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MA2265
-
DISCRETE MATHEMATICS

QUESTION BANK

Unit
-
I

LOGIC and PROOFS

PART
-
A

1.Using the truth table ,show that the proposition P
⋁⅂
(P⋀Q) is a tautology.

Ans :

P

Q

P

Q

P⋁⅂

T

T

T

F

T

T

F

F

T

T

F

T

F

T

T

F

F

F

T

T

Since⁡汬⁴he⁥ 瑩res in⁴he
resulting colu浮mis⁴rue Ⱐthe⁧iven⁰牯position⁩s⁡⁴au瑯logy.

2.Wha琠is⁮ gation⁯映瑨f⁳瑡瑥浥t琠
(∃x)(∀y)P(x,y).

Sol: given statement is (∃x)(∀y)P(x,y).

It’s negation is

[(
(∃x)(∀y)P(x,y)]

⇒(∃y)(∀x)⅂P(x,y).

3. Define Tautology With Example.

Ans: A stat
ement formula which is true always irrespective of the truth values of the individual
variables is called a tautology.

Ex: P⋁⅂P is a tautology.

4. Define

a rule of universal specification.

Ans: (x)A(x)⇨A(y), where A is any property ans x is a free
variableas.

5. Show

that the proposition p→q and ⅂p⋁q are logically equivalent.

Ans:†

p

q

p

p

q

P

q

T

T

F

T

T

T

F

F

F

F

From the truth table p→q,

P

q⁡re⁥ uivalen琮

wi瑨ou琠using⁴ru瑨 瑡ble⁳how⁴ha琠p

p
⇔⅂
p

䅮猺A

p

(q

p) reason

⇒⅂
p

(

q

p)

p

q
⇒⅂
p

q

((

p

p)
∨⅂
q

Assocaitive law

T
∨⅂
q

(

p
∨⅂
p

T)

T

p

T=T

And

p

(p

q)

P

(

p

q)

p

q
⇒⅂
p

q

⇒(
p
∨⅂
p)

q) (Assocative law)

T

q p
∨⅂
p

T

T

p

T

T

Therefore p

(q

p
⇔⅂
p

(p

q)

7. Show that (p

r))

((p

r)) ios⁡⁴ u瑯logy.

䅮A
: (p

(q

r))

((p

q)

(p

r))

((p

q)

(p

R))

((p

q)

(p

r))

⇒⅂
((p

q)

(p

r))

((p

q)

(p

r))

T

8.when
do you say that the two compound propositions are equivalent?

Ans
: Two statement formula A and B are equivalent iff A

B or A

B is a tautology . It is denoted by the
symbol A

B which is read as “A is equivalent to B”

To prove two statement formula A and B
are equivalent , we can use onle of the following method:

F

T

T

T

T

F

F

T

T

T

(i)

Using truth table, we show that truth values of both statements formulas A and B are same
for each 2
nd

combinations.

(ii)

Assume A, by applying various equivalence rules try to derive B and vice versa.

(iii)

P
rove A

B is a tautology.

9. Prove that p, p

qⱱ

r

䅮猺†
1) p

q rule P

2) q

r rule p

3) p

r Rule (P

q,q

r

p

r)

4) p Rule p

5) r Rule T (p,p

q

q)

10.Show that p

q⁡ d

p

q⁡牥⁬ogically⁥ uivalent.

䅮猺

††

P

Q

p

p

q

p

q

T

T

F

T

T

T

F

F

F

F

F

T

T

T

T

F

F

T

T

T

Hence p

q and

p

q are equivalent.

11. Define universal and
existential quantifiers.

Ans:
Quantifier is

one which is used to quantify the nature of variables.

There are 2 important quantifiers which are for “all” and for “some” where “some” means “
at least

one”.

The quantifier “ for all x” is called the universal
quantifiers. It is denoted by the symbol”(

x or (x))”

EXISTENTIAL QUANTIFIERS: the quantifier for “some x” is called the existential quantifier. It is denoted
by the symbol “(

x)”.

12.Give the converse and the
contra positive

of the implication “If it is r
aining, then I get wet”.

Ans
: let P: It is raining

Q: I get wet

Given statement is P

Q

It is converse is Q

P (ie) If I get wet then it is raining.

It is
contra positive

is

Q
→⅂
P (ie) If I do not get wet then it is not raining.

13.
symbolize the statement “ALL MEN ARE GAINTS”.

Ans
:
M(x) : X is a man

G(x) : x is a
gain
ts

Symbolic

form ; (

x)(M(x)

G(x))

14.Show

that(

M(砩x

H(s)

M(s)

䅮猺 1) ††(

x)(H(砩

M(砩 †† ††given⁰牥浩ses

†† †††
(2)† †䠨s)

M(s)
† †† †† †††

†† †††
(3)† ⁈ s) ††††† ††† † †† †Given⁰牥 ises

†† †††
(4)†⁍(s)† ††††† ††† ††  2) (3)⁍ dus⁰ nens

15.⁉s⁴he⁦ llowing
argumen琠valid?

I映瑡硥s⁡牥 lowered,⁴hen⁩nco浥mrises⁡ d⁩nco浥mrises.

Ta硥s⁡牥 lowered

The⁧iven⁦ r浵污 is ((P

p is⁮ 琠a⁴au瑯logyⰠ瑨e⁧iven⁡rgu浥m琠is⁮ 琠valid.

16.⁕ ing⁥ is瑥ttialⁱ anti晩ersⰠwri瑥⁴te⁥ u
ivalent⁦ r洠o映

x)(P(砩x

䅮猺†

p(砩x

17.⁗ i瑥⁴te
e硰ression in⁅nglish:†

n孱(n)

p(n)崠where⁰⡮)⁩s⁡ ⁥ en in瑥来爠andⁱ⡮)㵮⁩s
divisible⁢礠2 wi瑨in⁴he⁵niverse⁯映fll⁩n瑥ger.

