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Biyani's Think Tank Concept based notes
Discrete Mathematics (BCA Part-I)
Varsha Gupta M.Sc. (Maths) Revised by: Shiv Kishore Sharma
Lecturer Deptt. of Information Technology Biyani Girls College, Jaipur
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Discrete Mathematics
Published by :
Think Tanks Biyani Group of Colleges
Concept & Copyright :
Biyani Shikshan Samiti Sector-3, Vidhyadhar Nagar, Jaipur-302 023 (Rajasthan) Ph : 0141-2338371, 2338591-95 Fax : 0141-2338007 E-mail :
[email protected] Website :www.gurukpo.com; www.biyanicolleges.org
ISBN : 978-93-81254-37-3
Edition : 2011 Price :
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.
Leaser Type Setted by : Biyani College Printing Department
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4
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
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Discrete Mathematics
5
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.
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Content S.No. 1.
2.
Name of Topic 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 Trees 2.1 2.2 2.3 2.4
Definition and Properties of Trees Prim‟s Methods Tree Transversal 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
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Discrete Mathematics
S.No. 5.
7
Name of Topic Sets 5.1 5.2 5.3 5.4 5.5
Power Set Operations on Sets Symmetric Difference of Two Sets De-Margan‟s Law 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
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Chapter-1
Graph Theory Q.1
Draw simple graphs with one, two, three and four vertices.
Ans.: •V1
Simple graph with one vertex
Simple graph with two vertices
V1
V2
V3
Simple graph with three vertices V1
Simple graph with four vertices
V2
V4
V3
V1
V2
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Discrete Mathematics
Q.2
9
Show that if G = (V, E) is a complete bipartite graph with n vertices then the n2 total numbers of edges in G cannot exceed . 4
Ans.: Let Kp,q be a complete bipartite graph. The total no. of edges in K p,q is p.q and n total no. of vertices will be (p+q). If we take p = q = then in complete bipartite 2 n n n2 . graph K n2 , n2 no. of edges will be which is maximum (If two numbers 2 2 4 are equal then their product is maximum). Hence in a complete bipartite graph of n2 n vertices the no. of edges cannot exceed . 4 Q.3
Show that following two graphs are not isomorphic. V5 V1
V2
V3
V4
U6 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
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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)
Ans.: (i)
Walk
(ii) Trail
(iii) Path
(iv) Circuit
(v) Cycle
Walk : An alternating sequence of vertices and edges is called a Walk. It is denoted by ‘W’. Example : a
d e1
e4 b
e6 e5 e
e3 e2 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) :
(iii)
W = a e1 b e2 c e3 d is a trail.
Path : An open walk in which no vertex is repeated except the initial and terminal vertex is called a Path.
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Discrete Mathematics
11
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
3 33
2
1
2
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
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we must have a circuit going from V1 to V2, to V3, to V4, to V5 and to V1 i.e. a pentagon. V2 V1
V2 V3
V5
V1
V4
V5
(a)
V2 V3
V5
V4 (b)
V2 V1
V3
V1
V4
V3
V5
(c)
V4 (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
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Discrete Mathematics
13
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 K 5 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(v)
2e
v V
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(v)
2e
v V
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 matrix 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
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(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(v)
2e
v V
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 : 1V1
V2
V3
2V1 3-
Here degree of each vertex is one. So it is regular graph.
Degree of each vertex is two. V2
V4
V3
V1
V2
Degree of each vertex is two.
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Discrete Mathematics
15
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 = i.e. size of Kn =
n(n 1) 2
n(n 1) 2
Multiple Choice Question 1. The graph is : u v
(a) (b) (c) (d)
Simple graph Directed graph Directed multigraph Pseudograph
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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 :
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Discrete Mathematics
17
(a) 1 (b) 2 (c) 3 (d) 4 6. The length of path a,b,d,e,b in the following graph is : C
D
b
a
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
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(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 (c)
(b) 2 (d) 4
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Discrete Mathematics
19
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 (c)
1-d 11-a
2-b 12-d
(b) 7 (d) 28
3-d 13-a
4-a 14-b
5-a 15-b
6-b 16-b
7-a 17-b
8-b 18-b
9-b 19-c
10-b 20-b
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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 = v j 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
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Discrete Mathematics
21
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
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E(V1) = 2 E(V2) = 2
V3
E(V3) = 1 E(V4) = 2 (ii)
V4
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 Ans.:
Prove that every tree has either one or two centres. 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)
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Discrete Mathematics
23
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) Q.6
j(1)
k(2)
j (centre)
Find minimal spanning tree for the following weighted graph (use Prime‟s Method).
Ans.:
V1 3 V2
1 3
3
4 4
V3
2
5 1 4
V6 V5
2 V4
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V1
V2
V1
∞
3
V4
V5
V6
1 V6
4
1
V2
3
3∞
13
5
∞
∞
V3
3
3
∞
4
∞
4
V4
1
5
4
∞
2
∞
V5
4
∞
∞
2
∞
2
V6
1
∞
4
∞
2
∞
V2
VV 1 3 3
1
3
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 e
c f
g
h
d i
k
l
Pre-order
:
abeklmfgchdij
In-order
:
kelmbfgachidj
Post-order
:
klmefgbhcijda
j
m
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Discrete Mathematics
Q.8
25
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
a5 a6
a9
a10
a7
a11
a12
Full 3 – Ary Tree Q.9
Find the value of following prefix expression.
Ans.: *235/
234 23
*235 / 84 8/ 4
*2352 2 x3
*652 6 5
12
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a13
26
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 (c) tree is a simple graph graph
(b) A tree contains loops (d) A tree is a connected directed
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 :
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Discrete Mathematics
27
(a) hm (c) mh-1 5. The preorder traversal to tree T IS e k j I o
(b) hm+h (d) mh
p
T (a) j e k n o p (c) e j k n o p
(b) j e n k 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 (c) e j k n o p
(b) j e n k 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
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(a) j n o p k e (c) j k e n o p 9. A spanning tree of a graph G contains (a) All vertices of G (c) Maximum two vertices G
(b) j e k n o p (d) p o n k e j (b) At least one vertex of 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 (c) debfgihca 1-c
2-a
(b) dbeafeghi (d) none of the above 3-b
4-d
5-c
6-b
7-a
8-c
□□□
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9-a
10-c
Discrete Mathematics
29
Chapter-3
Number System Q.1
Convert (111001101)2 into decimal form and 96 into binary number.
