

e5 e2

e c

Figure (1)

Here W = ae1 b e2 c e3 d is a walk.

Walk is of two types :-

(a) Open Walk : If the end vertices of a walk are different then such a

walk is called Open Walk.

Example from fig.(1) : W = a e1 b e2 c e3 d is an open walk.

(b) Closed Walk : If a walk starts and end with same vertex then such

a walk is called closed walk.

Example from fig.(1) : W = a e6 e e5 b e1 a is a closed walk as it starts and end with same vertex a.

(ii) Trail : An open walk in a graph G in which no edge is repeated is called a

Trail.

Example from fig.(1) : W = a e1 b e2 c e3 d is a trail.

(iii) Path : An open walk in which no vertex is repeated except the initial and

terminal vertex is called a Path.

10

Example for fig.(1) : W = a e1 b e4 d e3 c is a path.

(iv) Circuit : A closed trail is called a Circuit.

Example for fig.(1) : W = a e1 b e5 e e6 a is a circuit.

(v) Cycle : A closed path is called a Cycle.

Example for fig.(1) : W = a e1 b e5 e e6 a is a cycle.

Q.5 Find the shortest path between the vertex a and z in the following graph.

b 5 d 5 f

4 7

a 2 33 z

3 4

c 6 e 5 g

Ans.: First we label the vertex a by permanent label 0 and rest by „∞‟. a b c d e f g h 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞

0 4 3 ∞ ∞ ∞ ∞ ∞

0 4 3 6 9 ∞ ∞ ∞

0 4 3 6 9 ∞ ∞ ∞ 0 4 3 6 7 11 ∞ ∞

0 4 3 6 7 11 12 ∞

0 4 3 6 7 11 12 18

0 4 3 6 7 11 12 16

Hence shortest path is a → c → d → e → g → z = 16

Q.6 Prove that K5 is non-planar.

Ans.: Let the five vertices of K5 be V1, V2, V3, V4 and V5. Since K5 is a complete graph so

every vertex of K5 is joined to every other vertex by means of an edge. Therefore

3 1 2


we must have a circuit going from V1 to V2, to V3, to V4, to V5 and to V1 i.e. a pentagon.

V2 V2

V1 V3 V1 V3

V5 V4 V5 V4

(a) (b)

V2 V2

V1 V3 V1 V3

V5 V4 V5 V4

(c) (d)

V2

V1 V3

V5 V4

(e)

Since vertex V1 is to be connected to V3 by means of an edge, this edge may be drawn inside or outside the pentagon (without intersecting the five edges drawn previously). Suppose we draw a line from V1 to V3 inside the pentagon. Now we have to drawn an edge from V2 to V4 and another one from V2 to V5. Since

12

neither of these edges can be drawn inside the pentagon without crossing over the already drawn edge. We draw both these edges outside the pentagon. Now the edge from V3 to V5 cannot be drawn outside the pentagon without crossing the edge between V2 to V4. Therefore V3 and V5 have to be connected with an edge inside the pentagon.

Now we have yet to draw an edge between V1 and V4. This edge cannot be placed inside or outside the pentagon without a crossover. Hence K5 is not a planar graph.

Q.7 State and prove Handshaking Theorem.

Ans.: Handshaking Theorem : The sum of degrees of all the vertices in a graph G is

equal to twice the number of edges in the graph.

Mathematically it can be stated as :

deg( ) 2

v V

v e

Proof : Let G = (V, E) be a graph where V = {v1, v2, . . . . . . . . . .} be the set of

vertices and E = {e1, e2, . . . . . . . . . .} be the set of edges. We know that every edge lies between two vertices so it provides degree one to each vertex. Hence each edge contributes degree two for the graph. So sum of degrees of all vertices is equal to twice the number of edges in G.

Hence deg( ) 2

v V

v e

Q.8 Explain Matrix Representation of Graphs.

Ans.: Although a pictorial representation of a graph is very convenient for a visual

study, other representations are better for computer processing. A mat rix is convenient and useful way of the representation of a graph to a computer for a graph. There are different types of matrices :

(i) Incidence Matrix

(ii) Circuit Matrix

(iii) Adjacency Matrix


(iv) Path Matrix etc.

Q.9 How many edges are there with 7 vertices each of degree 4? Ans.: In graph G, there are 7 vertices and degree of each vertex is 4. So sum of the

degrees of all the vertices of graph G = 7 x 4 = 28. According to Handshaking Theorem –

deg( ) 2

v V

v e

28 = 2e

e = 14

So, total no. of edges in G = 14.

Q.10 Define Regular and Complete Graph.

Ans.: Regular Graph : A simple graph G = (V, E) is called a Regular Graph if degree of

each of its vertices are equal.

Examples :

1-

V1 V2

Here degree of each vertex is one. So it is regular graph.

2-

V3

V1 V2

Degree of each vertex is two.

3-

V4 V3

V1 V2

Degree of each vertex is two.

14

Complete Graph : A simple graph G = (V, E) is called a Complete Graph if there

is exactly one edge between every pair of distinct vertices. A complete graph with n-vertices is denoted by Kn.

•K1

K2

K3

K4

K5

K6

In a complete graph Kn total no. of edges =( 1)

2

n n

i.e. size of Kn =( 1)

2

n n

Multiple Choice Question

1. The graph is :

u v

(a) Simple graph

(b) Directed graph (c) Directed multigraph

(d) Pseudograph


2. In the graph the vertices of equal degree are :

a

u v

d

e

(a) a and b

(b) b and c (c) b and d

(d) a and d

3. The number of edges in a graph with 10 vertices each of degree 6 are : (a) 60

(b) 120 (c) 15

(d) 30

4. The cycle C6 is also:

(a) a wheel

(b) a bipartite graph (c) not a complete graph

(d) a complete bipartite graph

5. The adjacency matrix for the graph is given below:

The value of x is :

16

(a) 1

(b) 2 (c) 3

(d) 4

6. The length of path a,b,d,e,b in the following graph is :

C b a

D e

(a) 2

(b) 4 (c) 3

(d) 5

7. The true statement is : (a) There is a simple path between every pair of distinct vertices of a connected

directed graph.

(b) There is a simple circuit between every directed graph (c) If there is a path then there is a circuit

(d) All the above statements are true.

8. In the following graph, the true statement is :

(a) No Euler path but Euler circuit exists (b) Euler path but no Euler circuit exists

(c) Euler circuit anct Euler path both exist (d) None of the above statement is true

9. In the following weighted simple graph the length of the shortest path from p to r is:

U 3 t

4 2 2 p 3 r 2 1

q 3 s

(a) 8 (b) 6


(c) 9 (d)10

10. The planner graph is : (a) K3.3 (b) Q3

(c) K4.4 (d) K5

11. Let G be a connected planner simple graph with e edges and v vertices. If r is

the number of regions in its planner representation, then: (a)r=2+e-v (b) r=2-e+v (c)r=2+e+v (d)r=e+v

12. The chromatic number of a planner graph is not greater than: (a) 3 (b) 5

(c) 2 (d) 4

13. If a connected planner simple graph has e edges and v vertices with v and no circuits of length 3, then

(a) e (b) e

(c) (d) e 14. If G is a connected planner simple graph with e edges and v vertices where

v 3 then:

(a) e (b) e

(c) (d) e 15. A graph is non planner if and only if it contains a sub graph homeomorphic to:

(a) C6 (b) Q3

(c) 2.2 (d) K3.3 or K5

16. The chromatic number of a complete bipartite graph is (a) 1 (b) 2

(c) (d) 4

17. How many colors are required to color properly the following graph.

(a) 1 (b) 2

(c) (d) 4

18

18. A graph has n vertices then how many edges contained by a Hamiltonian path.

(a) n-1 (b) n

(c) (d) n+2 19. In a graph there are 7 vertices each of degree 4 , then the number of edges in

the graph is : (a) 4 (b) 7

(c) (d) 28 20. Out degree of the vertex V4 in the following diagraph is :

V1 v5

v2

v3 v4

(a) 4 (b) 7

(c) (d) 28

1-d 2-b 3-d 4-a 5-a 6-b 7-a 8-b 9-b 10-b

11-a 12-d 13-a 14-b 15-b 16-b 17-b 18-b 19-c 20-b

□ □ □


Chapter-2

Trees

Q.1 Define a Tree. Prove that there is one path between every pair of distinct

vertices in a Tree „T‟.

Ans.: Tree : A Tree is a connected graph without any circuit i.e Tree is a simple graph.

•

Trees with one, two, three, and four vertices.

Proof : Since T is a connected graph. Let a and b be any two vertices of T. If it is possible let there are two different paths between the vertices a and b.

P = a u1, u2, . . . . . . . . . . um b

and Q = a v1, v2, . . . . . . . . . . vn b

are those two different paths between a and b.

In both these paths vertices after a can be common also. Let w be the first common vertex then for any i and j

W = ui = vj

Where i = 1, 2, . . . . . . . . . . m

j = 1, 2, . . . . . . . . . . n

Then we get a cycle a u1, u2, . . . . . . . . . . ui-1 , ui , vj-1 , vj-2 , . . . . . . . . . . v2, v1 a

20

which contradicts our assumption that T is a Tree. Hence there is only one path between a and b.

Q.2 A Tree with n-vertices has (n-1) edges.

Ans.: Let T be a tree having n vertices. We shall prove the theorem by mathematical

induction.

If n=1 then T contains only one vertex and 1 -1=0 edges. Hence the theorem is true for n=1.

Let it be true for k vertices. Now we shall prove it for (k+1) vertices. Since T is a connected graph so let P be a path of maximum length in T. P cannot be a circuit. Hence P contains atleast one vertex of degree one. Let this vertex be v. Now this vertex v and edge incident on it are eliminated from T so that we obtain a new tree T* which contains k vertices. According to our assumption T* contains (k-1) edges. Now if in T* the vertex v and edge is again included we again get T in which no. of edges are k. Hence the theorem is true for (k+1) vertices also.

Thus by mathematical induction theorem is true for all n N.

Hence proved.

Q.3 If G is an acyclic graph with n vertices and k connected components then G has (n-k) edges.

Ans.: Proof : Let G be an acyclic graph. Let G1, G2, . . . . . . . . . . GK be its k connected components. For every i (1 )i k ith component Gi has ni vertices then clearly -

n1 + n2 + n3 +. . . . . . . . . .+ nK = n

Again since every Gi is a tree. Hence no. of edges in every Gi will be (ni-1) so total no. of edges in G

= (n1 – 1) + (n2 – 1) + . . . . . . . . . . + (nk – 1)

= (n1 + n2 + . . . . . . . . . . + nk) – k

= n – k

Hence proved.

Q.4 Find eccentricity, centre radius and diameter of the following graph.

Ans.: (i) Eccentricity : V1 V2


E(V1) = 2

E(V2) = 2 V3

E(V3) = 1

E(V4) = 2 V4

(ii) Centre : Centre of the given graph is the vertex V3 because it has

minimum eccentricity.

(iii) Radius : Radius (eccentricity of the centre) = 1

(iv) Diameter : Maximum eccentricity = 2

Diameter of the given graph = 2

Q.5 Prove that every tree has either one or two centres.

Ans.: Let T be a tree if T contains only one vertex then this vertex will be centre of T. If

T contains two vertices then both vertices are centre of T. Now let T contains

more than two vertices. The maximum distance max.d (v, vi) from a given vertex

v to any other vertex vi occurs only when vi is a pendent vertex. Tree T must

have two or more pendent vertices. Delete all pendent vertices form T. The

resulting graph T‟ is still a tree in which the eccentricity of all vertices is reduced

by 1. Hence the centre of T will also be centre of T‟. From T‟, we can again

remove all pendent vertices and we get another tree T”. We continue this process

until we are left with a vertex or an edge. If a vertex is left then this vertex is the

centre and if an edge is left then both its end vertices are centre of T.

Example : f(6) g(6)

e(6) h(6)

d(5)

22

m(6)

a(6) b(5) c(4) j(3) k(4) l(5)

n(6)

i(5)

Removing all pendent vertices

d(4)

b(4) c(3) j(2) k(3) l(4)

Removing all pendent vertices

c(2) j(1) k(2) j (centre)

Q.6 Find minimal spanning tree for the following weighted graph (use Prime‟s Method).

Ans.:

V1

3 1

V2 3 4 V6

3 4 2

V3 5 1 V5

4 2

V4


V1

3 1

V2 V6

3 1

V3 V5

1

V4

Minimal Spanning Tree

Q.7 Write Pre-order, In-order and Post-order transversal of the following graph.