䅮猺⁁ll in瑥来rs⁤ivisible⁢y′⁡牥⁥癥 in瑥来rs

18.⁗ i瑥⁴te rules⁯映in晥rence.

o映瑨f⁰牯ceeding⁦ r浵污s⁩n⁴he⁤ riva瑩on.

19. Express the statement “If the moon is out then it is not snowing then ram goes out for a walk” in
symbolic form.

Ans: P: the moon is out, Q: it is snowing , R: ram goes out for a walk.

Symbolic form: (P
→⅂

R

20.”Every parrot is ugly”
-

Expres
s using quantifiers.

Ans:
universe

of discourse: set of all birds

P(x): x is parrot, Q(x)
: x is ugly

Symbolic form: (x)(P(x)

21. Negate the statement: “ For all real numbers X, if X>4, then

”.

Ans:A(x): x>4 B(x):

given statement (x)(A(x)

B(x))

(x)(

A(x)

B(x))

Negation of statement : (

x)(A(x)
⋀⅂
B(x))

22.Write the minterms and maxterms of three variables p,q,r.

Ans: the minterms are P

q

r,⅂
P

q

r,,

P
⋀⅂
q

r,,

P

q
⋀⅂
r,,⅂

P
⋀⅂
q

r,

P
⋀⅂
q
⋀⅂
r,

P

q
⋀⅂
r,⅂

P
⋀⅂
q
⋀⅂
r,

The max terms: P⋁q⋁r,

P⋁q⋁r,

P⋁⅂q⋁r,

P⋁q⋁⅂r,

P⋁⅂q⋁r,

P⋁q⋁⅂r,

P⋁⅂q⋁⅂r,

P⋁⅂q⋁⅂r,.

23.Define free and bound variable,

Ans:When a quantifier is used an a variable or when a varable is assigned a value to get a
proposition , the occurrence of the variabl
e is said to be bound or the variable is said to be a bound
variable.

An occurrence of a variable that is not bound by a quantifier or that is set equal to a particular value
is called a free
variable
.

Example: (∃x)P(x,y)
-

x is a bound variable

Y is a free variable.

24. Give the indirect proof of the theorem “ If 3n+2 is odd then n is odd”.

Ans:⁁ssum攠e桡琠n⁩ 敶en⸠.桥n⁮=㉫2f潲⁳潭攠ent敧敲⁫.

㍮+㈽㌨㉫⤫)=㙫+㈽㈨㍫+ㄩ=

in瑥t敲

∴⅂q→⅂p is true . Hence p→q is true

∴If 3n+2 is odd then n i
s odd.

PART
-
B

1.

Prove that the
following

argument is valid :
p→⅂q,r→q,r⇨⅂p.

2.

Determine the validity of the following argument: if 7 is less than 4, then 7 is not a prime
number,7 is not less

than 4. Therefore 7 is a prime number.

3.

Verify the validity of the following argument .Every living thing is a plant or an animal.
John’s gold fish is alive and it is not a plant. All animals have hearts. Therefore john’s gold

Fish

has a

he
art.

4. Show

th
at
(∀x)(p(x)→q(x)),(∃y)p(y)⇨(∃x)q(x).

㔮⁓h潷o

(q⋀r)

and (p⋁q)⋀(p⋁r)

Sh潷

, “

It is not sunny this afternoon and it is colder than yesterday”.

We

will go swimming only if it is sunny”. “If we do not g
o swimming , then we will take a
canoe trip” and “ If we take a canoe trip , then we will be home by sunset “ lead to the
conclusion “ we will be home by sunset”.

7. Find

the PDNF of the statement
, (q⋁(p⋀r))⋀⅂((p⋁r)⋀q)

㠮⁕se

(∃zQ(z)

∃x(p(x)→Q(x)) and ∃y P(y).

㤮⁕sing

in摩r散琠m整桯搠潦⁰牯潦‬⁤敲ev攠
p→⅂s

p→(q⋁r), q→⅂p.s→⅂r

㄰⸠1牯r攠瑨慴e

is irrational by having a proof using contradiction.

11.Show
that

∀x(p(x)⋁q(x))⇨
(∀xp(x))⋁(∃xQ(x))
by⁩湤楲 c琠m整桯搠o映灲潯昮

ㄳ⸠1i瑨t畴⁵uing⁴牵瑨⁴rb汥⁦ln搠瑨攠偃NF⁡ 搠偄NF⁯映
P→(Q⋀P)⋀(⅂P→(⅂Q⋀⅂R))

⸠卨潷

that (P→Q)⋀(R→S), (Q⋀M)⋀(S→N), ⅂(M⋀N) and (P→R)⇨⅂P.

15
. Verify

that validity of the following inference.

If one person is more successful thatn
another, then he has worked harder tom deserve success. Ram has not worked harder than
Siva. Therefore, Ram is not more successful than Siva.

16. Show that the premises
a→(b→c),d→(b⋀⅂c) and (a⋀d)

⸠.牯re

∀x(P(x)→Q(x)), ∀x(R(x)→⅂Q(x))⇨∀x(R(x)→⅂P(x)).

⸠.i瑨t畴

P→(Q→R)≡Q→(P⋁R).

⸠卨潷

that the conclusion ∀xP(x)→⅂Q(x) follows from the
premises
(P(x)⋀Q(x))→∀y(R(y)→S(y)) and
∃y(R(y)⋀⅂S(y)).