Ans.: (111001101) 2
1* 28 1* 27 1* 26 0* 25 0* 2 4 1* 23 1* 2 2 0* 21 1* 2 8
=
256 128 64 0 0 8 4 0 1 (461)10
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
Q.2
(96)10 = (110000) 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
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Q.3
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
Convert (175) 10 into hexadecimal number.
Ans.:
16
175
Remainder
16
10 . . . . . . . . . . . . 15 = F 10 . . . . . . . . . . . . 10 = A
Q.4
Find the sum of (1011)2 and (10111)2.
Ans.: 1 So, Q.5 Ans.:
(175)10 = (AF) 16
0
1
0
1
1
1
0
1
1
1
0
0
0
1
0
(1011)2 + (10111)2 = (100010) 2
Find fixed point representation for the decimal no. 3.056 E-5. 3.056 E-5
=
3.056 x 10-5
=
0.00003056
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Discrete Mathematics
Q.6
31
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 : 0.625 x8 5.000
Q.7 Ans.:
So,
0.625)10 = (.5)8
Hence,
(692.625)10 = (1264.5)8
Convert hexadecimal no. ABC.2 into decimal form. (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. (a) (c)
Is 2 an integer or a rational number Both Rational number
(b) Integer (d) None
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3. (a) (c)
Solution of the equation x+5=3 is a/an: Natural number (b) Integer No solution (d) None of the above
4. (a) (c)
Solution of the equation x2+1=0 is a : Real number Rational number
5. (a) (c)
Solution of the equation 2x=3 is a/an: Rational number (b) Integer None of the above (d) Natural Number
6.
The total number of symbols used to represent a number in Hexadecimal number system is : 16 (b) 15 6 (d) 8
(a) (c)
(b) Complex number (d) None of the above
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 (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
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33
11. The value of 67458+3768 is : (a) 71218 (c) 73433
(b) 73238 (d) None of the above
12. Addition of 10102 and 1102 is : (a) 100002 (c) 11012
(b) 11202 (d) None of the above
13. 100002-1012 is equal to : (a) 110112 (c) 1112
(b) 102 (d) None of the above
14. is : (a) a real number (c) an integer
(b) a rational number (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
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34
(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
□□□
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35
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 So, Q.2
0 0
1 1
0 0
0 1
1
1
+
1
0
0
2‟s complement is 0010100.
Find 8-bit 2‟s complement of (35)10.
Ans.:
(35)10 Let
X
=
(100011)2
=
(00100011)2
=
00100011
1‟s complement of X
=
11011100
2‟s complement of X
=
11011100 +1 11011101
So, Q.3 Ans.:
8-bit 2‟s complement of (35)10 is 11011101
Write BCD code for decimal number 12. (12)10
=
(0001 0010)BCD
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Q.4 Ans.: Q.5 Ans.:
Write Gray code of the no. (1111)2. Gray code of (1111)2 is 1000. Represent „SHORT‟ in EBCDIC code. 11100010
11001000
11010110
11011001
S
H
O
R
11100011
Multiple Choice Questions 1.
1‟s complement of the number 1011001 is (a) 0100110 (c) 0110010
(b) 1100110 (d) 0110011
2 . 2‟s complement of the number 1011001 is (a) 0100110 (c) 1110010
(b)0100111 (d) None of the above
3 . Natural BCD code of 4210 is : (a)01000010 (c)010010
(b)10010 (d) None of the above
4 . Natural BCD code of 4210 is : (a)01000010 (c)010010
(b)10010 (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 (c) 4 bits
(b) 6bits (d) None of the above
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T
Discrete Mathematics
7.
8-
9-
In ASCII, the symbol II stands for : (a) International information (c) Information Interchange
(b) International Information (d) International Interchange
Most of the errors occur at : (a) One bit position (c) Three bit position
(b) Two bit position (d) None of the above
The parity bit is : (a) (b) (c) (d)
10-
37
A bit at one position in any transmitted An extra bit attached to each code word for detecting the error An extra bit attached to each code word for correcting the error None of the above
Hamming code is a : (a) Gray code (c) Error correcting code
(b) Error deducting 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 (c) 010101002 1-a 11-d
2-b 12-a
3-d 13-d
4-a
5-c
(b) 101010112 (d) 010101012 6-a
7-c
8-a
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9-b
10-c
38
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
Ans.: (a)
(b) Intersection
(c) Difference
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
A.or.x B}
Example : If A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A B = {1, 2, 3, 4, 5, 6} U
A
B
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
A.and .x B}
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Discrete Mathematics
39
Example : If A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A B = { 3, 4}
U
A
B
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
Similarly,
A.and .x B}
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}
U
A
U
B
A
A–B Q.3
B B-A
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 For free study notes log on: www.gurukpo.com
A
B
40
Thus, A B ( A B) ( B A) = {x : x
A B}
A = {2, 3, 4} and
B = {3, 4, 5, 6}
A – B = {2}
B – A = {5, 6}
and
A B ( A B) ( B A) {2} {5,6} {2,5,6}
A B Q.4
State De Margan‟s Law.
Ans.: If A and B are any two sets then (i) Q.5
( A B)'
A ' B '
and
(ii)
( A B)'
A ' B '
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 – n( A B) n( A B ') n( A B) n( A ' B)
(1)
Again A ( A B ') ( A B) By sum rule – n( A B ') n( A B)
(2)
n( B) n( A B) n( A ' B)
(3)
n( A)
Similarly Now eq^(2) + eq^(3) gives n( A) n( B) n( A B) n( A B) n( A B ') n( A ' B)
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Discrete Mathematics
=>
41
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.