Ans.: a

b c d

e f g h i j

k l m

Pre-order : a b e k l m f g c h d i j

In-order : k e l m b f g a c h i d j

Post-order : k l m e f g b h c i j d a

V1 V2 V3 V4 V5 V6

V1 ∞ 3 3 1 4 1

V2 3 ∞ 3 5 ∞ ∞

V3 3 3 ∞ 4 ∞ 4

V4 1 5 4 ∞ 2 ∞

V5 4 ∞ ∞ 2 ∞ 2

V6 1 ∞ 4 ∞ 2 ∞

24

Q.8 Define m-ary and Full m-ary Tree. Gives an example for each.

Ans.: m-ary Tree : A rooted tree is said to be m-ary Tree if every internal vertex or

branch node has not more than m-children.

a

b c

d

e f

g h

2 – Ary Tree

Full m-ary Tree : A rooted tree is said to be Full m-ary Tree if every internal

vertex or branch node has exactly m-children.

a1

a3

a2 a4

a8 a9 a10

a5 a6 a7 a11 a12 a13

Full 3 – Ary Tree

Q.9 Find the value of following prefix expression.

Ans.:

*235/ 234

32

*235/ 84

8 / 4

*2352

2 3x

*652

6 5

12


1 2

3

Q.10 In any tree with 2 or more vertices there are atleast two pendent vertices.

Ans.: Let T be a tree with n vertices and (n-1) edges. Since all the edges are connected

with 2 vertices at a time. Hence the sum of the degree of all the vertices is –

= 2 x (no. of edges)

= 2 x (n - 1) = 2n – 2

Now, we have to prove that in tree T there are atleast two vertices of degree one and rest of vertices are of degree two or higher. Since no vertex in T has zero degree so let us assume that there is only one vertex of degree one and rest (n-1) vertices are of degree two or higher. Then the sum of the degrees of vertices is 1 + 2(n – 1) = 2n – 1 which is contradiction of (2n – 2). Hence there is another vertex of degree one.

If we take two vertices of degree 1 and remaining (n - 2) vertices of degree 2 or more than two then sum of the degrees of vertices = 2 + 2(n – 2) = 2n – 2 which is correct.

Hence proved.

Multiple Choice Question

1 . The correct statement is : (a) A tree contains multiple edges (b) A tree contains loops

(c) tree is a simple graph (d) A tree is a connected directed graph

2 . In a binary tree every internal vertex contains : (a) Exactly two children (b) Two children or more

(c) (d) Any number of children 3. A rooted m-ary tree of height h is balanced if all the leaves are at level.:

(a) h-2 (b) h or (h-1)

(c) (d)

4. The maximum number of leaves in an m-ary tree of height h are :

26

(a) hm

(b) hm+h

(c) mh-1

(d) mh

5. The preorder traversal to tree T IS e

k j

I o p

T

(a) j e k n o p

(b) j e n k o p

(c) e j k n o p

(d) j e k p o n

6. The in-order traversal of tree T1 is :

e k p

j n

o

T1 (a) j e k n o p

(b) j e n k o p

(c) e j k n o p

(d) j e k p o n

1. The post order traversal of tree T2 is :

e j

k p

n o T2


(a) j n o p k e

(b) j e k n o p

(c) j k e n o p

(d) p o n k e j 9. A spanning tree of a graph G contains

(a) All vertices of G

(b) At least one vertex of G

(c) Maximum two vertices G

(d) All the edges of G

10. The correct statement is : (a) A connected simple graph may not have a spanning tr (b) A simple graph is connected if and only if has a spanning tree

(c) A simple graph is connected if it has a spanning tree

(d) None of the above is true

11. What will be the right preorder of following graph:

a

b

h d e f i g

(a) abdechfthgi

(b) dbeafeghi

(c) debfgihca

(d) none of the above

1-c 2-a 3-b 4-d 5-c 6-b 7-a 8-c 9-a 10-c

□ □ □

28

Chapter-3

Number System

Q.1 Convert (111001101)2 into decimal form and 96 into binary number.

Ans.: 8 7 6 5 4 3 2 1 8

2(111001101) 1*2 1*2 1*2 0*2 0*2 1*2 1*2 0*2 1*2

= 256 128 64 0 0 8 4 0 1

10(461)

For converting 96 into binary number –

2 96 Remainders

2 48 . . . . . . . . . . . . . . . 0

2 24 . . . . . . . . . . . . . . . 0

2 12 . . . . . . . . . . . . . . . 0

2 6 . . . . . . . . . . . . . . . 0

2 3 . . . . . . . . . . . . . . . 1

1 . . . . . . . . . . . . . . . 1 (96)10 = (110000) 2

Q.2 Find the value of (195) 10 + (105)10 in binary code.

Ans.: Since (195 + 105) = 300


So, we have to find (300) 10

2 300 Remainders

2 150 . . . . . . . . . . . . 0

2 75 . . . . . . . . . . . . 0

2 37 . . . . . . . . . . . . 1

2 18 . . . . . . . . . . . . 1

2 9 . . . . . . . . . . . . 0

2 4 . . . . . . . . . . . . 1

2 2 . . . . . . . . . . . . 0

1 . . . . . . . . . . . . 0

1 . . . . . . . . . . . . 1 (195)10 +(105)10 = (100101100) 2

Q.3 Convert (175) 10 into hexadecimal number.

Ans.: 16 175 Remainder

16 10 . . . . . . . . . . . . 15 = F

10 . . . . . . . . . . . . 10 = A (175)10 = (AF) 16

Q.4 Find the sum of (1011)2 and (10111)2.

Ans.: 0 1 0 1 1

1 0 1 1 1

1 0 0 0 1 0

So, (1011)2 + (10111)2 = (100010) 2

30

Q.5 Find fixed point representation for the decimal no. 3.056 E-5.

Ans.: 3.056 E-5 = 3.056 x 10-5 = 0.00003056

Q.6 Convert the decimal number 692.625 into octal form.

Ans.: Integral Part :

8 692 Remainders

8 86 . . . . . . . . . . . . 4

8 10 . . . . . . . . . . . . 6

1 . . . . . . . . . . . . 2

1 . . . . . . . . . . . . 1 (692)10 = (1264)8

Fractional Part :

0 . 6 2 5

x 8

5 . 0 0 0 So, 0.625)10 = (.5)8

Hence, (692.625)10 = (1264.5)8

Q.7 Convert hexadecimal no. ABC.2 into decimal form.

Ans.: (ABC.2)16 = A x 162 + B x 161 + C x 160 + 2 x 16-1

= 10 x 256 + 11 x 16 + 12 x 1 + 2 x 0.0625

= 2748.125

Multiple Choice Questions

1Which of the following number -1,1,2,3 are natural number s?

(a) All (b) Only-1,2,3


(c) Only-1 (d) None

2. Is 2 an integer or a rational number

(a) Both (b) Integer

(c) Rational number (d) None

3. Solution of the equation x+5=3 is a/an:

(a) Natural number (b) Integer

(c) No solution (d) None of the above

4. Solution of the equation x2+1=0 is a :

(a) Real number (b) Complex number

(c) Rational number (d) None of the above 5. Solution of the equation 2x=3 is a/an:

(a) Rational number (b) Integer

(c) None of the above (d) Natural Number 6. The total number of symbols used to represent a number in Hexadecimal

number system is :

(a) 16 (b) 15

(c) 6 (d) 8 7. The base or radix of octal number system is :

(a) 7 (b) 8 (c) 2 (d) 16 8. The decimal number 25 is binary number system can be written as : (a) 100112 (b) 110012

(c) 101012 (d) None of the above 9. What is the normalized floating-point representation for the decimal number

15.854 ? (a) 1.5854E+1 (b) .15854E+2

32

(c) .015854E+3 (d) None of the above

10. Fill in the correct number at the place of question mark 1010102=?8 (a) 210 (b) 222

(c) 52 (d) None of the above 11. The value of 67458+3768 is : (a) 71218 (b) 73238 (c) 73433 (d) None of the above 12. Addition of 10102 and 1102 is : (a) 100002 (b) 11202 (c) 11012 (d) None of the above 13. 100002-1012 is equal to : (a) 110112 (b) 102

(c) 1112 (d) None of the above

14. is : (a) a real number (b) a rational number (c) an integer (d) None of the above 15. Which of the following statements is false?

(a) All the natural numbers are integers

(b) All the rational number are integers

(c) All the rational numbers are real number

(d) All the number are real numbers

16. The decimal equivalent of the bineary number (-10011)2 is:

(a) -12 (b) -15

(c) -19 (d) -21


17. The number (680)10 of the decimal system is equivalent to which number in octal system?

(a) (85)8 (b) (1012)8

(c) (1250)8 (d) (1300)8

18. The symbol N stand for the set of :

(a) Real numbers (b) Natural numbers

(c) Integers (d) Rational numbers

19. 13/5 is a :

(a) Irrational number (b) Rational number

(c) Integer (d) none of the above

20. Which is the decimal equivalent of the binary number 000100112

(a) 6 (b)11

(c) 19 (d) 35

Answer Key :

1-b 2-a 3-b 4-b 5-a 6-a 7-b 8-b 9-c 10-b

11-c 12-a 13-d 14-a 15-b 16-c 17-c 18-b 19-b 20-c

□ □ □

34

Chapter-4

Binary Arithmetic's

Q.1 Find 2‟s complement of the binary no. 1101100.

Ans.: 1‟s complement of 1101100 is 0010011

Adding 1 to this

0 0 1 0 0 1 1

+ 1

0 0 1 0 1 0 0

So, 2‟s complement is 0010100.

Q.2 Find 8-bit 2‟s complement of (35)10.

Ans.: (35)10 = (100011)2

= (00100011)2

Let X = 00100011

1‟s complement of X = 1 1 0 1 1 1 0 0

2‟s complement of X = 1 1 0 1 1 1 0 0

+ 1

1 1 0 1 1 1 0 1

So, 8-bit 2‟s complement of (35)10 is 11011101

Q.3 Write BCD code for decimal number 12.

Ans.: (12)10 = (0001 0010)BCD


Q.4 Write Gray code of the no. (1111)2.

Ans.: Gray code of (1111)2 is 1000.

Q.5 Represent „SHORT‟ in EBCDIC code.

Ans.: 11100010 11001000 11010110 11011001 11100011

S H O R T


1. 1‟s complement of the number 1011001 is (a) 0100110 (b) 1100110 (c) 0110010 (d) 0110011 2 . 2‟s complement of the number 1011001 is (a) 0100110 (b)0100111 (c) 1110010 (d) None of the above 3 . Natural BCD code of 4210 is : (a)01000010 (b)10010 (c)010010 (d) None of the above 4 . Natural BCD code of 4210 is : (a)01000010 (b)10010 (c)010010 (d) None of the above 5. The Gray code of a number whose binary representation is 1000 is : (a)0110 (b) 0111 (c)1100 (d) 0100 6. EBCDIC is a code with : (a) 8 bits (b) 6bits (c) 4 bits (d) None of the above

36

7. In ASCII, the symbol II stands for : (a) International information (b) International Information (c) Information Interchange (d) International Interchange

8- Most of the errors occur at : (a) One bit position (b) Two bit position (c) Three bit position (d) None of the above

9- The parity bit is :

(a) A bit at one position in any transmitted (b) An extra bit attached to each code word for detecting the error (c) An extra bit attached to each code word for correcting the error (d) None of the above

10- Hamming code is a : (a) Gray code (b) Error deducting code (c) Error correcting code (d) None of the above

11- 2‟s complement of the 2‟s complement of a number is : (a) Double the number (b) Half of the number (c) Ten times the number (d) The number it self

12- 1‟s complement of the number 010101012 is : (a) 101010102 (b) 101010112 (c) 010101002 (d) 010101012

12- 1‟s complement of the number 010101012 is : (a) 101010102 (b) 101010112 (c) 010101002 (d) 010101012

13- The Gray code of the number 01002 : (a) 101010102 (b) 101010112 (c) 010101002 (d) 010101012

1-a 2-b 3-d 4-a 5-c 6-a 7-c 8-a 9-b 10-c

11-d 12-a 13-d


Chapter-5

Sets

Q.1 Define Power Set. Write Power Set of A = {1, 2, 3}.

Ans.: Let B be a set then the collection of all subsets of B is called Power Set of B and is denoted by P(B).

i.e. P(B) = {S : S B}

If A = {1, 2, 3}

Then P(A) = {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

Q.2 Explain the following operations –

(a) Union (b) Intersection (c) Difference

Ans.: (a) Union : Let A and B be two sets then union of A and B which is denoted

as A B is a set of elements which belongs either to A or to B or to both A and B.