20
. Prove

the following

equivalences by proving the equivalences of the dual

((⅂P⋀Q)⋁(⅂P⋀⅂Q))⋁(P⋀Q)≡P.

21
. If

there was rain, then travelling was difficult. If they had umberalla then travelling
was
not

difficult. They had umbrella. Therefore, there was no rai
n. Show that these statements
constitute a valid argument.

22
. Show

that R⋁S is a valid conclusion from the premises(A⋀⅂B)→R⋁S, (C⋁D)→⅂H,
⅂H→(A⋀⅂B),(C⋁D).

23
. Using

CP rule , show that (x)(P(x)→Q(x))⇒(x)P(x)→(x)Q(x).

24
. Obtain

PDNF and PCNF for (P⋀Q)⋁(⅂P

Q)⋁(Q⋀R) without using truth tables.

25
. Show

that S is a valid inference from the premises P→⅂Q,Q⋁R,⅂S→P , and ⅂R.

26
. Verify

the validity of the inference . if one person is more successful than other another,
then he has worked harder to deserve succes
s. John has not worked harder than peter.
Therefore,

john is not successful than peter.

27.Using conditional proof, prove that ⅂p⋁q, ⅂q⋁r,r→s⇒p→s.

28.Use indirect method of proof to show that (x)(p(x)⋁Q(x))⇒(x)p(x)⋁(∃x)q(x).

29.Prove that (∃x)(A(x)⋁B(x))⇔(
∃x)A(x)⋁(∃x)B(x).

Unit
-
II

COMBINATORICS

1.
State the principle of mathematical induction.

Ans: if P(n) is any mathematical statement defined on a positive integer n, then p(n) is true for
n∊Z+ if ( ) P( ) is true (2) P(K)⇒p(k+ ) is true

2. Find the recurrence relation for the Fibonacci sequence
.

Ans: Fibonacci sequence 0,1,1,2
,3,5,8, 3…. Sa
tisfies the recurrence relation

+

And also satisifies the initial conditions f0=0and f1=1.

3.State Pigeon hole principle.

Ans: if (n+ ) pigeon occupies ‘n’ holes than atleast one hole has more than pigeon.

4. Solve:

, for k

,

.

Ans:the characterisitics equation is

r
-
3=0

r 3 ⇒

5
.
Give an indirect proof of the theorem”if 3n+2 is odd, then n is odd”.

Ans:⁷攠e牯r睥w瑨ts⁰牯r汥l⁢y⁴桥 m整e潤⁯映c潮o牡灯ri瑩ve

Suppose n is even⇒n 2k

3n+2=3(2k)+2

3n+2=2(3k+1)

∴3n+2 is even number

⇒3n+2 is not a odd number

6.write the generating function for the sequence 1,

,

,

.

Ans: generating function G(x)=1+ax+

+

G(x)=

for
|

|

7.Use math
ematical induction to show that
n

,

,

.
.

Ans: let P(n) : n

2

,

5
,

.
.

P(5) : 5

2

,

.

P(k) : k

2

,

Claim: P(K+1) is true

Multiply both sides by 2, we have

2k

2

2

,

(k+1)k

2

(k+1)

2

∴p(K+ ) is true.

Hence, by the principle of mathematical induction

n

2

for n 5, …

8. If seven colours are used to paint 50 bicycles , then show that at least 8 bicycles will be the same
colour.

Ans:

No of pifeon=m=no.of bicycl
es=50

No.

of holes=n=no of colours=7

Be generalized pigeon hole principle , we have
[

]
+

8

Therefore atleast 8 bicycles will have the same colour.

9.Find the recurrence relation which satisifies

+

(

)

Ans:

3

+

(

4
)

------------------
(1)

3

+

(

4
)

3

3

4

(

4
)

------------
(2)

3

+

(

4
)

9

3

+

(

4
)

-----------------
(3)

And 3+2
-
12(1)

+

.

10.Define Generating function

.

Ans: the generating function
for the sequence ‘s’ with terms

,

,

of real numbers is the
infinite sum.

G(x)=G(s,x)=

+

,

+

+

=

11.How many ways are there to form a committee , if the committee consists of 3 educanalist and 4
socialist, if there are 9 ed
ucanalist and 11 socialist.

Ans: the 3 educanalist can be choosen from 9 educanalist in
9

ways. The socialist can be choosen
from 11
socialist

in

ways.

By product rule

, the no. of ways to select , the committee is =
9

.

=

.

=27720

ways

12.if the sequence

.

, n

, then find the corresponding recurrence relation.

Ans;

3
.
2

,

3
.
2

,=

3
.

,

,⇒

2

,

13.Find the associated homogeneous solution for

+

.

Ans:it
is

associated homogeneous equation is

3

The
characteristic

equation r
-
3 ⇒r 3.

Hence

3

4. what is the value of ‘r’ if 5

Ans: 5

=60=5*4*3=5

⇒r 3.

15.How many integers between 1 to 100 that are divisible by 3 but not by7.

Ans: Let A and B denote the no. between 1
-
100 that are divisible by 3 and 7 respectively

|

|
=
[

]
=33,
|

|

[

]

4
,
|

|

4

The number of integers divisible by 3 but not by 7

=
|

|

|

|

33

4

29
.

16 . Obtain the generating function for the sequence {

where
?‡

Ans: G(x)=

3

(
3

)

+
3

+
(
3

)

+

(

3

)

PART
-
B

1. Use mathematical induction prove that

(

)
(

)

2. Prove

that

3.Show that
?‡

(

)
.

4.Show that
Û

for all n 4.

?w?ä?•‹•‰??ƒ–Š‡•ƒ–‹…ƒŽ?‹•†—…–‹‘•?’”‘˜‡?–Šƒ–?‹ˆ?• R?s?á–Š‡•??s?ä?s

+2.2

+3.3

E?å•?ä•

=(n+1)
?