□□□
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42
Chapter-6
Relations Q.1
Prove that – AxB
Ans.: A = {a, b}
BxA 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 Q.2
BxA
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,
1
2
and )
2
S such that ( 1 ,
R
(
1
2
2
1
2
,
1
)
2
R , then
)
R
R is symmetric on S. (iii)
Transitivity : Let
1
,
2
,
3
S such that ( 1 ,
2
)
R and (
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2
,
3
)
R , then
Discrete Mathematics
43
( 1,
2
)
R
(
3
)
R
2
,
Since
1
2
1
2 2
3
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) (a, b) R (c, d)
N x N be such that (a, b) R (c, d), then
ad
bc
cb da (by commutativty of multiplication on N) (c, d) R (a, b) Thus, (a, b) R (c, d) N
(c, d) R a, b) for all (a, b), (c, d)
Nx
So, R is symmetric on N x N. (iii)
Transitivity: Let (a, b), (c, d), (e, f) (c, d) R (e, f), then (a, b) R (c, d) and (c, d) R (e, f)
ad
N x N be such that (a, b) R (c, d) and
bc cf
de
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44
(ad )(cf ) (bc)(de) af
be
(a, b) R (e, f) Thus (a, b) R (c, d) and (c, d) R (e, f) b), (c, d), (e, f) N x N
(a, b) R (e, f) for all (a,
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=0xm 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 a
z such that
b (mod m) a–b=
a – b is divisible by m
m for all λ
b – a = (- )m
z
[ 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 then a
z such that a
b (mod m) a–b=
1
b (mod m) and b
a – b is divisible by m
m for some
1
z
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c (mod m),
Discrete Mathematics
45
b
c (mod m) b–c=
2
b – c is divisible by m
m for some
(a – b) + (b – c) = (a – c) = a
3
m
1
{
z
2
m+
1
2
+
2
m z}
(
3
=
1
+
2
)
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 A1 A2 A3 = {1, 2, 3, 4, 5, 7}
A
So sets A1 , A2 , A3 do not form a partition of A. (ii)
Since B1 B2 B3 = {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) (a, a) Thus (a, a) (ii)
S
[ R & S are reflexive]
R S
R S for all a A, so R S is a reflexive relation on A.
Symmetry : Let a, b (a, b)
A such that (a, b) R S
(a, b)
R S
R and (a, b)
S
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46
(b, a)
R and (b, a)
S
[ R & S are symmetric] (b, a) Thus (a, b)
R S
R S (b, a)
R S
R S is symmetric relation. (iii) Transitivity: Let a, b, c (a, b)
A such that (a, b)
R S and (b, c)
(a, b)
R and (a, b)
R S and (b, c)
R S 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)
(a, c)
R S
Thus (a, b)
S
R S and (b, c)
R S
(a, c)
So, R S is transitive Hence R S is an equivalence relation on A.
Q.7
R S
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
3
1
2
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R S
Discrete Mathematics
Q.8
47
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
(ii)
R2 = {(1, 5), (2, 4), (3, 6)} Clearly R2
(iii)
A x B. So, it is a relation from A to B. A x B. So, it is a relation from A to B.
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.
□□□
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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)
Q.2
If a, b (a)
Ans.: (a)
B
{1, 2, 3, 4} then which of the following are functions. f1 = {x, y : y = x + 1}
(b)
f2 = {(x, y) : x + y > 4}
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.
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Discrete Mathematics
Q.3
49
Prove that function f : Q→Q given by f(x) = 2x - 3
Ans.: (i)
x Q is a bijection.
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 x
y 3 2
Clearly for all y There exist x
2x – 3 = y
Q,
x
y 3 . Thus for all y 2
Q (domain) given
f(x) = f(
x
Q (Codomain)
y 3 such that 2
y 3 y 3 ) = 2( )–3=y 2 2
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
Ans.:
fog
(gof)(x)
Now (fog)(x)
=
g{f(x)}
=
g(x2)
=
2(x2) + 1
=
f{g(x)}
=
f(2x + 1)
=
(2x + 1)2
=
2x2 + 1
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50
Since 2x2 + 1 So, Q.5
(2x + 1)2 gof
fog
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,
xA (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}
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57
(d) none of the above 23. In following diagram shaded portion denote the set
A
B
(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 (d) none of the above 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
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58
□□□
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Discrete Mathematics
59
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.:
Q.3
q
p
T
F
F
T
T
F
F
T
F
F
T
F
T
T
F
T
T
T
F
F
T
T
T
T
T
p
q
T
T
T
p
q
q
p
(p
q)
( q
Using the truth table show that p
( p q) ( q
q
p)
Ans.: I p
II p
III q
IV q
V p→q
VI p q
VII q p
T
F
T
F
T
T
T
VIII ( p q) ( q
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T
p)
p)
60
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
T T F F
T F T F
T F F F
( p q)
p
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
IV
V
VI
VII
q
III p q
p
( p q)
p
q
( p ) ( q)
T
T
T
F
F
F
F
T
F
F
T
F
T
T
F
T
F
T
T
F
T
F
F
F
T
T
T
T
Column (IV) and (VII) are identical. (ii)
( p q) ( p) ( q)
I
II
p
q
III p q
IV
V
VI
VII
( p q)
p
q
( p ) ( q)
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61
T
T
T
F
F
F
F
T
F
T
F
F
T
F
F
T
T
F
T
F
F
F
F
F
T
T
T
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
(ii)
A,
x
A for which p(x) is true.
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.
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62
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
(5)
t
u
(i)
p
q
s
[by (3)]
p q (ii)
[by (1)]
r
p
[Modulus Ponens] [by (2)]
q r
(iii)
r
s
[Modulus Tollens] [by (4)]
r s
[Disjunctive Syllogism]
Hence treasure is buried under the flagpole.
Multiple Choice Questions 1.
2.
is logically equivalent to : (a) (c) p q is logically equivalent to :
(b) (d) )
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Discrete Mathematics
(a) (c) ) 3.
4.
5.
23
63
(b) (d) )
The converse of p q is : (a) (c) ) p q means: (a) (c) ) The argument relates to if p then q
(b) (d) ) (b) (d) )
(a) modus ponens (b) (c) ) (d) ) 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
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64
(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
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18-d
14-b
Discrete Mathematics
65
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.
2.
3.
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
( )
The antecedent in the conditional statement p q is: (a) p (b) q (c) (d) p
( )
q
For any two complex numbers z1 and z 2 . which is correct? (a)
z1 z1
z1
z2
(b)
z1 z1
z1
z2
(c)
z1 z2
z1
z2
(d)
z1 z2
z1
z2
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( )
66
4.
5.
6.
The decimal form of (101) 2 is: (5) 2 (a) (3)10 (c)
(b) (d)
(4)10
Binary numeral of 109 is: (1100111) 2 (a) (1101101) 2 (c)
(b) (d)
(1110011) 2
(6)10
( )
(1011011) 2
( )
(10000) 2 (1011) 2 is equal to:
(a) (b) (c) (d)
(1011) 2 (101) 2 (1110) 2 (111) 2
( )
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 (c) 010011
9.
10.
11.
If p (a) (c)
(b) (d)
0100110 0100111
q is an implication, then its contra positive is: (b) p q q (d) q p p
Which quantifier is used in (a) Universal (c) Propositional
p q
( )
( )
x, P ( x ) :
(b) (d)
Existential None of the above
Which is true? ( n N )(n 7 4) (a) ( n N )(n 2 6) (b) ( n N )(n 7 4) (c)
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( )
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(d) 12.