So, { : . . }A B x x Aor x B

Example : If A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A B = {1, 2, 3, 4, 5, 6}

A B

(b) Intersection : Intersection of A and B which is denoted as A B is a set

which contains those elements that belong to both A and B.

So, { : . . }A B x x Aand x B

U

A B

38

Example : If A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A B = { 3, 4}

A B

(c) Difference : Let A and B be two sets. The difference of A and B which is

written as A - B, is a set of all those elements of A which do not belongs to B.

So, { : . . }A B x x Aand x B

Similarly, { : . . }B A x x B and x A

Example : If A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A - B = { 3, 4} and B – A = {5, 6}

A – B B - A

Q.3 Define symmetric difference of two sets If A = {2, 3, 4} and B = {3, 4, 5, 6}, Find A B .

Ans.: Let A and B be two sets, the symmetric difference of A and B is the set

( ) ( )A B B A and is denoted by A B or A B

U

A B

U

A B

U

A B


Thus, ( ) ( )A B A B B A = { : }x x A B

A = {2, 3, 4} and B = {3, 4, 5, 6}

A – B = {2} and B – A = {5, 6}

( ) ( ) {2} {5,6} {2,5,6}A B A B B A

A B

Q.4 State De Margan‟s Law.

Ans.: If A and B are any two sets then

(i) ( ) ' ' 'A B A B and (ii) ( ) ' ' 'A B A B

Q.5 Prove the following relation –

( ) ( ) ( ) ( )n A B n A n B n A B

Ans.: If A and B are two sets then we know that

( ') ( ) ( ' )A B A B A B A B

Hence by sum rule –

U

A B

40

( ) ( ') ( ) ( ' )n A B n A B n A B n A B (1)

Again ( ') ( )A A B A B

By sum rule –

( ) ( ') ( )n A n A B n A B (2)

Similarly

( ) ( ) ( ' )n B n A B n A B (3)

Now eq^(2) + eq^(3) gives

( ) ( ) ( ) ( ) ( ') ( ' )n A n B n A B n A B n A B n A B

=> ( ) ( ) ( ) ( ) ( ') ( ' )n A n B n A B n A B n A B n A B

From eq^(1)

( ) ( ) ( ) ( )n A n B n A B n A B

Hence proved.

Q.6 State and prove Rusell‟s Paradox.

Ans.: See B.C.A. (Discrete Mathematics) CBH Pg. 4.22 Article No. 4.20.

□ □ □


Chapter-6

Relations

Q.1 Prove that –

A x B B x A

Ans.: A = {a, b} and B = {1, 2, 3}

A x B = { (a, 1) (a, 2) (a, 3) (b, 1) (b, 2) (b, 3)}

B x A= { (1, a) (1, b) (2, a) (2, b) (3, a) (3, b)}

Here A x B B x A

Q.2 Show that the relation „is congruent to‟ on the set of all triangles in plane is an equivalence relation.

Ans.: Proof : Let S be the set of all triangles in a plane and R be the relation on S

defined by 1 2( , ) R triangle 1 is congruent to triangle 2 .

(i) Reflexivity : for each triangle S , we have

( , ) R S

R is reflexive on S.

(ii) Symmetry : Let 1 and 2 S such that 1 2( , ) R , then

1 2( , ) R 1 2

2 1

2 1( , ) R

R is symmetric on S.

(iii) Transitivity : Let 1 2 3, , S such that 1 2( , ) R and 2 3( , ) R , then

42

1 2( , ) R

1 2

2 3( , ) R 2 3

Since 1 2 and 2 3 1 3 1 3( ) R

So, R is transitive.

Hence R is on equivalence relation.

Q.3 Let N be the set of all natural numbers and Let R be a relation on Nx N, defined by (a, b) R (c, d) ad = bc for all (a, b), (c, d) N x N. Show that R is an equivalence relation on Nx N.

Ans.: (i) Reflexivity : Let (a, b) be an arbitrary element of N x N, then

( , ) ,a b NxN a b N

ab ba

(a, b) R (b, a)

(by commutativity of multiplication on N)

Thus (a, b) R (b, a) for all ( , )a b NxN .

So, R is reflexive.

(ii) Symmetry : Let (a, b), (c, d) N x N be such that (a, b) R (c, d), then

(a, b) R (c, d) ad bc

cb da

(by commutativty of multiplication on N)

(c, d) R (a, b)

Thus, (a, b) R (c, d) (c, d) R a, b) for all (a, b), (c, d) N x N

So, R is symmetric on N x N.

(iii) Transitivity: Let (a, b), (c, d), (e, f) N x N be such that (a, b) R (c, d) and

(c, d) R (e, f), then

(a, b) R (c, d) ad bc

and (c, d) R (e, f) cf de


( )( ) ( )( )ad cf bc de

af be

(a, b) R (e, f)

Thus (a, b) R (c, d) and (c, d) R (e, f) (a, b) R (e, f) for all (a, b), (c, d), (e, f) N x N

So, R is transitive.

Hence, R being reflexive symmetric and transitive is an equivalence relation on N x N.

Q.4 Prove that the relation “congruence modules m” on the set z of all integers is an equivalence relation.

Ans.: (i) Reflexivity : Let a be an arbitrary integer, then

a – a = 0 = 0 x m

a – a is divisible by m

a a (mod m)

Thus, a a (mod m) for all a z

So, “congruence modules m” is reflective.

(ii) Symmetry : Let a, b z such that

a b (mod m) a – b is divisible by m

a – b = m for all λ z

b – a = (- )m [ λ z - λ z]

(b –a ) is divisible by m

b a (mod m)

So, “congruence modules m” is symmetric on Z.

(iv) Transitivity: Let a, b, c z such that a b (mod m) and b c (mod m),

then

a b (mod m) a – b is divisible by m

a – b = 1 m for some 1 z

44

b c (mod m) b – c is divisible by m

b – c = 2m for some

2 z

(a – b) + (b – c) = 1 m + 2 m

(a – c) = 3 m { 1 + 2 z} ( 3 = 1 + 2 )

a c (mod m)

So, “congruence modules m” is transitive.

Hence, “congruence modules m” is an equivalence relation.

Q.5 If A = {1, 2, 3, 4, 5, 6, 7} which of the following two is a partition giving rise to an equivalence relation.

(i) A1 = {1, 3, 5} A2 = {2} A3 = {4, 7}

(ii) B1 = {1, 2, 5, 7} B2 = {3} B3 = {4, 6}

Ans.: (i) Since 1 2 3A A A = {1, 2, 3, 4, 5, 7} A

So sets A1 , A2 , A3 do not form a partition of A.

(ii) Since 1 2 3B B B = {1, 2, 3, 4, 5, 7} =A and B1 , B2 , B3 are disjoint sets.

Hence B1 , B2 , B3 form a partition of A.

Q.6 If R and S are two equivalence relations on a set A, then prove that R S is an equivalence relation.

Ans.: It is given that R and S are equivalence relation on A. We have to show that R S is an equivalence relation.

(i) Reflexivity : Let a A then

(a, a) R and (a, a) S [ R & S are reflexive]

(a, a) R S

Thus (a, a) R S for all a A, so R S is a reflexive relation on A.

(ii) Symmetry : Let a, b A such that (a, b) R S

(a, b) R S (a, b) R and (a, b) S


(b, a) R and (b, a) S

[R & S are symmetric]

(b, a) R S

Thus (a, b) R S (b, a) R S

R S is symmetric relation.

(iii) Transitivity: Let a, b, c A such that (a, b) R S and (b, c) R S

(a, b) R S and (b, c) R S

(a, b) R and (a, b) S and (b, c) R and (b, c) S

(a, b) R and (b, c) R (a, c) R [R is transitive]

(a, b) S and (b, c) S (a, c) S [S is transitive]

(a, c) R and (a, c) S

(a, c) R S

Thus (a, b) R S and (b, c) R S (a, c) R S

So, R S is transitive

Hence R S is an equivalence relation on A.

Q.7 Represent the following relation by digraph.

R = {(1, 1), (1, 4), (1, 3), (2, 1), (2, 2), (3, 4), (4, 1)}

Ans.: Digraph is :-

4

1 2

3

46

Q.8 If A = {1, 2, 3} , B = {4, 5, 6} which of the following are relation from A to B. Give reason in support of your answer.

Ans.: (i) R1 = {(1, 4), (1, 5), (1, 6)}

Clearly R1 A x B. So, it is a relation from A to B.

(ii) R2 = {(1, 5), (2, 4), (3, 6)}

Clearly R2 A x B. So, it is a relation from A to B.

(iii) R3 = {(4, 2), (2, 6), (5, 1), (2, 4)}

Since (4, 2) R4 but (4, 2) A x B. So, it is not a relation.

□ □ □


Chapter-7

Functions

Q.1 A = {-1, 0, 2, 5, 6, 11} and B = {-2, -1, 0, 18, 28, 108} and f(x) = x2 – x-2 find f(A). Is

f(A) = B?

Ans.: f(-1) = (-1)2 – (-1) – 2 = 0

f(0) = 0 – 0 – 2 = -2

f(2) = (2)2 – 2 – 2 = 0

f(5) = (5)2 –5– 2 = 18

f(6) = (6)2 – 6 – 2 = 28

f(11) = (11) 2 – 11 – 2 = 108

f(A) = {f(x) : x A} = {f(-1), f(0), f(2), f(5), f(6), f(11)}

= {0, -2, 18, 28, 108}

We observe that -1 B but -1 f(A)

So f(A) B

Q.2 If a, b {1, 2, 3, 4} then which of the following are functions.

(a) f1 = {x, y : y = x + 1} (b) f2 = {(x, y) : x + y > 4}

Ans.: (a) Here f1 = {(1, 2), (2, 3), (3, 4)}

We observe that an element 4 of the given set has not mapped to any element of {1, 2, 3, 4}. So f1 is not function.

(b) f2 = {(1, 4), (4,1), (2,3), (3, 2), (2, 4), (4, 2), (3, 4), (4, 3)}

Here we observe that 2, 3, 4 are mapped to more than element of the set {1, 2, 3, 4}. So f2 is not a function.

48

Q.3 Prove that function f : Q→Q given by f(x) = 2x - 3 x Q is a bijection.

Ans.: (i) Injectivity : Let x, y be two arbitrary elements in Q then

f(x) = f(y) 2x – 3 = 2y – 3

2x = 2y x = y

So f is injective map.

(ii) Surjectivity : Let y be an arbitrary element of Q

f(x) = y 2x – 3 = y

3

2

yx

Clearly for all y Q, 3

2

yx . Thus for all y Q (Codomain)

There exist x Q (domain) given 3

2

yx such that

f(x) = f(3

2

y) = 2(

3

2

y) – 3 = y

Thus every element in the co-domain has its pre-image in x. So f is surjective.

Hence f : Q→Q is a bijection.

Q.4 If f : R→R such that f(x) = x2 and g : R→R such that g(x) = 2x + 1 then prove that

gof fog

Ans.: (gof)(x) = g{f(x)}

= g(x2)

= 2(x2) + 1 = 2x2 + 1

Now (fog)(x) = f{g(x)}

= f(2x + 1)

= (2x + 1)2


Since 2x2 + 1 (2x + 1)2

So, gof fog

Q.5 Define the following functions –

(1) Constant Function

(2) Identify Function

(3) Modulus Function

Ans.: Constant Function : A function f : R→R is a constant function if f(x) = c for all x R where C is a real constant.

y

f(x) = C

x‟ x

y‟

Identify Function : A function f : R→R is known as identify function if f(x) = x

for all x R

y

f(x) = x

x‟ 0 x

y‟

Modulus Function : A function f : R→R defined by

f(x) = x, x 0

-x, x < 0

is called the modulus function and is denoted by x .

y

50

y = -x y = x

x‟ x

y‟

Q.6 Prove that the identity function on a set A is a bijection.

Ans.: The identity function IA : A→A is defined as

IA (x) = (x) for all x A

(i) Injectivity : Let x, y be any two elements of A, then

IA (x) = IA (y)

x = y [By definition of IA]

So IA is an injective map.

(ii) Surjectivity : Let y A then there exist x = y A such that

IA (x) = x

= y

So IA is surjective map.

Hence IA is bijection.

Q.7 Show that the function f : R→R given by f(x) = Cos x for all x R is neither one-one nor onto.

Ans.: Injectivity : We know that f(0) = Cos 0 = 1 and f(2 ) = Cos 2 = 1

So different elements in R may have the same image.

Hence f is not an injective.