-
1

6.prove that in any group

of six people , there must be atleast 3 mutual friends or atleast 3 mutual
enemies.

7. If we select 10 points in the interior of an equilateral triangle of side1, show that there must be
atleast 2 points whose distance apart is less than

8. Show that

if any 11 numbers from 1 to 20 are chosen , then two of them will add up to 21.

9.

ppprove that mathematical induction , that for all n ,

+

is a multiple of 3.

10. Using the generating function , solve the difference equation

,

.

.

11.How many positive integer n can be formed using the digits 3,4,4,5,5,6,7 if n has to exceed
5000000?

12.Find the number of integers between 1
and

250 both
inclusive

that are divisible by any one of the
integers 2,3,5,7.

13.
there are 2500 students in a college , of these 1700 have taken a course in C, 1000 have

Taken a course Pascal and 550 have taken a course
in networking

. Further 750 have taken courses in
both C and paschal, 400 have taken courses in both C and
networki
ng,

and 275 have taken courses in
both Pascal and
networking,

If 200 of these students have taken courses in C,
Pascal

and
networking.(i) how many of these 2500 students have taken a course in any of these three courses C,
Pascal and networking?

(ii) How m
any of these 2500 students have not taken a course in any of these three courses C,

Pascal and networking

14. A box contain 6 white balls and five red balls. Find the number of ways four balls can be drawn
from the box if (i) They can be any colour(ii)
two must be white and two red. (iii) they must all be the
same colour.

15.if n pigeonhole are occupied by (kn+1) pigeons where k is positive integer, prove that atlest one
Pigeonhole is occupied by (k+1) or more Pigeons. Hence, find the minimum number of
m integers to
ne selected from S { ,2….9 so that the sum of two of the m integers are even.

16. Use mathematical induction to show that

+

+

.
.

,

• R?t?ä

17.Use
method

of generating function to solve the recurrence relation
?‡

+

n

given that
?‡

and
?‡

18.
What is the
maximum

number of students required is a discrete mathematics
class

to be sure that
at least six will receive the same

if there are five possible grades A,B,C,D,and F?

19.Find an explicit f
ormula for the Fibonacci sequence.

20.Solve D(K)
-
7D(k
-
2)+6D(k
-
3)=0 where D(0)=8, D(1)=6, and D(2)=22

21. Solve the recurrence relation
?‡

+

,

Using
generating

function.

22
. Find the sequence whose generating function is

using partial fra
ction.

23.how many positive integer not exceeding 1000 are divisible by 7 or 11?

24.A survey of 500 from a school produced the following information. 200 play vol
leyball, 120 play
hockey .60 play both volleyball and hockey . How many are not playing either volleyball or hockey?

UNIT
-
III

GRAPH THEORY

PART
-
A

1.
Define graph

Ans: A graph G (V,E,φ) consists of a non empty set V {

,

.
.

called the set of nodes of the
graph, E={

,

…. is said to be the set of edges of the graph, and φ is a mapping from the awet of
edges E to set of order or unordered pairs of elements of V.

2.
Define

self loop and parallel edges.

Ans: if there is an

edge from

then that edge is called self loop or simplyloop.

If two edges have same end points then the edges are called parallel edges.

3.Define simple graph

Ans;
A graph which has neither self loops not parallel edges is called a simple grap
h.

4 Name the types of graphs

Ans: 1.directed graph,2.undirected graph, 3.mixed graph, 4. Multigraph 5.pseudograph

5.Find the various degrees for

In degree out degree total degree

Deg
-
(a)=3

deg+(a
)=1

deg(a)=4

Deg
-
(b)=1

deg+(b)=2 deg(b)=3

Deg
-
(c)=2 deg+(c)=1 deg(c)=3

Deg
-
(d)=1

deg+(d)=3 deg(d)=5

6. State the handshaking theorem
.

Ans: Let G=(V,E) be an undirecte
d graph with e edges then

deg
(

)

2

The sum of degrees of all the vertices of an undirected graph is twice the number of deges of the
graph and hence even.

7. find the following sequence ,4,4,4,3,2 find if there exists a graph or not.

Ans:sum of the degree of all vetrtices}=4+4+4+3+2=17 which is an odd number

Such a graph does not exist

8. define regular graph.

Ans: if every vertex of a simple graph has the same degree, then the graph is called a regular graph.

If every vertex in a
regular graph has degree k, then the graph is called K
-
regualr.

9. Deifne Bipartite graph.

Ans: A graph G is said to be bipartite if its vertex set V(G) can be partitioned inot two disjoint non
empty sets

and

,

(

)
, such that every edge in

E(G) has one end vertex in V1 and
another vertex V2

°

Then G is a Bibaratite graph.

10. Define subgraph.

Ans:A graph H (V’ ,E’) is called a subgraph of G (V,E) if V’⊆V and E’⊆E. in otherwords a graph H is
said to
be a subgraph of G if all the vertices and all the edges of H are in G and if the adjacency is
preserved in H exactly as in G.

V ∘

V3∘

v2

V4

11. Define adjacency matrix of a simple graph.

Ans: Let G=(V,E) be a simple graph with n
-
verties {v ,v2…vn its adjacency matrix

is denoted by
A=[

]

and defined by A=[

]

=
{

12, Obatin adjacency matrix to represent the pseud
o graph show below.

Ans: Adjacency matrix A=[

]
=
[

3

2
3

2
2

2

]

13.Find incidence marix of

Ans: the incidence matrix

B=

]
=
[

]

14. Define
path

matrix.