67
( )
( n N )(n 2 6)
The law of syllogism is: p q (a)
p _____
p (b)
q
_____ q
q (c)
13.
p
(d)
p
q
q
q
r
p
q
_____
_____
p
q
r
P (Q R) ( P Q) ( P R) is known as: (a) Associative Law (b) (c) Detachment Law (d)
Idempotent Law 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= range (a) (c)
17.
If A is a subset of a universal set U, then A U equals to: (a) A (b) U1 (c) (d) A
18.
{2, 3}, B= {3, 4, 5} and R is a relation such that R= {(2, 3), (2, 4), (3, 5)}, then of R 1 is {2, 3, 4} (b) {2, 3} {3, 4, 5} (d) {2, 3, 5} ( )
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
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( )
68
(b) (c) (d) 19.
20.
21.
22.
23.
24.
25.
26.
reflexive, antisymmetric and transitive reflexive, symmetric and antisymmetric Only antisymmetric
If f : N (a) 2 (c) 4
N and g : N
N are defined as f ( x) (b) (d)
( ) x 2 and g ( x) 3 5
x 1 , then gof (2) is:
z is defined as f ( x) x 2 1 for all x z , where z A function f : z positive integers, then the function f is: (a) one-one onto (b) many one onto (c) many one into (d) one-one into
A relation R: A (a) AxB (b) BXA (c) A B (d) A B
( ) is a set of
( )
B is a subset of:
( )
If aRb aRc aRc , then the relation R is: (a) Reflexive (b) (c) (d) Transitive
Symmetric antisymmetric
( )
EBCDIC code expresses any character in low many binary numbers? (a) 4 (b) 6 (c) 8 (d) 16
( )
The gray code of the binary number 1101 is: (a) 1100 b) (c) 1001 (d)
( )
1011 1110
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
p (a) (c)
q is equivalent to: p q p q
(b) (d)
p q p q
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( )
( )
Discrete Mathematics
69
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) (b) 2m n 2m (c) (d) ( ) 2n 2mn
28.
A function is said to be a bijection if it is: (a) one-one (c) many one onto
29.
30.
31.
32.
33.
onto both one-one and onto
( )
The number of vertices in a graph is called (a) order of graph (b) (c) degree of graph (d)
size of graph length of graph
( )
This size of 5 regular graph with 20 vertices is: (a) 30 (b) (c) 50 (d)
40 100
( )
( 3) is a/an: (a) real number (c) complex number
(b) (d)
rational number irrational number
( )
Which is true of the following? (a) The trial is an open walk (c) The circuit is a closed walk
(b) (d)
The path ia a trial All of the above
( )
odd vertices even vertices
( )
A complete graph with n vertices has (a) (c)
34.
35.
odd edges even edges
(b) (d)
n(n 1) : 2 (b) (d)
A complete graph k n is plannar graph if and only if: (a) n>8 (b) n< 5 (c) n=6 (d) n>5
( )
A connected plannar grapg eith n vertices, e edges and r regions gives: (a) (b) 3r 2e e 3n (c) (d) All of the above 3n e 6
( )
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70
36.
37.
How many pendent vertices are there in a tree? (a) Two (b) (c) Three (d) Hamiltonian path ia a: (a) Euler circuit (c) Spanning tree ( )
(b) (d)
Atleast two Atleast three
( )
Rooted tree 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) (b) ( A B) ( A B) (c) (d) ( A B) ( B A) ( )
40.
An argument is valid means it has valid: (a) Hypothesis (c) Assumption
b) (d)
( A B) ( A B) ( A B) ( B A)
Logic 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. ( )
______
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71
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
Q.2
(a)
Convert ( FO)16 into octal form.
(b) (c)
Find the hexadecimal equivalent of (32.4)8 .31/ 2 . Evaluate : (11) 2 (111)2 (1111)2 (11111) 2
(a)
Check the validity of the following argument: r
p
q
q
r
_________ p
Q.3
Q.4
Q.5
(b)
Prove that: ( p q)
(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?
(a)
If R and S are equivalent relations on a set A, then prove that R equlvalence relation.
(b)
Show that the function f : R bijection
(a)
Find the shortest path between the vertices a and z in the following graph:
r
(p
r ) (q
r)
S is also an
R defined by the f ( x) 3x3 7 for all x
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R is a
72 5
b
d
5
f 7
4 2
a
3
1
4
3
5 c
(b)
z
2
6
e
g
Are the following graphs isomorphic ? Explain your answer: V5
V2
V1
V1
V3
Q.6
(a)
V4
V2
V4
V3
Find the minimal spanning tree of the following graph: V1 11
1
V2
6
9
2
8
7
3
10
V3 5
V5
4 V4
(b)
Prove that a tree with n vertices has n-1 edges.
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73
Discrete mathematics Paper-114 OBJECTIVE PART- I Year - 2010 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.
2.
3.
The largest four digit number in hexadecimal system is: (a) 7777 (b) 1111 (c) FFFF (d) 9999
( )
The value of the number 11.0011 E-112 is: (a) 3.1875 (b) (c) –3.1875 (d)
.398437 –0.398437
( )
(A (a) (c)
A A'
( )
B)' is equal to: A' B' A B
(b) (d)
B 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) (c) 2 (d)
6.
1 None of the above
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
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( )
( )
74
7.
8.
9.
10.
11.
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
( )
(p (a) (c)
( )
q) is logically equivalent to: p q p q
(b) (d)
p (q
(g o f)–1 is equal to: (a) (f o g)–1 (c) g–1 o f1
(b) (d)
f1 o g1 fog
If A = (8n – 7n–1/n (a) A B (c) A=B
N) and B = (49 (n–1)/n N then: (b) B A (d) None of the above
q p)
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 n (c) m (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
( )
Degree of isolated vertex is: (a) 0 (c) Infinite
( )
14.
15.
(b) (d)
Number of edges in a complete graph Kn is: (a) n (b)
1 None of the above
2n
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Discrete Mathematics
(c)
75
(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
( )
Conversion the binary number 1011100012 into the octal number system is: (a) 6518 (b) 7618 (c) .2358 (d) 561g
( )
10101012 + 111001012 is equal to: (a) (01001000)2 (c) (11010111)2
( )
18.
19.
20.
21.
22.
23.