Surjectivity : Since the value of Cos x lie between -1 and 1, it follows that the

range of f(x) is not equal to its co-domain so f is not surjection.


Q.8 Show that f : N→N defined by

f(x) = 1

2

n, if n is odd

2

n, if n is even

is many one function.

Ans.: We observe that

1 1

(1) 12

f and 2

(2) 12

f

Thus 1 2 but f(1) = f(2) so f is many-one function.

Q.9 Let f : R→R and g : R→R be defined by f(x) = x + 1 and g(x) = x – 1, show that

fog = gof = IR

Ans.: Since f : R→R such that f(x) = x + 1 and g : R→R such that g(x) = x – 1

fog (x) = f[g(x)]

= f[x - 1]

= (x – 1) + 1

= x (1)

gof(x) = g[f(x)]

= g[x + 1]

= (x + 1) - 1

= x (2)

IR(x) = x (3)

So by equation (1), (2) and (3), we find that

fog = gof = IR

52

Q.10 Check whether the function f(x) = x is injective and surjective or not?

Ans.: Given function f(x) = x

(i) Injectivity : Since 1 R and -1 R

1 = 1 and 1 = 1

So different elements in R may have the same image.

Hence f is not an injection.

(iii) Surjectivity : Since x = x and x = x

So all negative and positive elements are mapped to positive element.

Hence it follows that range of f(x) is not equal to its codomain. So f is not bijection.


1. Two sets A and B are said to be disjoint if:

(a) (b)

(c) (d) None of the above

2. In the following venn diagram the shaded portion denotes the set:

(a) B-A (b)A

(c) A (d) A-B

3. Which of the following is true for two sets A and B?

(a) AxB=BxA (b)A-B=B-A

(c) A (d) None of the above

4. Which of the following is false for two sets A and B:

(a) A (b)


(c) A-B=B-A (d) None of the above

5. Which of the following is correct?

(a) A reflexive relation is always an identity relation

(b) An identity relation is always reflexive relation

(c)A reflexive relation is always equivalence relation

(d)None of the above

6. The relation of equality (=) in the set of real numbers is :?

(a) Reflexive

(b) Symmetric

(c) Transitive

(d) All(a) , (b) and (c)

7. If R ={(1,2),(2,3)} is a relation, then R-1

is given by?

(a) {(2,3),(1,2)

(b) {(2,1),(3,2)} (c) {(1,2)

-1,(2,3)

-1}

(d) None of the above

8. If R is a relation defined from the set A to set B, then :

(a) R AxB

(b) R=AxB

(c) R BxA


9. A relation R that is reflexive, anti symmetric and transitive is called: (a) Equivalence relation

(b) Partial order relation (c) Total order relation


10. The set { x:x is an integer and 0 x } is equal to : (a) {1,2}

54

(b) {0,1,2} (c) {0,1}


11. If X= {a, b,c}, Y={1,2,3,4} and f={a,2),(b14),(c,2)}, then domain of f is : (a) X

(b) Y (c) {2,4}


12. If A={ , b}, then A- is equal to : (a) A

(b) { (c) {b}


13. A function f is said to be bijection if it is : (a) One-one

(b) Onto (c) Both one-one and onto


14. The cardinality of the set {1,2…….,n} where n is a positive integer is :

(a) n

(b) infinite (c) n(n+1)/2


15. The set of all natural numbers N is : (a) finite

(b) Countable infinite (c) Uncountable infinite

(d) None of the above 16. The set of all integer Z is :

(a) finite



(d) None of the above 17. The set of all real numbers R is :

(a) finite



18. If A={a,b,c,d} and B={d,f,e,g} then A = (a) {a,b,c,d}

(b) {d,e,f,g} (c) {a,b,c,d,e,f,g}


19. Which one of the following is true

(a) A > A

(b) A A

(c) A A (d) None of the above

20. If A={a,b,c,d} and B={1,2,3,4} then which relation is a function from A to B (a) R={(a,1), (b,2), (a,3) ,(c,4)}

(b) R={(a,2), (b,2); (c,3)} (c) R={(a,1), (b,1); (c,2);(d,3)}

(d) None of the above 21. If A={a,b,c,d} and B={1,2,3,4} then which is bijection

(a) f={(a,1), (b,1), (c,2) ,(d,4)}

(b) f={(a,2), (b,3); (c,4),(d,3)} (c) f={(d,1), (b,2); (b,3);(a,4)}

(d) f={(a,1), (d,2); (b,2);(c,1)}

22. The set A={x:x (a) {1,2,3}

(b) {0,1,1,2,3} (c) {-3,-2,-I,0,1,2,3}

56


23. In following diagram shaded portion denote the set

(a) A

(b) A’ ’

(c)

(d) ( )’

24. If A is subset of a universal set U , them A U

(a) A

(b) U (c) 0


30. In the set of straight lines in the plane, R is the relation “paralle” then R is: (a) not reflexive

(b) antisymmetric (c) not transitive

(d) an equivalent relation

ANSWER KEY:

1-b 2-d 3-c 4-c 5-a 6-d 7-a 8-b 9-b 10-a

11-a 14-a 15-a 16-a 17-b 18-b 19-c 20-c 21-b 22-c

23-d 24-c 25-d 26-a 27-b 28-d 29-b 30-d

B A


Chapter-8

Proportional Calculus

Q.1 Write converse, inverse and contraposition of the implication “If I am hungry,

then I will eat”.

Ans.: Let p = “I am hungry”

and q = “I will eat”

Converse : If I will eat then I am hungry.

Inverse : If I am not hungry, then I will not eat.

Contraposition : It I will not eat, then I am not hungry.

Q.2 Complete truth table for

(p→q)↔( q→ p)

Ans.:

p q p q q p q p ( ) ( )p q q p

T

T

F

F

T

F

T

F

T

F

T

T

F

T

F

T

F

F

T

T

T

F

T

T

T

T

T

T

Q.3 Using the truth table show that

( ) ( )p q p q q p

Ans.:

I II III IV V VI VII VIII

p p q q p→q p q q p ( ) ( )p q q p

T F T F T T T T

58

T

F

F

F

T

T

F

T

F

T

F

T

F

F

T

F

T

T

T

F

T

F

F

T

Column (V) and Column (VIII) are identical. Hence the statement is true.

Q.4 Show that ( )p q p is a tautology.

Ans.:

p q p q ( )p q p

T

T

F

F

T

F

T

F

T

F

F

F

T

T

T

T

Hence it is tautology.

Q.5 State and prove De-Margan‟s Law by Proportional Calculus.

Ans.: The following statements are known as De-Margan‟s Law :

(i) ( ) ( ) ( )p q p q

(ii) ( ) ( ) ( )p q p q

(i) ( ) ( ) ( )p q p q

I II III IV V VI VII

p q p q ( )p q p q ( ) ( )p q

T

T

F

F

T

F

T

F

T

F

F

F

F

T

T

T

F

F

T

T

F

T

F

T

F

T

T

T

Column (IV) and (VII) are identical.

(ii) ( ) ( ) ( )p q p q

I II III IV V VI VII

p q p q ( )p q p q ( ) ( )p q


T

T

F

F

T

F

T

F

T

T

T

F

F

F

F

T

F

F

T

T

F

T

F

T

F

F

F

T

Here Column (IV) and (VII) are identical.

Hence Proved.

Q.6 If A = {1, 2, 3, 4, 5, 6, 7, 8}, then examine the truth value of the following predicates –

(i) x A, p(x) = x + 3 = 9

(ii) x A, p(x) = x + 4 = 13

Ans.: (i) If x = 6 then x + 3 = 9 is true.

Since 6 A , x A for which p(x) is true.

(ii) x + 4 = 13 is true only for x = 9

But 9 A

the given predicate p(x) is false.

Q.7 Using proper logical statement variables and logical connectives answer the

following question –

A person got a note about some treasure hidden in the vicinity of his home. If

the statements are taken as true then where is treasure hidden?

(a) If this house is next to lake, then treasure is not in the kitchen.

(b) If there in the front yard is neem, then the treasure is in the kitchen.

(c) This house is next to a lake.

(d) The tree is in the front yard is neem, or the treasure is buried under the

flagpole.

(e) If the tree in the backyard is an oak, then the treasure is in the garage.

Ans.: We assign symbols to the proportions as given below :-

p : House is next to a lake.

60

q : Treasure is in the kitchen.

r : Tree in the front yard is neem.

s : Treasure is buried under the flagpole.

t : Tree in the backyard is oak.

u : Treasure is in the garage.

Then the given data can be put as follows :-

(1) p q

(2) r p

(3) p

(4) r s

(5) t u

(i) p q [by (1)]

p [by (3)]

q [Modulus Ponens]

(ii) r p [by (2)]

q

r [Modulus Tollens]

(iii) r s [by (4)]

r

s [Disjunctive Syllogism]

Hence treasure is buried under the flagpole.

Multiple Choice Questions 1. is logically equivalent to :

(a) (b)

(c) (d) ) 2. p q is logically equivalent to :


(a) (b)

(c) ) (d) )

3. The converse of p q is :

(a) (b)

(c) ) (d) )

4. p q means:

(a) (b)

(c) ) (d) )

5. The argument relates to if p then q

(a) modus ponens (b)

(c) ) (d) ) 23 For the argument

If today is Holi Dahan , then today is full moon day. Today is Holi Dahan Today is full moon day.

The fact that this argument form is valid is called

(a) modus ponens (b)

(c) )

(d) )

22. An argument is valid means that it has valid:

(a) Logic (b) Hypothesis

(c) Form (d) Assumption

21. The contra positive of a conditional “if p then q” is :

(a) Logic (b) Hypothesis

62

(c) Form (d) Assumption

18. A proposition (or statement) is a sentence that is :

(a) True or false (b) False+

(c) True (d) True or false but not both

14. What will be the contra positive of p

(a) q (b)

(c) ) (d) None of the above

Answers key:

1-c 2-a 3-d 4-c 5-b 23-c 22-b 21-d 18-d 14-b


BACHELOR OF COMPUTER APPLICATIONS

(Part-I) EXAMINATION

(Faculty of Science) (Three – Year Scheme of 10+2+3 Pattern)

Discrete mathematics

Paper-114

OBJECTIVE PART- I

Year - 2011 Time allowed : One Hour Maximum Marks : 20

This question paper contains 40 multiple choice questions with four choices and student will

have to pick the correct one (each carrying ½ mark).

1. Which is the false statement? (a) All the integers are real numbers

(b) All the integers are rational numbers (c) All the natural numbers are rational numbers

(d) All the rational numbers are integers ( )

2. The antecedent in the conditional statement p q is:

(a) p (b) q (c) (d) p q ( )

3. For any two complex numbers 1z and 2z . which is correct?

(a) 1 1 1 2z z z z (b) 1 1 1 2z z z z

(c) 1 2 1 2z z z z (d) 1 2 1 2z z z z ( )

64

4. The decimal form of 2(101) is:

(a) 2(5) (b)

10(4)

(c) 10(3) (d) 10(6) ( )

5. Binary numeral of 109 is:

(a) 2(1100111) (b) 2(1110011)

(c) 2(1101101) (d) 2(1011011) ( )

6. 2 2(10000) (1011) is equal to:

(a) 2(1011)

(b) 2(101)

(c) 2(1110)

(d) 2(111) ( )

7. What is the normalized floating point representation for the decimal number 88.95?

(a) 8.8954E+1 (b) .88954E+22n–1 (c) .08954E+2 (d) .08954E+1 ( )

8. 2’s complement of the number 101101 is: (a) 1001100 (b) 0100110

(c) 010011 (d) 0100111 ( )

9. If p q is an implication, then its contra positive is:

(a) p q (b) q p

(c) q p (d) p q ( )

10. Which quantifier is used in , ( )x P x :

(a) Universal (b) Existential (c) Propositional (d) None of the above ( )

11. Which is true? (a) ( )( 7 4)n N n

(b) ( )( 2 6)n N n

(c) ( )( 7 4)n N n


(d) ( )( 2 6)n N n ( )

12. The law of syllogism is:

(a) _____

p q

p

q

(b) _____

p

p q

q

(c) (d)

_____

p q

q r

p r

_____

q

p q

q

( )

13. ( ) ( ) ( )P Q R P Q P R is known as:

(a) Associative Law (b) Idempotent Law (c) Detachment Law (d) Distributive Law ( )

14. Let P (n) be a statement for mathematical induction technique, then n belongs to: (a) Natural number (b) Integer

(c) Real number (d) Complex number ( ) 15 If A= {1, 2, 3}, then cardinality of the set P (A) is:

(a) 3 (b) 4 (c) 8 (d)