Ans: If G=(V,E)

be a simple digraph in which
|

|

and the nodes of G are assumedto be ordered
an n

matrix P whose elements are given by

{

Is called the path matrix of the graphG.

15.. Find the adjacency matrix of the following graphs. Hence find the degree of the vertices

,
v3,v6

A=

]

[

]

deg(v1)= sum of the entries in 1 st row=3

Deg(v3)= sum of the entries in 3
rd

row=3

Deg(v6)=sum of the entries in 6
th

row=0

16.what are the necessary conditions for

G1and G2 to be isomorphic
.

Ans: If G1 and G2 are isomorphic then G1 and G2 have (i) the same number of vertices

(ii) the same number if edges

(iii) an equal number of vertices with a given degree

17. if a graph has n vertices and a vertex “v” is connected

to a vertex ‘w’ then there exists a path
from v to w of length not more than (n
-
1) .

Ans:let v u1,u2…um
-
1,w be a path in G from v to w.

By definition of the path , the vertices v,u1,u2…um
-
1 and w all are distinct.as G contains only n vertices
it follows

that m+1

n.

m

n
-
1.

18. if the simple graph G has v vertices and e edges howm many edges does

have?

Ans:we know that
|

(

|

(

)

|

(

|
+
|

(

)
|

(

)

|

(

|
+

(

)

|

(

|

(

)

-
e

(

)

edges

19. define strongly connected and weakly connected
.

Ans: A simple digraph is said to strongly connected, if for any pair of nodes of the graph both the nodes
of the pair are r
eachable from one another.

Weakly connected: We call a digraph is weakly connected, if it is connected as an undirected graph in
which the direction of the edges is neglected.

20 . State the conditions for Eulerian cycle.

Ans: An eulerian circuit or cycl
e should satisfies the following conditions.

(i) starting and ending points (vertices) are same.

(ii) cycle should contain all the edges of graph but exactly once.

21. check the given graph is
Euler

graph

or not.

Ans: deg(v1)=2, deg(v2)=4, deg(v3)=2,
deg(v4)=2 , deg(v5)=4.

Since all the vertices is of even degree, by he above theorem the given graph is euler graph.

PART
-
B

1. PROVE that a connected graph G is Eulerian if and only if all the vertices
are of even degree.

2.Show that graph G is disconnected if and only if its vertex set V can be partitioned into two nonempty
subset V1 and V2 such that there exists as no edges in G whose one end vertex is in V1 and the other
V2.

3.
how many paths of len
gth four are there from a to d in the simple graph G given below.

4.show that the complete graph with n vertices

has a Hamiltonian circuit whenever n

3.

5.Determine whether the graphs G and H given below are isomorphic.

6. Prove that an undirected
graph

has an even number of vertices of odd degree.

7.Determine which of the following graphs are bipartite and which are not. If a graph is bipartite, state if
it is completely bipart
ite.

8. Using circuit , examine whether the following pairs of graphs G1, G2 given below are isomorphic or
not;

9.Prove that the maximum number of edges in a simple graph disconnected graph G with n vertices and
K components is
(

)
(

)

1
0.
Find an aeuler path or an euler circuit. If it exists in each of the three graphs below. If it does not
exists, explain why?

11. Let G be a simple indirected graph with n vertices .let u and v be two non adjacent vartices in G
suxch that
deg(u)+deg(v)

n in G show that the G is Hamiltonian
if and only if G+uv is Hamiltonian.

12.Draw the graph with 5 vertices A, B, C,D and E such that the deg(A)=3 , B ia n odd vertex , deg (c)=2
and D and E are adjacent.

13.Find all the connected sungraph
obtained from the graph given in the following figure, by deleting
each vertex.List ou the simple paths from A to in each sub graph.

14.Draw the complete graph K5 with vertices A,B,C,D,E draw all complete subgraph of K5 with 4
vertices.

15.If all th
e vertices of an undirected graph ar each of degree K, show that the number of edges of the
graph is a multiple of K.

16.the adjacency matrices of two pairs of graph as given below. examine the isomorphism of G and H by
finding a permutation matrix.

=
|

|

[

]

17.Show that the number of vertices of odd degree in an undirected graph is even.

18.Prove that if a graph G has not more than two vertices of odd degree, then there can be Euler path in
G.

19.the maximum number of deges in a simple graph with n vertices is

(

)

20.Prove that a simple graph with n vertices must be connected if it has more than
(

)
(

)

edges.

21.Let G be a simple graoph with n vertices . show that if S(G)

[

]
, then G
is connected where S(G) is
minimum degree of the graph G.

Unit
-
IV
-
ALGEBRAIC STRUCTURE

Part
-
A

1.define
semi group

with example

Ans: If S is nonempty set
s TOGETHER WITH THE BINARY OPERATION * satisfying the
following
properties(i)a*b=b*a (ii) (a*b)*c=a*(b*c) a,b,c

S is called semigroup it is denoted by (S,*).

Ex: Let X be any nonempty set. Then the set of all functions from X to X
is the set

, is a semigroup
w.r.to* , the composition of functions.

2. Defi
ne
monoid with

example.

Ans
:
Let M be a n on
-
empty set with a binary operation * on it.Then M is called a monoid for the
operation* if (i) * is associative (ie) (a*b)*c=a*(b*c)

(ii) ther exists an element e

M such that e*a=a*e

a

M

a semigroup with an
identity element is a
monoid.

Ex:N={0,1,2…} then (N.+),(N,x) are monoids.

3. Give an example of a semigroup but not a monoid.

Ans: If E is the set of positive even numbers , then (E,+) and (E,*) are the semigroup but not monoid.

4.
De
fine normal subgroup of

a groupwith example

Ans;A subgroup H of G a group (G,*) is called a normal subgroup if a*H=H*a

a

G.