(b) (d)
The composite mapping f o g of the maps f : R x2, is : (a) x2 sin x (b) 2 (c) sin x (d)
(100111010)2 (00010111)2 R, f(x) = sin x and g : R
R, g(x) =
(sin x)2 ( )
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}
( )
If A = {1,2,3,4}, B = {3,4,5,6}, then A–B =: (a) {1,2} (b) (c) (d)
{5,6} {1,2,3,4,5,6}
( )
If A = {1,2,3,4} B = {2,3,6,7} C = {2,5,8} then A (a) {2} (b) (c) {2,3,6,7} (d)
(B C) =: {1,2,3,4} {2,5,8}
( )
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76
24.
Shaded portion of the following Venn diagram denotes: A
(a) (c) 25.
26.
27.
28.
29.
30.
A B A–B
B
(b) (d)
A B B–A
( )
(b) (d)
Symmetric Antisymmetric
( )
The void relation on a set A is: (a) reflexive (c) reflexive and symmetric
(b) (d)
symmetric and transitive Reflexive and transitive
( )
If R1 = {(x,y)/ y,x,y R}, then R1 is: (a) Partial order relation (c) both
(b) (d)
equivalence relation none of the above
( )
If R1 {(x,y)/x2 +y2 = 1, x,y (a) Reflexive (c) Transitive
R}, then R1 is :
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
( )
Completes bipartite graph K3,3 is: (a) Planner graph (c) both
( )
(b) (d)
non-planer graph None of the above
This graph is: V3
V2 e2
e5
e1
e6 e3
V1
e4
V4
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(a) (b) (c) (d) 31.
32.
33.
34.
35.
36.
37.
38.
39.
77
Eulerian Eulerian and Hamiltonian Eulerian but not Hamiltonian Hamiltonian but not Eulerian
( )
If f(x) = loge x, the f(xy) is : (a) f(x) + f(y) (c) f (x) – f(y)
(b) (d)
f (x) f(y) f(x) / f (y)
( )
is a : (a) Natural Number (c) Irrational Number
(b) (d)
Rational Number Imaginary Number
( )
The argument is valid if: (a) p q is tautology (c) p q is fallacy
(b) (d)
q is false None of the above
( )
Number of vertex in a wheel Wn is: (a) n (c) 2n
(b) (d)
(n +1) n2
( )
A tree with n vertices has exactly…………edges. (a) n (b) (c) 2n (d)
n +1 n –1
( )
For n N, 32n + 7 is divisible by: (a) 16 (c) 8
(b) (d)
9 6
( )
The negation of a conditional statement p p q (a) q p (c)
q is equal to : (b) q p (d) p^ p
( )
Number of maximum leaves in an m-ary tree of height h is: (a) hm (b) hm–1 (c) mh (d) mh–1
( )
How many edges are there in a graph with ten vertices each of degree 6? (a) 30 (b) 60 (c) 20 (d) 10
( )
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78
40.
Which of the following is true ? (a) domain range (c) range codomain
(b) (d)
codomain codomain
domain 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)
______
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Discrete Mathematics
79
DESCRIPTIVE PART- II Year- 2010 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, 561, into hexadecimal number system.
(b)
Write the full form of BCD and EBCDIC. Why was BCD code extended to EBCDIC?
(c) Q.2
Q.3
Q.4
Q.5
Write the bit pattern for the word CISTEMS using the ASCII-7 coding scheme.
Test the validity of the argument p p^q rvs q s r (b)
Prove that P q = (p ^ q ) v ( p ^ ˜q)
(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.
(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 (x).
(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.
R, f(x) = 2x + 5, then prove that f is one-one onto function. Also find f-1
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80
(b)
Find the shortest path between the vertices a and f, in the following weighted graph.
b
5
d 6
4 a
f
2
8 1
3
2 5 10
C
Q.6
e
(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
1 4
4
a
3
1
3
d 4
5 f
3
e
__________
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Discrete Mathematics
81
Discrete mathematics Paper-114 OBJECTIVE PART- I Year - 2009 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.
2.
3.
4.
5.
6.
(17)10 in binary system is: (a) (1001)2 (c) (10001)2
(b) (d)
(100001)2 (01001)2
( )
0.5452E3 – 0.5424E3 is: (a) 0.0024E3 (c) 0.002E3
(b) (d)
0.0028E3 0.003E3
( )
1 None of these
( )
(A (A
( )
If A = { , { (a) 0 (c) 2 (A (a) (c)
}}, then cardinal number of A is: (b) (d)
B)' is equal to: (A' B') (A' B')
(b) (d)
B) B')
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 (g o f)–1 is equal to: (a) (f o g)–1 (b) g–1 o f-1
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( )
82
(c) (d) 7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
f–1 o g–1 fog
( )
Degree of pendant vertex is: (a) 0 (c) infinite
(b) (d)
1 none of the above
( )
The number of vertices of odd degrees in a graph G is always. (a) even (b) odd (c) zero (d) rational number
( )
Number of edges in a graph with 7 vertices each of degree 4 will be: (a) 28 (b) 14 (c) 7 (d) 4
( )
A tree with n vertices has exactly……………..edges. (a) n (b) n –1 (c) n +1 (d) n –2
( )
(25.6)8 can be represented as: (a) (21)10 (c) (21.75)10
( )
(b) (d)
(12.75)10 (168)10
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
( )
(00010111)2 + (00010001)2: (a) (01001000)2 (c) (11010111)2
(b) (d)
(00101000)2 (00010111)2
( )
Hamming code is: (a) error – finding code (c) error – correcting code
(b) (d)
error – detecting code error – approximation code
( )
(b) (d)
q q
( )
Contra positive statement of p q (a) p (c) q p {a,b}
q is :
p p
{{a,b} } =
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Discrete Mathematics
(a) (c) 17.
18.
19.
20.
21.
83
{a,b} {a,b, {a,b}}
(b) (d)
A connected graph is a tree if: (a) e=v (c) e = v–1
1
(b) (d)
Infinite
( )
e = v +1 none of the above
( )
Conversion of the decimal number (58)10 into hexadecimal is: (a) (3B)16 (b) (2A)16 (c) (3A)16 (d) (2B)16
( )
If A = {a,c,d}, B = {c,d,e}, then A – B = (a) {a, e} (c) {a}
( )
(b) (d)
{e} {c,d}
Shaded portion of the following venn diagram denotes:
(a) (c)
23.
( )
Incoming degree of the root of the rooted tree is: (a) 0 (b) (c) 2 (d)
A
22.