16. If A= {2, 3}, B= {3, 4, 5} and R is a relation such that R= {(2, 3), (2, 4), (3, 5)}, then

range of 1R is (a) {2, 3, 4} (b) {2, 3}

(c) {3, 4, 5} (d) {2, 3, 5} ( )

17. If A is a subset of a universal set U, then A U equals to: (a) A (b) U1

(c) (d) A ( )

18. If ,a b N such that a is a divisor of b, then the relation “Is a divisor of” is:

(a) reflexive, symmetric and transitive

66

(b) reflexive, antisymmetric and transitive

(c) reflexive, symmetric and antisymmetric (d) Only antisymmetric ( )

19. If :f N N and :g N N are defined as 2( )f x x and ( ) 1g x x , then (2)gof is:

(a) 2 (b) 3 (c) 4 (d) 5 ( )

20. A function :f z z is defined as 2( ) 1f x x for all x z , where z is a set of

positive integers, then the function f is: (a) one-one onto (b) many one onto

(c) many one into (d) one-one into ( )

21. A relation R: A B is a subset of:

(a) A x B (b) B X A (c) A B

(d) A B ( )

22. If aRb aRc aRc , then the relation R is: (a) Reflexive (b) Symmetric

(c) Transitive (d) antisymmetric ( )

23. EBCDIC code expresses any character in low many binary numbers?

(a) 4 (b) 6 (c) 8 (d) 16 ( )

24. The gray code of the binary number 1101 is: (a) 1100 b) 1011

(c) 1001 (d) 1110 ( ) 25. If the statement p is false and q is true, then p q is:

(a) False (b) True

(c) True or False (d) True or False ( ) 26. p q is equivalent to:

(a) p q (b) p q

(c) p q (d) p q ( )


27. If A and B be two finite sets having m and n elements respectively, then number of relation from A to B are:

(a) 2m n (b) 2m

(c) 2n (d) 2mn ( )

28. A function is said to be a bijection if it is:

(a) one-one (b) onto (c) many one onto (d) both one-one and onto ( )

29. The number of vertices in a graph is called (a) order of graph (b) size of graph

(c) degree of graph (d) length of graph ( )

30. This size of 5 regular graph with 20 vertices is: (a) 30 (b) 40 (c) 50 (d) 100 ( )

31. ( 3) is a/an:

(a) real number (b) rational number (c) complex number (d) irrational number ( )

32. Which is true of the following? (a) The trial is an open walk (b) The path ia a trial

(c) The circuit is a closed walk (d) All of the above ( )

33. A complete graph with n vertices has ( 1)

2

n n:

(a) odd edges (b) odd vertices (c) even edges (d) even vertices ( )

34. A complete graph nk is plannar graph if and only if:

(a) n > 8 (b) n < 5

(c) n = 6 (d) n > 5 ( ) 35. A connected plannar grapg eith n vertices, e edges and r regions gives:

(a) 3 2r e (b) 3e n

(c) 3 6n e (d) All of the above ( )

68

36. How many pendent vertices are there in a tree?

(a) Two (b) Atleast two (c) Three (d) Atleast three ( )

37. Hamiltonian path ia a: (a) Euler circuit (b) Rooted tree

(c) Spanning tree (d) None of the above

( ) 38. A graph with 2n vertices of degree one, 3n vertices of degree two and n vertices of degree

three is called: (a) tree (b) minimal spanning tree

(c) Hamilton circuit (d) Complete graph ( ) 39. The symmetric difference of two sets A and B is:

(a) ( ) ( )A B A B (b) ( ) ( )A B A B

(c) ( ) ( )A B B A (d) ( ) ( )A B B A

( )

40. An argument is valid means it has valid: (a) Hypothesis b) Logic

(c) Assumption (d) Premises ( )

Answer Key

1. ( ) 2. ( ) 3. ( ) 4. ( ) 5. ( ) 6. ( ) 7. ( ) 8. ( ) 9. ( ) 10. ( )

11. ( ) 12. ( ) 13. ( ) 14. ( ) 15. ( ) 16. ( ) 17. ( ) 18. ( ) 19. ( ) 20. ( )

21. ( ) 22. ( ) 23. ( ) 24. ( ) 25. ( ) 26. ( ) 27. ( ) 28. ( ) 29. ( ) 30. ( )

31. ( ) 32. ( ) 33. ( ) 34. ( ) 35. ( ) 36. ( ) 37. ( ) 38. ( ) 39. ( ) 40. ( )

______


DESCRIPTIVE PART- II

Year- 2011

Time allowed: 2 Hours Maximum Marks : 30

Attempt any four questions out of the six. All questions Carry 7½ marks each.

Q.1 (a) Convert 16( )FO into octal form.

(b) Find the hexadecimal equivalent of 1/ 2

8(32.4) .3 .

(c) Evaluate : 2 2 2 2(11) (111) (1111) (11111)

Q.2 (a) Check the validity of the following argument:

_________

r

p q

q r

p

(b) Prove that: ( ) ( ) ( )p q r p r q r

Q.3 (a) Prove by mathematical induction that the sum of the cubes of three consecutive

integers is divisible by 9(nine)

(b) A survey shows that 74% of the Indians like apples, whereas 68% like oranges.

What percentage of the Indians like both apples and oranges?

Q.4 (a) If R and S are equivalent relations on a set A, then prove that R S is also an equlvalence relation.

(b) Show that the function :f R R defined by the 3( ) 3 7f x x for all x R is a

bijection

Q.5 (a) Find the shortest path between the vertices a and z in the following graph:

70

b 5 d 5 f

7

z

4

gc e6

5

2132

3

a

4

(b) Are the following graphs isomorphic ? Explain your answer:

V2

V4

V3

V1

V5

V4

V1

V3

V2

Q.6 (a) Find the minimal spanning tree of the following graph:

V5

111

6

4

V4

5

V3

8

9

3

10

V2

7

V1

2

(b) Prove that a tree with n vertices has n-1 edges.



Paper-114

OBJECTIVE PART- I




1. The largest four digit number in hexadecimal system is: (a) 7777 (b) 1111

(c) FFFF (d) 9999 ( )

2. The value of the number 11.0011 E-112 is: (a) 3.1875 (b) .398437 (c) –3.1875 (d) –0.398437 ( )

3. (A B)' is equal to:

(a) A' B' (b) A B (c) A B (d) A' B' ( )

4. If R = {(1,4), (1,8), (3,4,) ,(3,6), (3,8)} is a relation, then the range of R is equal to: (a) (1,3) (b) (1,3,4)

(c) (4,6,8) (d) (3,4,6) ( )

5. If A = { , { }}, then cardinal number of A is: (a) 0 (b) 1 (c) 2 (d) None of the above ( )

6. Relation R = {(1,1), (2,2) (3,3)} is: (a) Only reflexive

(b) Only symmetric (c) Only reflexive and symmetric (d) Reflexive, symmetric and transitive ( )

72

7. Number or reflexive relations formed on a set of n elements: (a) 2n (b) 2n–1 (c) 2n(n-1) (d) None of the above ( )

8. (p q) is logically equivalent to:

(a) p q (b) p q

(c) p q (d) (q p) ( )

9. (g o f)–1 is equal to:

(a) (f o g)–1 (b) f1 o g1 (c) g–1 o f1 (d) f o g ( )

10. If A = (8n – 7n–1/n N) and B = (49 (n–1)/n N then: (a) A B (b) B A

(c) A = B (d) None of the above ( )

11. Which of the following is the empty set? (a) {x/x is a real number and x2 – 1 = 0} (b) {x/x is a real number and x2 + 1 = 0}

(c) {x/x is a real number and x2 – 9 = 0} (d) {x/x is a real number and x2 = x +2} ( )

12. Let A and B be two finite sets having m and n elements, respectively. Then the total number of mappings from A to B is:

(a) mn (b) 2mn (c) mn (d) nm ( )

13. Let f : R R be defined by f(x) = 3x–4 then f(x) is:

(a) (b)

(c) 3x +4 (d) None of the above ( )

14. Degree of isolated vertex is: (a) 0 (b) 1

(c) Infinite (d) None of the above ( )

15. Number of edges in a complete graph Kn is: (a) n (b) 2n


(c) (d) n (n–1) ( )

16. If G is connected planner graph with n vertices, e edges and r regions then n–e + r is equal to : (a) 0 (b) 1

(c) 2 (d) 3 ( )

17. A vertex v in a nontrivial tree T is a cut vertex is and only if: (a) d (v) < 0 (b) d (v) > 0 (c) d (v) < 1 (d) d (v) > 1 ( )

18. Conversion the binary number 1011100012 into the octal number system is:

(a) 6518 (b) 7618 (c) .2358 (d) 561g ( )

19. 10101012 + 111001012 is equal to: (a) (01001000)2 (b) (100111010)2

(c) (11010111)2 (d) (00010111)2 ( ) 20. The composite mapping f o g of the maps f : R R, f(x) = sin x and g : R R, g(x) =

x2, is :

(a) x2 sin x (b) (sin x)2

(c) sin x2 (d) ( )

21. Which one of the following is not a function? (a) {(x,y)}/x,y R, x2 = y} (b) {(x,y)/x, y R, y2 = x}

(c) {(x,y)/x, y R, x =y3} (d) {(x,y)/x, y R, y = x3} ( )

22. If A = {1,2,3,4}, B = {3,4,5,6}, then A–B =: (a) {1,2} (b) {5,6}

(c) (d) {1,2,3,4,5,6} ( )

23. If A = {1,2,3,4} B = {2,3,6,7} C = {2,5,8} then A (B C) =: (a) {2} (b) {1,2,3,4}

(c) {2,3,6,7} (d) {2,5,8} ( )

74

24. Shaded portion of the following Venn diagram denotes:

A B

(a) A B (b) A B (c) A – B (d) B – A ( )

25. If R1 {(x,y)/x2 +y2 = 1, x,y R}, then R1 is : (a) Reflexive (b) Symmetric

(c) Transitive (d) Antisymmetric ( ) 26. The void relation on a set A is:

(a) reflexive (b) symmetric and transitive (c) reflexive and symmetric (d) Reflexive and transitive ( )

27. If R1 = {(x,y)/ y,x,y R}, then R1 is: (a) Partial order relation (b) equivalence relation

(c) both (d) none of the above ( )

28. If G is a simple planer graph, then there is a vertex is a vertex v in G such that: (a) deg (v) 1 (b) deg (v) 3 (c) deg (v) 5 (d) deg (v) 7 ( )

29. Completes bipartite graph K3,3 is:

(a) Planner graph (b) non-planer graph (c) both (d) None of the above ( )

30. This graph is: V2

e1

V1

e2

e3

V3

e6

V4

e5

e4


(a) Eulerian

(b) Eulerian and Hamiltonian (c) Eulerian but not Hamiltonian

(d) Hamiltonian but not Eulerian ( )

31. If f(x) = loge x, the f(xy) is :

(a) f(x) + f(y) (b) f (x) f(y) (c) f (x) – f(y) (d) f(x) / f (y) ( )

32. is a : (a) Natural Number (b) Rational Number (c) Irrational Number (d) Imaginary Number ( )

33. The argument is valid if: (a) p q is tautology (b) q is false

(c) p q is fallacy (d) None of the above ( )

34. Number of vertex in a wheel Wn is: (a) n (b) (n +1) (c) 2n (d) n2 ( )

35. A tree with n vertices has exactly…………edges.

(a) n (b) n +1 (c) 2n (d) n –1 ( )

36. For n N, 32n + 7 is divisible by: (a) 16 (b) 9

(c) 8 (d) 6 ( ) 37. The negation of a conditional statement p q is equal to :

(a) p q (b) q p

(c) q p (d) p ̂ p ( )

38. Number of maximum leaves in an m-ary tree of height h is:

(a) hm (b) hm–1 (c) mh (d) mh–1 ( )

39. How many edges are there in a graph with ten vertices each of degree 6? (a) 30 (b) 60

(c) 20 (d) 10 ( )

76

40. Which of the following is true ? (a) domain range (b) codomain domain (c) range codomain (d) codomain range ( )

Answer Key

1. (c) 2. (c) 3. (a) 4. (c) 5. (c) 6. (d) 7. (c) 8. (b,c) 9. (b) 10. (a)

11. (b) 12. (b) 13. (c) 14. (a) 15. (c) 16. (c) 17. (d) 18. (d) 19. (b) 20. (c)

21. (b) 22. (a) 23. (b) 24. (b) 25. (b) 26. (b) 27. (a) 28. (c) 29. (b) 30. (c)

31. (a) 32. (c) 33. (a) 34. (b) 35. (c) 36. (c) 37. (c) 38. (c) 39. (a) 40. (c)

______



Year- 2010



Q.1 (a) Convert, 561, into hexadecimal number system.