Ex: consider the group (Z,+) and H={...

5. Find the all subgroups of (

,
+

)

Ans:

{
[

]

{
[

]
,
[

]
,
[
2
]
,
[
3
]
,
[
4
]

are the
subgroup of (

,
+

)

6. what do you mean left coset.

Ans:
the left coset of a subgroup H is defined by a*H {a*h/h∊H

7. Define abelian group.

Ans: a group (G,*) is called abelian if a*b b*a ∀a,b ∊G
(ie) * is commutative in G.

Ex: (

,
+

)

group.

8. The inverse of every element ina group is unique.

Ans: let (G,*) be a group with identity element e, Let b and c be inverse of an element a∊G

a*b=b*a=e,

⇒a*c c*a e

⇒b b*e

=b*(a*c)=e*c=c

9.Define semigroup homomorphism.

Ans:A mapping f:(S,*)→(T,∆) is called a semigroup homomorphism if f(a*b) f(a)

f(b) ∀a,b∊S
where (s,*) and (T,∆) are two semigroups.

10. If ‘a’ is generator of a cyclic group G, then show that

is

also a generator of G.

Ans:
(

)

{
(

)

{

=
{

=a

11.Prove that the identity of a sungroup is the same as that of the group
.

Ans:let G be a group let H be a subgroup of G.

Let e and e’ be the identity elemmments in G and H

Now if a

H,
then a

G and ae=a

+

0

1

2

3

4

0

0

1

2

3

4

1

1

2

3

4

0

2

2

3

4

0

1

3

3

4

0

1

2

4

4

0

1

2

3

Again if a

H, then ae’=a

ae=ae’

e=e’

12.state the lagrange’s theorem in
𝛅

group theory.

Ans Let G be a finite group of order ‘n’ and H be any subgroup G. then the order of H divides the order of
G.(ie) O(H)/O(G)

13. Define homomorphism a
nd osimorphism between two algebraic systems.

Ans: semigroup homomorphism: Let (S,*) and (T,
𝛥
) be any two set algebraic system with binary
operation * and
𝛥

respectively

A mapping f:S→T isomorphism : A one one , onto algebraic homomorphism is called a
n
isomorphism.

14.when is a group (G,*) is called abelian?

Ans:in a group (G,*) , if a*b b*a for all a,b ∊G, then the group (G,*) is called an abelian group.

15.If a and b are any two elements of a group (G,*) show that G is an abelian group if and only if

(

)

Ans:
assume that G is abelian a*b b*a⇒

(

)

(

)

=a*[(a*(b*b)]=a*[(a*b)*b]=a*[(b*a)*b]

=(a*b)*(a*b)=(a*b)
2

Conversely,assume that
(

)

=

(a*b)*(a*b)=(a*a)*(b*b)

⇒a*[b
*(a*b)]=a*[a*(b*b)]

⇒b*(a*b) a*(b*b) [left cancellation]

⇒b*a a*b[right cancellation]

⇒G is abelian

16.Let [M,*,

]

be a monoid and a

?y

.If a invertible , then show that its inverse is unique.

Ans:let b and c be elements of a monoid M

Such that

a*b b* a e⇒a*c c*a e⇒b b*e b*(a*c) (b*a)*c e*c c⇒b c

17.State any two properties of a group.

Ans: (i) the identity element of

a group is unique

(ii)the inverse element of a group is unique

18
. Define

a commutative ring.

Ans: the ring (R,+,•) is called a
commutative

ring, ifab ba for a,b∊R.

19
. Obtain

all the distinct left coset of{(0),(3)} in the group (

,
+

)

and find

their union
.

Ans:

={0,1,2,3,4,5}

H={0,3}

0+H={0,3}, 1+H={1,4}, 2+H={2,5}, 3+H={0,3}=H

4+H={4,1}=1+H, 5+H=2+H

∴ +H, +H, and 2+H are three distinct left coset of H. their union is

2 . Show that the set of all elements ‘a’ of a group (G,*) such
that a*x x*a for every x∊G is a subgroup
o映G.

Ans:clearly ex xe x ∀x∊G

∴e∊H and H is non empty now let a, b∊H then ax xa &bx xb

Now bx xb⇒

(

)

(

)

(

)

(

)

Xb
-
1
=b
-
1
x

Now (ab
-
1
)x=a(b
-
1
x)=a(xb
-
1
)=(ax)b
-
1
=(xa)b
-
1
=x(ab
-
1
)

(ab
-
1
)x=x(ab
-
1
)hence ab
-
1

H

Therefore H is a
subgroup

PART
-
B

1. Prove that for any
commutative

monoid (M,*) , the set of
idempotent

elements of M, forms

A

submonoid.

2. Prove that the intersection of two subgroups of a group G is also a subgroup of
G.

3. If

f
:
(G,*)→(G,∘)

is⁡ g牯r瀠桯m潭潲灨ism‬ 瑨敮e灲潶攠瑨慴⁫敲e敬⁯映映fs⁡ n潲oa氠
s畢u牯異

㐮4
Sh潷

semi⁧牯rp

semi⁧牯rp

㔮⁓h潷

(

,

)

is

an abelia
n group where * defined
by
a*b=

for all
a,b∊

6. Prove

that the set
A={1,w,

}

is an anelian group of order 3 under usual multiplication , where
1,w,

are cube roots of unity and

7.Let
S=Q

be the set of all ordered pairs of rational numbers and given by
(a,b)*(x
,y)=(ax,ay+b)
(i) check (S,*) is a
semi group
. Is it commutative?

(ii) Also find the identity element of S.

8. Prove

that the
identity of

a group is
unique. and

the inverse element of a group is unique.