{{a,b}}
A Ac
B
B
U
(b) (d)
A B A–B
( )
If aRb ^ bRc aRc, then relation R is: (a) Reflexive (c) Transitive
(b) (d)
Symmetric Antisymmetric
( )
If f (x) = eax, the f (x +y) is : (a) f (x) + f(y) (c) af (x) f (y)
(b) (d)
f (x) f (y) f (x)/f (y)
( )
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84
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 (d) p ( ) p q
25. (a) (c) 26.
27.
28.
29.
30.
31.
natural number irrational number
(b) (d)
rational number imaginary number
( )
Number of different edges in the complete graph Kn is: (a) n (b) n (n–1) (c)
(d)
( )
P = 2 > 3, q = Earth does not rotate, then: (a) p is true (c) p q is true
(b) (d)
q is true p ^ q is false
( )
The argument is valid if: (a) P q is fallacy (c) q is false
(b) (d)
p q is tautology none of the above
( )
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 Degree of each vertex of complete graph k4 is: (a) 2 (b) (c) 3 (d)
( )
4 5
This graph is:
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Discrete Mathematics
85 a
e1
e2 e3
b
e4
c
c5
d
(a) (b) (c) (d) 32.
33.
34.
35.
36.
37.
38.
Eulerian Eulerian and Hamiltonian Eulerian but not Hamiltonian Hamiltonian but not Eulerian
( )
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
( )
A connected graph that contains no cycle is: (a) simple graphs (b) (c) tree (d)
planar graph circuit
( )
Which of the following is true? (a) domain range (c) co-domain domain
(b) (d)
range co-domain co-domain range
( )
A function is called bijection if it is: (a) only one-one (c) one-one and onto
(b) (d)
only onto None of the above
( )
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
( )
The universal quantifier of P (x) is denoted by : (a) x P (x) (b) (c) x P (x) (d)
x x
( )
A relation R: A (a) A B (c) AxB
BxA A B
P (x) P (x)
B is subset of : (b) (c)
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( )
86
39.
40.
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
If F: R
B}
( )
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)
__________
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Discrete Mathematics
87
DESCRIPTIVE PART- II Year- 2009 Time allowed: 2 Hours Maximum Marks : 30 Attempt any four questions out of the six. All questions Carry 7½ marks each. Q.1
Q.2
(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.
(a)
Prove that: (p q) ^ (P
(b)
r)
p
(q ^ r)
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
Q.4
Q.5
(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)
(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 g (x) Find f o g and g o f.
(a)
Prove that a connected graph G is Eulerian if and only if the degree of every vertex is even.
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R is defined by :
88
(b)
Find the shortest path between the vertices V1 and V8 in the following weighted graph: V2
2
V3
4 2
3
2
1
V1
V8 V4 4 3
V5 7
1 V6
Q.6
5
6
V7
(a)
Prove that a tree with n vertices has exactly (n–1) edges.
(b)
Write a short note on minimal spanning tree.
_________
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Discrete Mathematics
89
Discrete mathematics Paper-114 OBJECTIVE PART- I Year - 2008 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.
2.
The number 2.444 ……….is called a or an: (a) Natural number (b) (c) Rational number (d)
4.
n +2
( )
( )
The decimal equivalent of the binary number (101111)2 is:
(74)10 (46)10
(b) (d)
(64)10 (47)10
( )
Conversion of the decimal number (47)10 into hexadecimal is:
(a) (c) 6.
(d)
The range of unsigned integers stored in N bits computer is from zero to: (a) 2N (b) 2N–1 N+1 (c) 2 (d) None of these (a) (c)
5.
( )
If n is composite integer, then n has prime divisor less than or equal to: (a) (b) n2 (c)
3.
Integer Irrational number
(67)16 (2 F)16
(b) (d)
The binary number of a octal number (7)8 is : (a) (110)2 (b) (c) (1101)2 (d)
(3 E)16 (2 A)16
( )
(101)2 (111)2
( )
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90
7.
8.
9.
10.
11.
12.
13.
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}
( )
The difference of {1,3,5} and {1,2,3} is the set: (a) {1} (b) (c) {3} (d)
{2} {5}
( )
What is the power set of the empty set : (a)
(b)
{ }
(c)
(d)
None of these
( )
Equal elements Different elements
( )
{ , { }}
Two sets are equal if any only if they have: (a) Same elements (b) (c) Unequal elements (d)
The solution of the linear congruence 3x = 4 (mod 7) is: (a) 2 (b) 4 (c) 6 (d) 8
( )
The identity law is: (a) A =A (c) A =
( )
(b) (d)
The universal quantifier of P (x) is denoted by : (a) x P (x) (b) (c) x P (x) (d)
A U=U All of the above
x
x P (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:
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Discrete Mathematics
91 A
B
U
C
(a) (c) 16.
A A
B B
C C
(b) (d)
A A
B B
C C
Let A and B be two non empty sets, then a binary relation R: A (a) B (b) BxA A (c) AxB (d) B A
( ) B is a subset of: ( )
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) (c) Antisymmetric (d)
19.
20.
21.
Symmetric Transitive
( )
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
( )
If f (x) = log, x then f (x) is: (a) f (x) + f (y) (c) F (x) . f (y)
(b) (d)
f (x) – F (y) f (x) / f (y)
( )
If f : N Z, f (x) = x2, then f is called: (a) many one function (c) onto function
(b) (d)
one-one function one one onto function
( )
Z given by f (x) = x2 –2x –3, then range of f is: (b) {–4, –3, 5, 8} (d) {–4, –3, 0, 5 ,6} ( )
22.
If A = {–2, –1, 0, 1, 2} and f : A (a) {5, 6, –3, –4} (c) {–4, –3, 0, 5}
23.
For the principle of mathematical induction the statement should include n as:
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92
(a) (c) 24.
If f: R
a natural number any rational number
(b) (d)
an integer an irrational number
( )
R be defined by f (x) = 2x –3, then f–1 (x) is:
(a)
(b)
(c) 25.
26.
27.
28.
29.
30.
31.
x+2 (d) ( ) 3 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 ( ) Negation of the statement (a) x y P (x,y) (c) x y P (x, y)
x
y, P (x, y) is: (b) (d)
The bit strings of the set {1,2,4,6,9} is: (a) 1110101010 (c) 1101010010
(b) (d)
x, y P (x and y) x y P (x, y)
1100011010 1001011001
( )
( )
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
( )
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
( )
A vertex of degree zero is called: (a) Pendent vertex (c) Point to point vertex
(b) (d)
Isolated vertex Simple vertex
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
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( )
( )
Discrete Mathematics
32.
Maximum number of edges in a simple graph with n vertices is : (a) n (n +1) (b) n (n –1) (c)
33.