(b) Write the full form of BCD and EBCDIC. Why was BCD code extended to

EBCDIC?

(c) Write the bit pattern for the word CISTEMS using the ASCII-7 coding scheme.

Q.2 Test the validity of the argument p

p ^ q r v s

q s r

(b) Prove that

P q = (p ^ q ) v ( p ^ ˜q)

Q.3 (a) If A, B, C and D are any four sets, then prove that : (A X B) (C X D) = (A C ) X (B B)

(b) Prove by the principle of mathematical induction that 9n –8n–1 is divisible by 64,

for all integers n 2.

Q.4 (a) The relation R defined on a non-empty set A is antisymmetric if and only if R

R-1 IA.

(b) If f : R R, f(x) = 2x + 5, then prove that f is one-one onto function. Also find f-1

(x).

Q.5 (a) Define Euler graph.

Prove that if G is simple planar graph, then there is a vertex v is G such that deg

(v) 5.

78

(b) Find the shortest path between the vertices a and f, in the following weighted

graph.

C e

fa

4

2

82

10

6

b 5 d

3

5

1

Q.6 (a) Define Binary tree. Prove that if T is a binary tree with a n vertices and of height h then.

h + 1 n 2 h+1 –1 (b) Find the minimal spanning tree of the graph.

b 2 c

2

da

33

4

1

f 3 e

14

54

__________



Paper-114

OBJECTIVE PART- I



have to pick the correct one (each carrying ½ mark). 1. (17)10 in binary system is: (a) (1001)2 (b) (100001)2

(c) (10001)2 (d) (01001)2 ( )

2. 0.5452E3 – 0.5424E3 is: (a) 0.0024E3 (b) 0.0028E3 (c) 0.002E3 (d) 0.003E3 ( )

3. If A = { , { }}, then cardinal number of A is: (a) 0 (b) 1

(c) 2 (d) None of these ( ) 4. (A B)' is equal to:

(a) (A' B') (b) (A B) (c) (A' B') (d) (A B') ( )

5. Relation R = {(1,1), (1,2) (1,3) (2,2) (2,3), (3,3)} is : (a) Only reflexive

(b) Only symmetric (c) Only reflexive and transitive

(d) Reflexive, symmetric and transitive ( ) 6. (g o f)–1 is equal to:

(a) (f o g)–1 (b) g–1 o f-1

80

(c) f–1 o g–1

(d) f o g ( ) 7. Degree of pendant vertex is:

(a) 0 (b) 1 (c) infinite (d) none of the above ( )

8. The number of vertices of odd degrees in a graph G is always. (a) even (b) odd

(c) zero (d) rational number ( )

9. Number of edges in a graph with 7 vertices each of degree 4 will be: (a) 28 (b) 14 (c) 7 (d) 4 ( )

10. A tree with n vertices has exactly……………..edges.

(a) n (b) n –1 (c) n +1 (d) n –2 ( )

11. (25.6)8 can be represented as: (a) (21)10 (b) (12.75)10

(c) (21.75)10 (d) (168)10 ( ) 12. If n(A) = r, then total number of non empty subsets of A are:

(a) 2r (b) 2r–1 (c) 2r –1 (d) 2r +1 ( )

13. (00010111)2 + (00010001)2 : (a) (01001000)2 (b) (00101000)2

(c) (11010111)2 (d) (00010111)2 ( )

14. Hamming code is: (a) error – finding code (b) error – detecting code (c) error – correcting code (d) error – approximation code ( )

15. Contra positive statement of p q is :

(a) p q (b) q p

(c) q p (d) q p ( )

16. {a,b} {{a,b} } =


(a) {a,b} (b) {{a,b}}

(c) {a,b, {a,b}} (d) ( )

17. Incoming degree of the root of the rooted tree is: (a) 0 (b) 1 (c) 2 (d) Infinite ( )

18. A connected graph is a tree if: (a) e = v (b) e = v +1

(c) e = v–1 (d) none of the above ( ) 19. Conversion of the decimal number (58)10 into hexadecimal is:

(a) (3B)16 (b) (2A)16 (c) (3A)16 (d) (2B)16 ( )

20. If A = {a,c,d}, B = {c,d,e}, then A – B = (a) {a, e} (b) {e}

(c) {a} (d) {c,d} ( )

21. Shaded portion of the following venn diagram denotes:

A B U

(a) A B (b) A B (c) Ac (d) A – B ( )

22. If aRb ^ bRc aRc, then relation R is:

(a) Reflexive (b) Symmetric (c) Transitive (d) Antisymmetric ( )

23. If f (x) = eax, the f (x +y) is :

(a) f (x) + f(y) (b) f (x) f (y) (c) af (x) f (y) (d) f (x)/f (y) ( )

82

24. If P: It is raining and q : team wins, then "if the team does not win, then it is not raining is:

(a) p q (b) q p

(c) q p (d) p q ( )

25. (a) natural number (b) rational number (c) irrational number (d) imaginary number ( )

26. Number of different edges in the complete graph Kn is: (a) n (b) n (n–1)

(c) (d) ( )

27. P = 2 > 3, q = Earth does not rotate, then:

(a) p is true (b) q is true (c) p q is true (d) p ^ q is false ( )

28. The argument is valid if: (a) P q is fallacy (b) p q is tautology

(c) q is false (d) none of the above ( )

29. Graph G = (V, E) is called infinite graph if:

(a) Only set V is infinite (b) Only set G is infinite (c) Both sets V and G are finite

(d) Both sets V and G are infinite ( )

30. Degree of each vertex of complete graph k4 is: (a) 2 (b) 4 (c) 3 (d) 5

31. This graph is:


a

e2

e3

e1

e4 c5

d

cb

(a) Eulerian

(b) Eulerian and Hamiltonian (c) Eulerian but not Hamiltonian

(d) Hamiltonian but not Eulerian ( )

32. If G be a connected planer graph with v vertices, e edges and r region, then: (a) v + e – r =2 (b) e + r – v = 2 (c) v + r – e = 2 (d) v + r + e = 2 ( )

33. A connected graph that contains no cycle is:

(a) simple graphs (b) planar graph (c) tree (d) circuit ( )

34. Which of the following is true? (a) domain range (b) range co-domain

(c) co-domain domain (d) co-domain range ( ) 35. A function is called bijection if it is:

(a) only one-one (b) only onto (c) one-one and onto (d) None of the above ( )

36. If R be an equivalence relation in A, then R–1 in the set A is: (a) Partial order relation (b) Equivalence relation

(c) Anit-symmetric relation (d) None of the above ( )

37. The universal quantifier of P (x) is denoted by : (a) x P (x) (b) x P (x)

(c) x P (x) (d) x P (x) ( )

38. A relation R: A B is subset of :

(a) A B (b) B x A (c) A x B (c) A B ( )

84

39. If x is element of A B then:

(a) A B = {x x A ^ x B}

(b) A B = {x x A ^ x B}

(c) A B = {x x A x B}

(d) A B = {x x ^ x B} ( )

40. If F: R R be defined by f (x) = 5 x + 7, then F –1 (x) is:

(a) (b)

(c) (d) ( )

Answer Keys

1. (c) 2. (b) 3. (c) 4. (c) 5. (c) 6. (c) 7. (b) 8. (a) 9. (b) 10. (b)

11. (a) 12. (b) 13. (b) 14. (c) 15. (b) 16. (c) 17. (a) 18. (c) 19. (c) 20. (a)

21. (d) 22. (c) 23. (b) 24. (c) 25. (c) 26. (c) 27. (d) 28. (b) 29. (a) 30. (c)

31. (b) 32. (c) 33. (c) 34. (d) 35. (c) 36. (b) 37. (c) 38. (c) 39. (a) 40. (b)

__________



Year- 2009



Q.1 (i) Convert (0.65625)10 into binary system.

(ii) Encode the following in ASCII code: 'How are you”?

(iii) Explain the error – detecting and error correcting codes.

Q.2 (a) Prove that: (p q) ^ (P r) p (q ^ r)

(b) Test the validity of the argument :

If you do every problem in this book,

Then you will learn discrete mathematics,

You have learned discrete mathematics,

You did every problem in this book.

Q.3 (a) Prove by mathematical induction that : P (n) = 1.2 + 2.22 + ………..n.2n = (n – 1)2n+1 +2; for n N

(b) For the sets A,B, C prove that :

A X (B C) = (A X B) (A X C) Q.4 (a) Show that a relation R defined in a set A is symmetric if and only if R = R–1

(b) The function f: R R is defined by f (x) = x2 +3 and g : R R is defined by :

g (x)

Find f o g and g o f.

Q.5 (a) Prove that a connected graph G is Eulerian if and only if the degree of every vertex is even.

86

(b) Find the shortest path between the vertices V1 and V8 in the following weighted

graph: V2 2 V3

4

2 3

V5

76

V7

V8V1

1

2

V4

3

4

5

1

V6 Q.6 (a) Prove that a tree with n vertices has exactly (n–1) edges.

(b) Write a short note on minimal spanning tree.

_________



Paper-114

OBJECTIVE PART- I




1. The number 2.444 ……….is called a or an:

(a) Natural number (b) Integer (c) Rational number (d) Irrational number ( )

2. If n is composite integer, then n has prime divisor less than or equal to:

(a) (b) n2

(c) (d) n +2 ( )

3. The range of unsigned integers stored in N bits computer is from zero to: (a) 2N (b) 2N–1

(c) 2N+1 (d) None of these ( ) 4. The decimal equivalent of the binary number (101111)2 is:

(a) (74)10 (b) (64)10 (c) (46)10

(d) (47)10 ( ) 5. Conversion of the decimal number (47)10 into hexadecimal is:

(a) (67)16 (b) (3 E)16 (c) (2 F)16 (d) (2 A)16 ( )

6. The binary number of a octal number (7)8 is : (a) (110)2 (b) (101)2

(c) (1101)2 (d) (111)2 ( )

88

7. An element x belongs to the difference of a and B, this tells us:

(a) A – B= {x/x A ^ x B}

(b) A – B = {x/x A ^ x B} (c) A – B = {x/x A x B}

(d) A – B = {x/x A x B} ( )

8. The difference of {1,3,5} and {1,2,3} is the set: (a) {1} (b) {2}

(c) {3} (d) {5} ( ) 9. What is the power set of the empty set :

(a) (b) { }

(c) { , { }} (d) None of these ( )

10. Two sets are equal if any only if they have:

(a) Same elements (b) Equal elements (c) Unequal elements (d) Different elements ( )

11. The solution of the linear congruence 3x = 4 (mod 7) is: (a) 2 (b) 4

(c) 6 (d) 8 ( )

12. The identity law is: (a) A = A (b) A U = U

(c) A = (d) All of the above ( )

13. The universal quantifier of P (x) is denoted by :

(a) x P (x) (b) x P (x)

(c) x P (x) (d) x P (x) ( )

14. Let Q (x, y) denote the statement 'x = y +3:' what is the truth value of the proposition Q

(3,0) (a) True (b) False (c) Empty (d) Not exist ( )

15. Shaded Portion of the following Venn diagram denotes:


A B

C

U

(a) A B C (b) A B C

(c) A B C (d) A B C ( )

16. Let A and B be two non empty sets, then a binary relation R: A B is a subset of:

(a) A B (b) B x A

(c) A x B (d) A B ( )

17. If the sets A and B having n element, then total number of relation defined from A to B are: (a) 2n (b) 2n2

(c) n (d) n2 ( )

18. If aRb ^ bRa a=b, then the relation R is called:

(a) Reflexive (b) Symmetric

(c) Antisymmetric (d) Transitive ( ) 19. If R is an equivalence rlation in a set, A, then the relation R–1 in the set a is:

(a) An equivalence relation (b) a partial order relation (c) an ant symmetric relation (d) None of these ( )

20. If f (x) = log, x then f (x) is: (a) f (x) + f (y) (b) f (x) – F (y)

(c) F (x) . f (y) (d) f (x) / f (y) ( )

21. If f : N Z, f (x) = x2, then f is called:

(a) many one function (b) one-one function (c) onto function (d) one one onto function ( )

22. If A = {–2, –1, 0, 1, 2} and f : A Z given by f (x) = x2 –2x –3, then range of f is:

(a) {5, 6, –3, –4} (b) {–4, –3, 5, 8} (c) {–4, –3, 0, 5} (d) {–4, –3, 0, 5 ,6} ( )