9. The

necessary and sufficient condition that a non
-
e
mpty subset H of a group G to be subgroup is
a,b∊H⇒a*

H.

10
. let

G be a group and

are subgroup of G.then

is also a subgroup.

11
. Prove

that
if f:G→G’

is 愠桯mom潲灨ism⁴桥n
Ker昽{e}

if映映fis‱
-

.

s瑡te

and prove that lagrange’s theorem
.

13
.

let

G and G’ be any two grou
ps with identity element eand e’ respectively. If f:G→G’ be a
homomorphism , then ker(f) is a normal subgroup
.

14
.

State

and prove that
fundamentals

theorem on homomorphism of groups
.

15
.

Every

subgroup of an abelian group is normal.

16
. A

subgroup H of a group G is normal if
x*h*x
-
1
=H

for
all x∊G.

⸠.ny

⸠.潲

a
2
=e

with
a≠e, then G is abelian
.

.

L整

f:G→G’

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s畢u牯r瀠o映f⁡ 搠
䜯K

is⁩som潲o桩c⁴漠瑨攠im慧攮

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{
[

]
,
[

]
,
[

]
,
[

]

is a commutative ring with respect to the binary
operation addition modulo amd multiplicat
ion modulo
+

21
. Let

(S,*)

be a
semi group

. then prove that there exist a homomorphism
g:S→

Where (

,

) is

a semigroup of function from S to S under the operation of composition.

22
. Prove

that every finite of order n is isomorphic to a

permutation group of order n.

23
. If

* is
binary

operation on he set R of real numbers defined by x*y=x+y+2xy

(i) find {R,*} is a
semi group
(ii) find the identity element if it exist(iii) which elements has inverse
and what are they.

24.
if (G,*) is an abe
lian
group, show

that
(

)

25
. Show

that (Z,+,

) is an integral domain where Z is the set of all integers.

26
. Let

(G,*) be a group and a∊G. let f:G→G be a given by f(x) a*x*a
-
1
, for every x∊G. prove that f is
an isomorphism of G onto G.

UNIT
-
V

LATTICES AND BOOLEAN ALGEBRA

PART
-
A

1.define partial order relation with example.

Ans: Let X be any set R be a relation defined on X. then R is said to be a
Partial order relation if it
satisfies reflexive, antisymmetric and transitive relation subset
relation ⊆ is a partial order relation.

For example consider any three sets(i) since any set is a subset of itself ,A⊆A, therefore ⊆ is
reflexive(ii) If A⊆B and B⊆A then A B therefore ⊆ is anti symmetric.

(iii) If A⊆B and B⊆C then A⊆C therefore ⊆is transit
ive. ⊆is a partial order relation

2,define partial order set.

Ans: A set together with a partial order relation defined on it is called patially ordered set or poset.

Ex: let R be the set of rela numbers . the relation “less than orequal to” or ≤ is a part
ial order on R

Therefore (R,≤) is a poset.

3.draw the hasse diagram for (

(

)
,

)

{

,

,

Ans: A={a,b,c}

(

)

{{a},{b], {c}, {a,b}, {a,c}, {b,c},{a,b,c}}

4. draw the hasse diagram for {(a,b)/a divides b} on
{1,2,3,4,6,8,12}

Ans: Let
R={(1,2),(1,3),(1,4),(1,6),(1,8),(1,12)<(2,4),(2,6),(2,8),(2,12),(3,6),(3,12),(4,8),(4,12),(6,12)}

5.which elements of the poset {(2,4,5,10,12,20,25},/} are maximal and which are minimum.

Ans: the relation R is
R={(2,4),(2,10),(2,12),(2,20),(4,12),(4,20),(5,10),(5,20),(5,25),(10,20)}

It’s hasse diagram

The maximal elements are 12,20 and 25

The minimal element are 2 and 5.

6.T
abulate th
e peopertices of lattices.

Ans: idempotenet law: a∧a a , a∨a a

Commutative law: a∧b b∧a , a∨b b∨a

Associative law (a∧b)∧c a∧(b∧c) , (a∨b)∨c a∨(b∨c)

Absorption law : a∧(a∨b) a a∨(a∧b) a.

7.
check the pentagon latt
ice or

lattice is modular or not.

Ans: consider (a,b,c) clearly a≤c

Now LHS a∨(b∧c) a∨ a

RHS (a∨b)∧c ∧c c.

LHS a≠c RHS

If a≤c then

a∨(b∧c)≠(a∨b)∧c

∴M condition is not satisfied

N5 or pentagon lattice is not a modular lattice.

8.show that in a la
ttice if a≤b and c≤d, then a∧c≤b∧d.

Ans a≤b⇒a∧b a

c≤d⇒c∧d c

claim: a∧c≤b∧d

it is enough to prove that

(a∧c)∧(b∧d) a∧c

Now, LHS (a∧c)∧(b∧d)

a∧(c∧b)∧d

a∧(b∧c)∧d

(a∧b)∧(c∧d) a∧c

Therefore
(a∧c)∧(b∧d) a∧c

⇒a∧c≤b∧d

9.
Define Sublattice with example
.

Ans: Let (L,∧,∨) be a lattice and Let S⊆L be a subset of L. then (S,∧,∨) is a sublattice of (L,∧,∨) iff S is
closed under both operation ∧and ∨

(ie) ∀a,b ∊s⇒a∧b∊sand a∨b∊s

Ex: (

,
D) is a sub
lattice of (

,

)

10.define direct product of lattice.

Ans: let (L,*,

) and (S,∧,∨) be two lattice the algrbraic system (L

,

,
+
)

in which the binary
operation + and • on L

are such that for any (

,

)

and (

,

)

in L

S

,

+

,

is called the direct product of the lattice (L,*

), and

(S,∧,∨).