34.
93
(d)
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
( )
( )
For the following two graphs, we have:
(a) (b) (c) (d)
Isomorphic graphs Planar graphs Isomorphic and planer graphs 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
( )
A graph containing no simple circuit but not connected is called a: (a) (b) forest tree (c) (d) 12 binary tree
( )
37.
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94
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: A
C
C
B
B
C
B
A
A
D
T1
D
D
D
D T3
(a) (c) 40.
T1 T3
A graph G is a tree if and only if it is: (a) Minimally connected (c) Not connected`
(b) (d)
T2 All T1, T2 and T3
( )
(b) (d)
Maximally connected 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)
_________
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Discrete Mathematics
95
DESCRIPTIVE PART- II Year- 2008 Time allowed: 2 Hours Maximum Marks : 30 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.
Q.2
Q.3
Q.4
( v)
Convert the decimal number 28.4 into an equivalent number in octal system.
(a)
Let S be the set of all number and R is a relation in S defined by ' a 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.
(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?
(a)
Prove that p
q uis a valid argument.
(b)
If U = {x : x
8, x
N} A = {x : x
the bit string for the set, A, B, A Q.5
6, x
B, A
b'. Prove that
N} and B = {3,4,5,7,8} then find B, A, B and A
B.
(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:
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96 V4
e4
V3 e7
e5
e3 b
e6
e2 V5
1
e1
V1
Q.6
a
V2
Find the minimal spanning tree of the following labeled connected graph: V1 5 V5
V2 9
7
4
1 6
V3
2 V2
8
3 V1
.
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Discrete Mathematics
97
Discrete mathematics Paper-114 OBJECTIVE PART- I Year - 2007 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. (a) (c) 2.
3.
4.
A bit has how many possible values? (a) (c)
a real number none of the above
( )
(b) (d)
Two Eight
( )
( )
What is the decimal expansion of the (2 AEOB) 16?
(175629)10 (175827)10
(b) (d)
(175627)10 (175829)10
( )
(b) (d)
(30071)8 (3007)8
( )
(b)
neither true nor false
Base 8 expansion of (12345)10 is given by:
(a) (c) 6.
(b) (d)
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
(a) (c) 5.
a rational number an irrational number
(3071)8 (2071)8
A proposition is a statement that is : (a) either true or false
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98
(c) 7.
8.
9.
11.
12.
13.
14.
15.
(d)
Let p and q be propositions. In the implication P (a) Hypothesis (b) (c) Consequence (d) The contra positive of p (a) p q (c) p q
q is the proposition: (b) (d)
false
( )
q, p is called the: Conclusion None of the above
( )
q q
p p`
( )
neither true nor false
( )
a variable is called a Boolean variable if its value is: (a) (b) true false (c)
10.
true
either true or false
(d)
Bitwise XOR of the bit string 0110110110 and 1100011101: (a) 1010111010 (b) 1010101011 (c) 1110101011 (d) 11101010
( )
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
( )
Which is de Morgan's law? (a) p q p q (c) (p q) p q
(b) (d)
(p (p
q) q)
q
q p
Let Q (x,y) denote "x + y =0". What is the truth value of the quantification (a) (b) true false (c) neither true nor false (d) none of the above If n is a positive integer then n3 –n is divisible by: (a) 2 (b) (c) 4 (d) The statement n P (n) is true if P (1) is true and as known as: (a) Principle of mathematical induction (b) Second principle of mathematical induction (c) Rule of inference
3 5
q
( ) y
x (x, y)? ( )
( ) n [p (n)
p (n + 1)] is true, known
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Discrete Mathematics
(d) 16.
99
( )
None of the above
The set {x/x A ^ x (a) A B (c) A–B
B} tells us as: (b) (d)
A
B ( )
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 (a) A (B C ) (c) (C B) A
19.
20.
21.
22.
23.
24.
If f (x) f (y) whenever x (a) One–one (c) One-one onto
(B
C) is equivalent to: (b) (C B ) (d) (C B)
A B
y then the function f is called: (b) on to (d) one–one into
( )
( )
The function f (x) = x2 is: (a) one–one (c) one–one onto
(b) (d)
on to neither one one nor onto
( )
If f(x) = ex then f (x–y) is: (a) f (x) – f (y) (c) f (x) . f (y)
(b) (d)
f (x) +f (y) f (x)/f (y)
( )
A relation "=" (equipment to) defined in any set A is: (a) Reflexive relation (b) Symmetric relation (c) Transitive relation (d) Equivalence relation
( )
A partial order relation is not a: (a) Symmetric relation (c) Transitive relation
( )
If an function f : R F (x) = 1 –1
x x
Q Q
(b) (d)
Reflective relation Transitive relation
R be defined as follows:
then f (
) is :
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100
(a) (c) 25.
26.
27.
28.
29.
30.
31.
32.
–1 none of the above
1 1 and –1
(b) (d)
If f : A B and g : B (a) gof : A B (c) gof: B B
C such that A = C, then which is true: (b) gof : A A (d) gof: A C
( )
( )
What is cardinality of the set {x x is a subset of the set {a,b}} : (a) Two (b) Three (c) Four (d) Infinite
( )
A compound proposition that is sometimes true and sometimes false is called: (a) Contradiction (b) Tautology (c) Contingency (d) Logical equivalence
( )
The bit string of the set {1,3,5,7,9}is : (a) 1111100000 (c) 1001001011
( )
(b) (d)
1011001101 1010101010
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|
( )
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)
( )
How many edges are there in a graph with ten vertices each of degree six? (a) 10 (b) 20 (c) 30 (d) 16
( )
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
( )
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Discrete Mathematics
101
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
a
b
c
d
c
(a) a
e
d
(b) b a
e
b
c d
c
d
(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 (c) point vertex
37.
(b) (d)
Pendant vertex Simple vertex
A connected undirected graph with no simple circuits is known as a: (a) (b) Directed Graph Planar graph (c) (d) Tree Weighted graph
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( )
( )
102
38.
39.
40.
Every non-=trivial tree has at least two vertices of degree: (a) one (b) two (c) three (c) zero
( )
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
( )
A visiting of the vertices of a tree is called : (a) decision tree (b) (c) tree traversal (d)
( )
ordered tree 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)
__________
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Discrete Mathematics
103
DESCRIPTIVE PART- II Year- 2007 Time allowed: 2 Hours Maximum Marks : 30 Attempt any four questions out of the six. All questions carry 7½ marks each.