23. For the principle of mathematical induction the statement should include n as:

90

(a) a natural number (b) an integer

(c) any rational number (d) an irrational number ( ) 24. If f: R R be defined by f (x) = 2x –3, then f–1 (x) is:

(a) (b)

(c) x+2 (d) ( )

3

25. If p is ' roses are red' and q is 'violets blue', then 'roses are not red, or violets are not blue' is:

(a) (p q) (b) P q (c) P ^ q (d) None of these ( )

26. Negation of the statement x y, P (x, y) is: (a) x y P (x,y) (b) x, y P (x and y)

(c) x y P (x, y) (d) x y P (x, y) ( ) 27. The bit strings of the set {1,2,4,6,9} is:

(a) 1110101010 (b) 1100011010 (c) 1101010010 (d) 1001011001 ( )

28. P: I am hungry and q : I will eat. ' If I am hungry then I will eat' stands for : (a) p q (b) q p

(c) p q (d) p q ( )

29. The compound statement p q is true if: (a) At least one of p or q is true

(b) Both p and q are true (c) Either p is true or q is true but not both

(d) Conjunction of p and q is true ( ) 30. A vertex of degree zero is called:

(a) Pendent vertex (b) Isolated vertex (c) Point to point vertex (d) Simple vertex ( )

31. The sum of the degree all all vertices in a graph G is equal to: (a) Number of edges in G

(b) twice the number of edge in G (c) One more the number of edges in G

(d) two more the number of edges in G ( )


32. Maximum number of edges in a simple graph with n vertices is : (a) n (n +1) (b) n (n –1)

(c) (d) ( )

33. A graph with a closed path that includes every vertex exactly once is called is: (a) Hamiltonian graph (b) Simple graph

(c) Diagraph (d) Euler graph ( ) 34. For the following two graphs, we have:

(a) Isomorphic graphs (b) Planar graphs

(c) Isomorphic and planer graphs (d) None of the above ( )

35. Let G be a connected planer graph with n vertices, edges and r regions, then which of the

following is true:

(a) 3r 2e (b) e 3n (c) r = e–n+2 (d) All of the above ( )

36. Let A = {1,2,3} and B = {2,4,5,7}, then [A B] is given by: (a) 3 (b) 4

(c) 6 (d) 12 ( )

37. A graph containing no simple circuit but not connected is called a: (a) tree (b) forest

(c) binary tree (d) 12 ( )

92

38. Let T be a sub graph of a connected graph G, IF T is a tree and it contains all the vertices

of G, then T is called : : (a) a spanning tree (b) skeleton (c) maximal tree subgraph (c) all of the above ( )

39. Identify binary tree from the following three trees T1, T2 and T3 :

BC

A

D

BC

A

D

BC

A

D

T1 D DT

3

(a) T1 (b) T2

(c) T3 (d) All T1, T2 and T3 ( ) 40. A graph G is a tree if and only if it is:

(a) Minimally connected (b) Maximally connected (c) Not connected` (d) Without pendant vertices ( )

Answer Keys

1. (c) 2. (a) 3. (b) 4. (d) 5. (c) 6. (d) 7. (a) 8. (d) 9. (b) 10. (a)

11. (c) 12. (d) 13. (b) 14. (a) 15. (d) 16. (c) 17. (d) 18. (c) 19. (a) 20. (a)

21. (c) 22. (c) 23. (a) 24. (b) 25. (b) 26. (d) 27. (c) 28. (a) 29. (a) 30. (b)

31. (b) 32. (d) 33. (a) 34. (c) 35. (c) 36. (c) 37. (b) 38. (d) 39. (d) 40. (a)

_________



Year- 2008


Attempt any four questions out of the six. All questions carry 7½ marks each.

Q.1 Solve the following:

(i) Convert (101.001)2 in decimal number system

(ii) Convert (14.875)10 into binary system.

(iii) Write the binary representation of –47 in one's complement scheme

(iv) Write 0.0007 into equivalent normalized floating point number with mantissa and

exponent.

( v) Convert the decimal number 28.4 into an equivalent number in octal system.

Q.2 (a) Let S be the set of all number and R is a relation in S defined by ' a b'. Prove that

R is a partial order relation.

(b) Let p be the proposition that the sum of the first n odd numbers is n2. Prove it

byusing the mathematical induction principle.

Q.3 (a) Let the functions f: R R, g : R be defined by f (x) = 2x +1, g (x) = x2 –2 find

the formula for gof and fog. (b) Out of 1000 students, 750 offer Hindi and 400 offer English. How many can offer

Hindi only and how many can offer English only?

Q.4 (a) Prove that p q uis a valid argument.

(b) If U = {x : x 8, x N} A = {x : x 6, x N} and B = {3,4,5,7,8} then find

the bit string for the set, A, B, A B, A B, A, B and A B.

Q.5 (a) Show that e 3 v –6. Where v vertices and e (>2) edges are of a loop – free

connected planar graph. (b) Find the incidence matrix of the following diagraph:

94

e4

e6

e1

V4

e5

V1 V2

V3

e7

e2

e3 V5b

a

1

Q.6 Find the minimal spanning tree of the following labeled connected graph:

V1

5

2

V2

9

4

7

V2

V1

6 V3

3

1V5

8

.



Paper-114

OBJECTIVE PART- I




1.

(a) a rational number (b) a real number (c) an irrational number (d) none of the above ( )

2. A bit has how many possible values? (a) (b) Two

(c) (d) Eight ( )

3. Let x, y and z be integers. Then which is true: (a) if x/y and x/z, then x (y +z)

(b) if x/y, then x/yz for all integers z (c) if x/y and y/z then x/z

(d) all of the above ( ) 4. What is the decimal expansion of the (2 AEOB) 16?

(a) (175629)10 (b) (175627)10

(c) (175827)10 (d) (175829)10 ( )

5. Base 8 expansion of (12345)10 is given by:

(a) (3071)8 (b) (30071)8

(c) (2071)8 (d) (3007)8 ( )

6. A proposition is a statement that is : (a) either true or false (b) neither true nor false

96

(c) true (d) false ( )

7. Let p and q be propositions. In the implication P q, p is called the:

(a) Hypothesis (b) Conclusion (c) Consequence (d) None of the above ( )

8. The contra positive of p q is the proposition:

(a) p q (b) q p

(c) p q (d) q p ` ( )

9. a variable is called a Boolean variable if its value is:

(a) true (b) false

(c) either true or false (d) neither true nor false ( )

10. Bitwise XOR of the bit string 0110110110 and 1100011101: (a) 1010111010 (b) 1010101011 (c) 1110101011 (d) 11101010 ( )

11. P and q propositions that always have the same truth value are called:

(a) Logically equivalent (b) p q is a tautology

(c) p q (d) All the above are true ( )

12. Which is de Morgan's law? (a) p q p q (b) (p q) q q

(c) (p q) p q (d) (p q) p q ( )

13. Let Q (x,y) denote "x + y =0". What is the truth value of the quantification y x (x, y)?

(a) true (b) false

(c) neither true nor false (d) none of the above ( ) 14. If n is a positive integer then n3 –n is divisible by:

(a) 2 (b) 3 (c) 4 (d) 5 ( )

15. The statement n P (n) is true if P (1) is true and n [p (n) p (n + 1)] is true, known

as known as: (a) Principle of mathematical induction

(b) Second principle of mathematical induction

(c) Rule of inference


(d) None of the above ( )

16. The set {x/x A ^ x B} tells us as: (a) A B (b) A B (c) A – B (d) ( )

17. How many elements in the set of Cartesian product of A = {1,2}and B = {a,b,c} are :

(a) 2 (b) 3 (c) 6 (d) 5 ( )

18. Let A, B and C be three sets A (B C) is equivalent to: (a) A (B C ) (b) (C B ) A

(c) (C B) A (d) (C B) B ( ) 19. If f (x) f (y) whenever x y then the function f is called:

(a) One–one (b) on to (c) One-one onto (d) one–one into ( )

20. The function f (x) = x2 is: (a) one–one (b) on to

(c) one–one onto (d) neither one one nor onto ( )

21. If f(x) = ex then f (x–y) is: (a) f (x) – f (y) (b) f (x) +f (y) (c) f (x) . f (y) (d) f (x)/f (y) ( )

22. A relation "=" (equipment to) defined in any set A is: (a) Reflexive relation (b) Symmetric relation

(c) Transitive relation (d) Equivalence relation ( )

23. A partial order relation is not a: (a) Symmetric relation (b) Reflective relation (c) Transitive relation (d) Transitive relation ( )

24. If an function f : R R be defined as follows:

F (x) = 1 x Q

–1 x Q then f ( ) is :

98

(a) 1 (b) –1 (c) 1 and –1 (d) none of the above ( )

25. If f : A B and g : B C such that A = C, then which is true:

(a) gof : A B (b) gof : A A

(c) gof: B B (d) gof: A C ( )

26. What is cardinality of the set {x x is a subset of the set {a,b}} : (a) Two (b) Three (c) Four (d) Infinite ( )

27. A compound proposition that is sometimes true and sometimes false is called:

(a) Contradiction (b) Tautology (c) Contingency (d) Logical equivalence ( )

28. The bit string of the set {1,3,5,7,9}is : (a) 1111100000 (b) 1011001101

(c) 1001001011 (d) 1010101010 ( ) 29. The bit string of the set {1,3,5,7,9} is:

(a) |A B|= |A|–|B|+|A B| (b) |A B| = |A|+|B|–|A – B|

(c) |A B| = |A|+|B|+|A B|

(d) |A B| = |A|+|B|–|A B| ( )

30. If f: R R is a function such that f (x) = 4x +3 for x R, then f–1 (x) is: (a) 3–4 x (b) 3x+4

(c) (x – 3) (d) (x – 3) ( )

31. How many edges are there in a graph with ten vertices each of degree six?

(a) 10 (b) 20 (c) 30 (d) 16 ( )

32. An undirected graph has an even number of vertices: (a) of odd degree (b) of even degree

(c) of zero degree (d) None of the above ( )


33. A connected multi graph has an Euler path but not an Euler circuit if and only if it has

exactly: (a) two vertices of odd degree (b) two vertices of even degree (c) (a)and (b) are true (d) None of the above ( )

34. Which is of the following graph has a Hamilton circuit?

a b

ce

d

a b

cd

a b

cd

c

a b

de

(a) (b)

(d)(c) ( )

35. If G is a connected planar simple graph with four vertices, then number of edges are less

than or equivalent to: (a) Four (b) Five

(c) Six (d) Seven ( ) 36. A vertex of degree one is called:

(a) Isolated vertex (b) Pendant vertex (c) point vertex (d) Simple vertex ( )

37. A connected undirected graph with no simple circuits is known as a: (a) Planar graph (b) Directed Graph

(c) Weighted graph (d) Tree ( )

100

38. Every non-=trivial tree has at least two vertices of degree:

(a) one (b) two (c) three (c) zero ( )

39. The number of vertices in a full ary tree with I internal vertices are: (a) mi +1

(b) mi – 1 (c) m + i (d) m – i ( )

40. A visiting of the vertices of a tree is called :

(a) decision tree (b) ordered tree (c) tree traversal (d) tree balance ( )

Answer Keys

1. (a) 2. (d) 3. (d) 4. (b) 5. (b) 6. (a) 7. (a) 8. (b) 9. (c) 10. (b)

11. (d) 12. (d) 13. (a) 14. (b) 15. (a) 16. (b) 17. (c) 18. (a) 19. (a) 20. (d)

21. (d) 22. (d) 23. (a) 24. (b) 25. (b) 26. (a) 27. (a) 28. (d) 29. (d) 30. (d)

31. (c) 32. (a) 33. (a) 34. (c) 35. (c) 36. (b) 37. (d) 38. (b) 39. (a) 40. (c)

__________



Year- 2007



Q.1 Solve the following:

(a) Find the product of (110)2 and (101)2

(b) Convert the decimal 43.375 into its binary equivalent.

(c) Convert the hexadecimal number 39.B8 to an octal number.

(d) Represent – 13 in binary form.

(e) To write hexadecimal number to a decimal number,

Show that 12 AF16 = 478310

Q.2 (a) Prove that a relation R defined in a set A is a symmetric relation if and only if R = R– 1

(b) Show that the propositions p (q ^ r) and (p q ) ^ ( P r) are logically equivalent.

Q.3 (a) Use set builder notation and logical equivalences to show that A B = A B.

(b) If f : R R be given by f (x) = x2 +2 and g : R R be given by g (x) = 1–

find (gof) x and (fog)x, R being the set of real numbers.