11.check the given lattice is complemented lattice or not.

Ans:

Since b∧c a and b∨c , b∧a a, and b∨a b.

12.show that the

a chain of three or more elements is not complemented.

Ans: let (L,∧,∨) be a given chain. We know that, in a chain any 2 elements are comparable.

Let ,x and be any three elements of (L,∧,∨) with is the least element and is the greatest
element we h
ave ≤x≤ . Now ∧x , and ∨x x.

∧x x, and ∨x .

In both case , x does not have any complement hence,any chain with three or more elements is not
complemented.

13.
simplify the expression (a.b)+(b+c)

Ans: (a.b)+(b+c)=(a.b)+(b+c)=b(a+1)+c=b.1+c=b+c

14.
IN
Lattices (L,≤) ,prove that a∧(a∨b) a for all a,b∊L.

Ans:⁳ nc攠
a∧b

is the GLB of {a,b}

a∧b≤a
--------------
(1)

obviously a≤a
-------------------------------
(2)

from (1) and (2) we have

a∨(a∧b)≤a

by the definition of LUB we have

a≤(a∧b) a

similarly we can prove that a∧(a∨b) a.

15.Define a Boolean algebra.

Ans: A complemented distributive lattice is called Boolean algebra

(ie) A bollean algebra is distributive lattice with ‘ ’ element and element in which every element
has a com
plement.

Or equivalently , a Boolean algebra is a non empty set with 2 binary operation ∧ and ∨ and is
satisified by the following conditions ∀a, b,c∊L

. a∧a a

a∨a a

2.a∧b b∧a a∨b b∨a

3.a∧(b∧c) (a∧b)∧c

a∨(b∨c) (a∨b)∨c

4. a∧(a∨b) a a∨(a∧b) a

5.a
∨(b∧c) (a∨b)∧(a∨c)

a∧(b∨c) (a∧b)∨(a∧c)

There exists element 0 and 1
such that a∧ and a∨ a, a∧ a and a∨

∀a∊L there exists corresponding element a’ in L such that a∧a’ and a∨a’ .

16.when is a lattice said to be bounded?

Ans: Let (L,∧,∨) be a giv
en lattice. If it has both 0 element and 1 element then it is said to be
bounded lattice. It is denoted by (L,∧,∨, , )

17. when is a lattice called complete?

Ans: A lattice is called complete, if each of its non empty subset has a glb and lub.

18.check
whether the poset {(1,3,6,9)} and {(1,5,25,125),D} are lattices or not. justify your claim.

Ans; the hasse diagram for the poset {(1,3,6,9),D} is

Here lub{ ,9 ∨9 does not exist.

∴given poset is not lattice.

(2) the hasse diagram for {(1,5,25
,125),D} is

For any pair of nodes both glb and lub exist.

∴Given pose is lattice.

19.show that the boolean algebra a

̅
+

̅

=0 iff a=b

Ans : assume that a=b then prove that a

̅
+

̅

=0

⇒a

̅
+

̅

=a.

̅
+

̅

.

̅
+

̅
.

+

a

̅
+

̅

=0

Conversely, assume tha a

̅
+

̅

=0

⇒a+

a

̅
+

̅

=a

⇒a+

a

̅
=a

⇒(a+

̅
)
.
(

+

)

1.(a+b)=a

⇒a+b a
----------------
(1)

Consider a

̅
+

̅

=0

a

̅
+

̅

+

=b

a

̅
+

=b

(a+b).(b+

̅
)

(a+b).1=b

⇒a+b b
---------------
(2)

From 1 and 2 a=b

20. give an example of a lattice

which is modular but not distributive.

Ans:

is an example of modular lattice bit nor
-
distributive lattice

Part
-
b

1.consider X=[1,2,3,4,6,12}
R={(a,b)/ a/b} find LUB and GLB for the poset (X,R).

2.is the poset (

,
/
)

.

3.let (L,∧,∨) be a given
lattice. then

for any a,b ,c∊L, a∨b b∨a and a∧b b∧a.

4.every finite lattice is bounded.

5.state and prove that Isotonicity property of
lattice.

6.State and prove that distributive
inequality

of lattice.

7.state and prove the necessary and sufficient condition for a lattice to be modular.

8.in any distributive lattice (L,∧,∨) ∀a,b,c∊L prove that a∨b a∨c, a∧b a∧c⇒b c

9.in any distributive
lattice (L,∧,∨) prove that (a∨b)∧(b∨c)∧(c∨a) (a∧b)∨(b∧c)∨(c∧a)

10.
state and prove that Demorgan’s law of lattice.

11.prove that in a complemented distributive lattice, complement is unique.

12.in a complemented , distributive lattice, show that the followi
ng are equivalent a≤b⇔a∧b’

⇔a’∨b ⇔b’≤a’.

13.prove that , in any Boolean algebra

̅
+

̅
+

̅

̅

+

̅

+

̅

14.simplify the expression z(y+z)(x+y+z).

15
. Apply

Demorgan’s theorem to the following expression (i)
(

+

̅
)
(

̅
+

)

(ii)
(

+

+

)

7. Draw the hasse diagram representing the partial ordering {(A,B):A⊆B on the power set P(S)
whe牥⁓={a,bⱣ} 晩nd⁴he maxima氠Ⱐminima氠g牥瑥s琠and 汥as琠o映瑨e poset
.

8.simplify the Boolean expression a’.b’c+a.b’c+a’.b’.c’ using Boolean algebra indenti
es.

19.state and prove that modular inequality/

20.in any Boolean algebra prove that the following statements are equivalent (i)a+b=0

(2) a.b (3)a’+b (4) a.b’

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