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
Q.3
(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 equivalent.
(a)
Use set builder notation and logical equivalences to show that A
(b)
If f : R
(q ^ r) and (p q ) ^ ( P r) are logically
R be given by f (x) = x2 +2 and g : R
B=A
B.
R be given by g (x) = 1–
find (gof) x and (fog)x, R being the set of real numbers. Q.4
(a)
(b)
Use mathematical induction to show that: 1+2 +22 +………+2n = 2 n+1 –1for all non – negotiable integers n. How many paths of length four are from a to d in the following simple graph:
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104
Q.5
a
b
d
c
(a)
If G is a connected simple graph with e edges and v vertices where v then prove that e 3v, –6.
(b)
Determine whether the following graphs G and H are isomorphic.
µ1
µ3
µ1
µ2
µ2
µ5
µ6
µ6 µ3
µ4 Q.6
µ5
µ4
(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
c
d
e
f
h
I
j
m
k
I
g
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3v,
Discrete Mathematics
105
Discrete mathematics Paper-114 OBJECTIVE PART- I Year - 2006 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.
is: (a) (c)
2.
3.
a rational number an integer
(b) (d)
a real number none of the above
( )
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
( )
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
( )
(11011)2 + (10011)2 is equal to: (a) (45)10 (c) (48)10
( )
6.
(b) (d)
(46)10 (49)10
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106
7.
Shaded portion following Venn diagram denotes: A
B µ
(a) (c) 8.
9.
A B A–B
(b) (d)
If A is subset of a universal set U, then A (a) A (c) If a, b (a) (b) (c) (d)
A B B–A
, then A U equals: (b) U (d) None of the above
N such that a is a divisor of b then the relation "is a divisor of" is: Reflexive, symmetric, transitive Reflexive, anti-symmetric, transitive Symmetric, anti-symmetric, transitive 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) (d) not transitive an equivalent relation
13.
For two distinct non-void sets A and B, (A (a) Ac Bc (c) (A B)c
)c is equal to : (b) (Ac Bc) (d) A–B
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( )
( )
Discrete Mathematics
14.
15.
107
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
( )
Which of the following diagrams defines a function from A = {a,b,c} into B = {x, y, z}? A
B
A
B
a
x
a
x
b
y
b
y
c
z
c
z
(i)
(ii)
A
B
A
B
a
x
a
x
b
y
b
y
c
z
c
z
(iii)
(a) (c)
(i) (iii)
(iv)
(b) (d)
(ii) (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) pv q (c) p ^ q (d) p v q ( )
17.
p v (q ^ r) is equivalent to: (a) (p v q) v (q ^ r) (c) (p ^ q) v (q ^ r)
18.
19.
The statement p (a) p v q (c) p^q
(b) (d)
(p v q) ^ (p ^ r) (p ^ q) ^(q ^ r)
q is logically equivalent to the proposition: (b) pvq (d) p ^ q
( )
( )
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
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108
(c)
q
p
(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) (c) uncountable (d)
countable infinite 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
D
(a) (c) 25.
E
4 6
F
(b) (d)
5 7
Number of edges in a graph is : (a) equal to the sum of the degress of all vertices
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( )
Discrete Mathematics
(b) (c) (d) 26.
109
double of the sum of the degress of all vertices half of the sum of the degrees of all vertices having no relation with the sum of the degrees of all vertices
( )
Out degree of the vertex V4 in the following diagram is: V1
V1
V2
V1
(a) (c)
4
V1
(b) (d)
3 ( )
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
( )
The number of edges in a tree with its vertices is: (a) n (b) n +1 (c) n–1 (d) 1
( )
29.
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) (c) (A–B) C (d)
32.
(A–B) C none of the above
In the following weighted graph, the minimum distance between A and f is:
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( )
110 B
D
1
2
A
5
2
3
6
4 1
C
(a) (c)
33.
E
15 10
(b) (d)
( )
The following rooted tree represents:
+ x
34.
11 9
y
z
2 2
(a)
[{x–y}–3] –
(b)
(c)
(x –y)3
(d)
– (x –y)3 – (z2 +2)
( )
The following graph is: V1 e1 V2
e2 V6
e5 e3
e4 V3
(a) (c)
Hamiltonian Both Hamiltonian and Eulerian
(b) Eulerian (d) Neither hamiltonian nor Eulerian ( )
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Discrete Mathematics
35.
36.
37.
38.
39.
40.
111
A void set is: (a) Non– existing (c) A sub – set of all sets
(b) (d)
having zero as an element not a sub–set of any set
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) If f: X Y and g : Y (a) g–1 o f-1 (c) (g o f)–1
Z are bijections then (f o g)–1 equals: (b) f–1 o g–1 (d) gof
if q p, then : (a) p q q (c) p
(b) (d)
q q
( )
( )
( )
p p
( )
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
( )
Bijection is: (a) Only into function (c) a one-one onto function
(b) (d)
only onto function 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)
__________
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112
DESCRIPTIVE PART- II Year- 2006 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 (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 Z where f (x) = x2 + x for all x
Q.2
Let f: Z
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.
(a)
Construct truth tables for the following compound propositions. (p q (ii) (p q) ^ ( p q ) q)
Q.4
(c)
Z, then show that f is a many one function.
Find the shortest path between the vertices a and z in the following weighted graph. b
5
d
5
f 7
4 a
z
2
2 3
1 4
3 5 C
Q.5
(a)
6
e
g
Show that, if G is simple planar graph, then there is vertex V in G such that deg (V) 5.
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Discrete Mathematics
(b)
113
Show that in the following graph. Is not Hamiltonian while (ii) is Hamiltonian e1 V2
V1
V1 e1 e4
e2
e3
V3 V4
V5 e2 e4 V3
Q.6
e3
e5
e6
V2 V4
(a)
Show that a tree with a vertices has exactly (n–1) edges.
(b)
Construct a minimum spanning tree for the following weighted graph.
2
4 11 2 1
9
3
8 10
7
5 6
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114
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.
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Discrete Mathematics
115
Reference Books 1. 2. 3. 4. 5.
Discrete Mathematics and Its Applications by Kenneth Rosen Discrete Mathematics by Gary Chartrand and Ping Zhang Discrete Mathematics with Applications by Susanna S. Epp Discrete Mathematics by Kevin Ferland DiscreteMathematicsby Norman L. Biggs
Websites:1. 2. 3. 4. 5.
www.graphtheory.com www.purplemath.com www.mathwarehouse.com www.wikipedia.org www.geeksforgeeks.org
***********
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