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
⅂
⡐
⋀
儩
P⋁⅂
⡐
⋀
儩
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positions⁴au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
⋁
qre uivalen琮
㘮
wi瑨ou琠using⁴ru瑨 瑡blehow⁴ha琠p
→
⡱
→
p
⇔⅂
p
→
⡰
→
焩
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
→
焩
→
⡰
→
r)) ios⁴ 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
⇒
爮
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(
∀
砩(䠨H)
→
M(砩x
⋀
H(s)
⇒
M(s)
䅮猺
1) ††(
∀
x)(H(砩
→
M(砩 †† ††given⁰牥浩ses
†† †††
(2)† †䠨s)
→
M(s)
† †† †† †††
啓 rule om
1)
†† †††
(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
†
债† 瑡硥s牥oweredⰠq: Inco浥ises
Theiven r浵污 is ((P
→
焩
⋀
焩
→
p is 琠a⁴au瑯logyⰠ瑨eivenrgu浥m琠is 琠valid.
16.⁕ ing is瑥ttialⁱ anti晩ersⰠwri瑥⁴te u
ivalent r洠o映
⅂
嬨
∀
x)(P(砩x
䅮猺†
⅂
嬨
∀
砩(倨x)崽(
∃
砩
⅂
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rsivisibley′牥癥 in瑥来rs
18.⁗ i瑥⁴te rules映in晥rence.
創汥⁐㨠R䄠premisesayen瑲oduced琠 ny⁰ in琠in⁴he rivation.
創汥⁔: ⁁ r浵污⁓ 浡礠ben瑲oducedneriva瑩on映fs⁴au瑯logically i浰mied礠anynerore
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
→⅂
儩
→
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)
→
儨砩Q
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=
敶敮
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
瑨慴⁰
⋁
(q⋀r)
and (p⋁q)⋀(p⋁r)
慲攠汯aic慬汹 畩v慬an琮
㘮
Sh潷
瑨慴⁴桥t
桹h潴o敳is
, “
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
瑨攠in摩牥c琠m整eo搠瑯⁰牯r攠e
桡琠瑨攠e潮c汵si潮o
(∃zQ(z)
景fl潷o om⁴桥⁰牥ris敳
†
∃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
晲om⁴桥⁰牥mis敳
p→(q⋁r), q→⅂p.s→⅂r
慮a
瀮
⸠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))
ㄴ
⸠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)
慲攠anc潮sis瑥t琮
ㄷ
⸠.牯re
瑨慴t
∀x(P(x)→Q(x)), ∀x(R(x)→⅂Q(x))⇨∀x(R(x)→⅂P(x)).
ㄸ
⸠.i瑨t畴
畳ing⁴牵瑨⁴rb汥⁰牯r攠瑨慴e
⅂
P→(Q→R)≡Q→(P⋁R).
ㄹ
⸠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
2MARK WITH ANSWER
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ly⁴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
?è
+2.2
?è
+3.3
?è
E?å•?ä•
?è
=(n+1)
?
?è

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
?Ÿ
,
?Ÿ
.
.
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
grade
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
∘
嘵
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
Ans: adjacency matrix
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
?Ò
?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
潦⁇o
㐮4
Sh潷
瑨慴⁴桥tcom灯siti潮o潦o
semi牯rp
桯mom潲灨osms汳漠
semi牯rp
桯m潭潲灨ism.
㔮⁓h潷
瑨慴t
(
,
)
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‱

ㄮ
ㄲ
.
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.
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⸠.ny
瑷漠物杨琠cos整映䠠in⁇牥 敩瑨敲isj潩n琠潲摥ntic慬.
ㄸ
⸠.潲
慮a牯r瀠䜬Gi映
a
2
=e
with
a≠e, then G is abelian
.
ㄹ
.
L整
f:G→G’
b攠愠e潭om潲o桩sm 潦牯rps⁷it栠步牮敬⁋⸠瑨敮e灲潶攠瑨e琠䬠Ks n潲m慬a
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瑨慴⁴桥整e
{
[
]
,
[
]
,
[
]
,
[
]
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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