Q.4 (a) Use mathematical induction to show that: 1+2 +22 +………+2n = 2 n+1 –1for all non – negotiable integers n.

(b) How many paths of length four are from a to d in the following simple graph:

102

a b

cd Q.5 (a) If G is a connected simple graph with e edges and v vertices where v 3v,

then prove that e 3v, –6.

(b) Determine whether the following graphs G and H are isomorphic.

µ1 2

3

5

4

6

µ

µ

µ

µ

µ

µ1

2

3

4

6µ

µ

µ

µ5µ

Q.6 (a) Show that there are at most mh leaves in an m–ary tree of height h.

(b) Use a breadth – first search to find a spanning tree for the following graph:

a b Ic

d e gf

h I j

m k



Paper-114

OBJECTIVE PART- I




1. is:

(a) a rational number (b) a real number (c) an integer (d) none of the above ( )

2. Which of the following statements is false?

(a) All the natural numbers are integers (b) All the rational numbers are integers

(c) All the rational number are real numbers

(d) All the integers are real numbers ( )

3. The decimal equivalent of the binary number (–10011)2 is: (a) –12 (b) –15 (c) –19 (d) –21 ( )

4. The number (680)10 of the decimal system equivalent to which number in octal system?

(a) (85)8 (b) (1012)8 (c) (1250)8

(d) (1300)8 ( )

5. EBCDIC code expresses any character in hot many binary numbers? (a) 2 (b) 4

(c) 8 (d) 16 ( ) 6. (11011)2 + (10011)2 is equal to:

(a) (45)10 (b) (46)10 (c) (48)10

(d) (49)10 ( )

104

7. Shaded portion following Venn diagram denotes:

A B

µ

(a) A B (b) A B (c) A – B (d) B – A ( )

8. If A is subset of a universal set U, then A , then A U equals: (a) A (b) U

(c) (d) None of the above ( )

9. If a, b N such that a is a divisor of b then the relation "is a divisor of" is: (a) Reflexive, symmetric, transitive

(b) Reflexive, anti-symmetric, transitive (c) Symmetric, anti-symmetric, transitive

(d) Reflexive, symmetric, anti-symmetric ( ) 10. If f: N N and g: N – N are defined as f (x) = x2 and g (x) = x + 1, then (g o f) (2) is :

(a) 2 (b) 3

(c) 4 (d) 5 ( )

11. A function F: z+ Z+ is defined as f (x) x2 +1 for all x Z+ where z+ is a set of positive integers, then the function F is:

(a) one–one into (b) many–one onto (c) many–one into (d) one–one into ( )

12. In the set of straight lines in the plane, R is relation "parallel" R is: (a) not reflexive (b) anti-symmetric

(c) not transitive (d) an equivalent relation ( )

13. For two distinct non-void sets A and B, (A )c is equal to : (a) A

c B

c (b) (Ac Bc)

(c) (A B)c (d) A – B ( )


14. R = {(1,1), (2,3), (3,2)} is a relation in X = {1,2,3} then R–1 : (a) does not exist (b) is an equivalent relation (c) equals R (d) is only reflexive ( )

15. Which of the following diagrams defines a function from A = {a,b,c} into B = {x, y, z}?

a

b

c

x

y

z

A B

a

b

c

x

y

z

A B

a

b

c

x

y

z

A B

a

b

c

x

y

z

A B

(i) (ii)

(iv)(iii) (a) (i) (b) (ii)

(c) (iii) (d) (iv) ( )

16. Let p be "Sita speaks English" and q be "Sita speaks German" then "Sita speaks English

but not German" is described by:

(a) p ̂ q (b) p v q

(c) p ̂ q (d) p v q ( )

17. p v (q ^ r) is equivalent to:

(a) (p v q) v (q ̂r) (b) (p v q) ̂(p ̂r)

(c) (p ̂q) v (q ̂r) (d) (p ̂q) (̂q ̂r) ( )

18. The statement p q is logically equivalent to the proposition:

(a) p v q (b) p v q

(c) p ^ q (d) p ̂q ( )

19. If p denotes "It is cold" and q denotes "it rains" then the statement" it reins only if it is

cold" can be written symbolically as: (a) p q (b) p p

106

(c) p q (d) p p ( )

20. If p denotes "He studies" and q denotes" He will pass" then the negation of the statement "If he studies, he will pass" can be written symbolically as:

(a) p q (b) q p

(c) (p q) (d) (q p) ( )

21. The set of all not-negative even integers is: (a) finite (b) countable infinite (c) uncountable (d) none of the above ( )

22. If |P| denotes the cardinality of the finite set P then for any such non – void finite disjoint

sets P and Q : (a) |P Q|=|P| – |Q| (b) |P Q| |P|+|Q|

(c) |P Q|=|P|+|Q| (d) |P Q| |P|+|Q| ( )

23. For the principle of mathematical induction the statement should include as: (a) an integer (b) any rational number

(c) a natural number (d) an irrational number ( )

24. In the following graph the degree of vertex B is:

A B

C

DE F

(a) 4 (b) 5 (c) 6 (d) 7 ( )

25. Number of edges in a graph is :

(a) equal to the sum of the degress of all vertices


(b) double of the sum of the degress of all vertices

(c) half of the sum of the degrees of all vertices (d) having no relation with the sum of the degrees of all vertices ( )

26. Out degree of the vertex V4 in the following diagram is:

V1V1

V1

V1

V2

(a) 4 (b) 3

(c) (d) ( )

27. In a graph there are 7 vertices each of degree 4, and then the number of edges in the graph is:

(a) 4 (b) 7 (c) 14 (d) 28 ( )

28. If G is a connected planar graph with n vertices, e edges, and r regions, then: (a) n + r = 2 + e (b) n + r = 2 – e

(c) n + r = e – 2 (d) none of the above ( ) 29. The number of edges in a tree with its vertices is:

(a) n (b) n +1

(c) n–1 (d) 1 ( )

30. In an acyclic graph with n vertices and k connected components, the number of edges is: (a) n + k +1 (b) n + k

(c) (d) ( ) 31. For arbitrary sets A,B and C (A–B) – C equals: (a) A – (B C) (b) (A–B) C

(c) (A–B) C (d) none of the above ( )

32. In the following weighted graph, the minimum distance between A and f is:

108

B D

C E

A

1

2

4

5 3

1

2

6

(a) 15 (b) 11 (c) 10 (d) 9 ( )

33. The following rooted tree represents:

x y z 2

+2

(a) [{x–y}–3] – (b) –

(c) (x –y)3 (d) (x –y)3 – (z2 +2) ( )

34. The following graph is:

V1

V3

V2 V6

e1 e2

e5

e4e3

(a) Hamiltonian (b) Eulerian (c) Both Hamiltonian and Eulerian (d) Neither hamiltonian nor Eulerian ( )


35. A void set is:

(a) Non– existing (b) having zero as an element (c) A sub – set of all sets (d) not a sub–set of any set ( )

36. If f: R R is a function such that f (x) = 2x –3 for x R, R f–1 (x): (a) is not defined (b) will have different range (c) is (3 –2 x) (d) is ½ (x +3) ( )

37. If f: X Y and g : Y Z are bijections then (f o g)–1 equals:

(a) g–1 o f-1 (b) f–1 o g–1 (c) (g o f)–1 (d) g o f ( )

38. if q p, then :

(a) p q (b) q p

(c) p q (d) q p ( )

39. Tautology is a proposition which is:

(a) false under all circumstances (b) false under a few circumstances

(c) true under all circumstances (d) true under a few circumstances ( )

40. Bijection is: (a) Only into function (b) only onto function

(c) a one-one onto function (d) a many-one function ( )

Answer Keys

1. (d) 2. (b) 3. (c) 4. (c) 5. (c) 6. (b) 7. (d) 8. (a) 9. (b) 10. (d)

11. (d) 12. (d) 13. (b) 14. (c) 15. (c) 16. (a) 17. (b) 18. (d) 19. (a) 20. (a)

21. (b) 22. (c) 23. (c) 24. (c) 25. (c) 26. (b) 27. (c) 28. (a) 29. (c) 30. (c)

31. (c) 32. (d) 33. (c) 34. (a) 35. (c) 36. (d) 37. (a) 38. (c) 39. (c) 40. (c)

__________

110


Year- 2006



Q.,1 (a) Convert (111001101)2 into decimal form.

(b) Represent "SHORT" in EBCDIC code.

(c) Find the value of (195)10 + (105)10 in binary code.

(d) Convert (175)10 to hexadecimal number.

(e) Show that (1101.01011)2 = (15.26)8

Q.2 Let f: Z Z where f (x) = x2 + x for all x Z, then show that f is a many one function.

Q.3 (a) If C (A) = m, C (B) = n and C (A B) = r, where C (P) is cardinal number of the

set P and r < m + n, Find C (A B).

(b) Define a symmetric and an ant symmetric relation and give an example in each case.

Q.4 (a) Construct truth tables for the following compound propositions. (p q) q (ii) (p q) ^ ( p q )

(c) Find the shortest path between the vertices a and z in the following weighted graph.

C g

za

4

2 2

7

b

5

f

3

13

6e

4

5d5

Q.5 (a) Show that, if G is simple planar graph, then there is vertex V in G such that deg (V) 5.


(b) Show that in the following graph.

Is not Hamiltonian while (ii) is Hamiltonian

V3 V4

V5

e4 e5

e2e3

V1V2

e1

e6

V1

e1

e4

V3

V4

e2e3

V2

Q.6 (a) Show that a tree with a vertices has exactly (n–1) edges.

(b) Construct a minimum spanning tree for the following weighted graph.

42

11

9

8

75

2

1 3

610

112

Keyterms

EBCDIC:- extended binary coded decimal interchange code

BCD:- binary coded Deccimal

ASCII: american standard code for information interchange

A circuit is a trail that begins and ends on the same vertex.

A cycle is a path that begins and ends on the same vertex. A trail is a walk that does not pass over the same edge twice. A trail might visit the same vertex

twice, but only if it comes and goes from a different edge each time.

A path is a walk that does not include any vertex twice, except that its first vertex might be the same as its last.

a graph consists of a finite nonempty set. whose elements are the vertices (or nodes) of the graph, together with a (possibly empty) set of unordered pairs of distinct vertices called edges (or

arcs). (A single one of the vertices is called a vertex.) Graphs:-

Graphs that have arrows added to each edge are called directed graphs or digraphs (pronounced "DYE-graphs"). The arrows show that the edge has a direction associated with it.

Forest:-

A forest is a graph with no cycles. It may or may not be connected. So a single tree is a forest,

and a forest consists of one or more trees

Kuratowski's theorem:-

Kuratowski's theorem is the statement that a graph is planar if and only if it has no subgraph that can be "collapsed" to the complete graph K(5) or the bipartite graph K(3, 3)

Function:-

A function from set S to set T is a set of ordered pairs, the first element of which is in S and the second of which is in T, such that no first element occurs in pairs with two different second elements.


Reference Books

1. Discrete Mathematics and Its Applications by Kenneth Rosen 2. Discrete Mathematics by Gary Chartrand and Ping Zhang

3. Discrete Mathematics with Applications by Susanna S. Epp

4. Discrete Mathematics by Kevin Ferland 5. DiscreteMathematicsby Norman L. Biggs

Websites:-

1. www.graphtheory.com 2. www.purplemath.com

3. www.mathwarehouse.com

4. www.wikipedia.org 5. www.geeksforgeeks.org

***********

http://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073383090/ref=sr_1_3?ie=UTF8&qid=1362199779&sr=8-3&keywords=discrete+mathematics

http://www.amazon.com/Discrete-Mathematics-Gary-Chartrand/dp/1577667301/ref=sr_1_4?ie=UTF8&qid=1362199779&sr=8-4&keywords=discrete+mathematics

http://www.amazon.com/Gary-Chartrand/e/B0027MLJP4/ref=sr_ntt_srch_lnk_4?qid=1362199779&sr=8-4

http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0534359450/ref=sr_1_6?ie=UTF8&qid=1362199779&sr=8-6&keywords=discrete+mathematics

http://www.amazon.com/Susanna-S.-Epp/e/B001IGHMF6/ref=sr_ntt_srch_lnk_6?qid=1362199779&sr=8-6

http://www.amazon.com/Discrete-Mathematics-Kevin-Ferland/dp/0618415386/ref=sr_1_5?ie=UTF8&qid=1362199779&sr=8-5&keywords=discrete+mathematics

http://www.graphtheory.com/

http://www.purplemath.com/

http://www.mathwarehouse.com/

http://www.wikipedia.org/

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