This document contains a model exam for discrete mathematics with two parts: Part A contains 10 multiple choice questions and Part B contains 5 long answer questions worth 16 marks each. The exam covers topics in discrete mathematics including logic, sets, functions, relations, proof techniques like mathematical induction, combinatorics, graph theory, groups, lattices, and Boolean algebra. It provides the framework and questions for a summative assessment of these core concepts in discrete mathematics.

1. Immanuel Arasar J J College of Engineering
Model Exam
Sub: Discrete Mathematics Marks: 50
Branch: III- CSE Date:29/09/2015
PART-A (10*2=20)
1.Define a conditional and bi-conditional statement
2.Show that }
,
{ 
 is not functionally complete.
3.What is a relation?
4.Give an example of a relation which is both symmetric and transitive.
5.State Lagranges theorem
6.In (G,* ) is a group in which (a* b)2
=a2
*b2
for all a,bG, Prove that (G,*) is abelian
7.In a ring, show that (–a)b= a(–b) = – (ab) for all elements a and b of the ring.
8.What is meant by a homomorphism?
9.Define a chain and Give an example
10.Prove that every distributive lattice is modular.
PART-B (5*16=80)
11.a) i)Show that the statements (p →q)  (p →r) and p → (q  r) are logically equivalent.
ii) Show that t is a valid conclusion from the premises p s
s
r
r
q
q 
→
→
→ ,
,
, and p t

(OR)
b) i)Prove that by induction method t
t
p
q
p
q 

→
 ,
,
ii) Show that the premises are inconsistent p r
p
s
r
s
q
q 

→
→
→ ,
,
,
12. a) i)Use mathematical induction prove that, 
1
=
i
Ai
n =
n
i
Ai
1
=
, where A1,A2,…..An are subsets of
U and n 2
ii)Solve an = 4(an-1 +an-2) with initial conditions a0 =1, a1 =1
(OR)
b) i)Solve an+1 - 8an +16 an-1 =4n
,n 1
 with the initial conditions a0 =1 ,a1 =8
ii) Solve s(k) + 3s(k-1) -4s(k-2) =0 ,k 2 and s(0) = 3 ,s(1) =-2
13.a) i)Necessary and sufficient conditions for Euler circuit and path
ii) A non trivial connected graph has atmost two vertices of odd degree
(OR)
b)i) DefineBi partite graph, Regular graph,Isomorphism of graph,connected graph,Eulerian graph.
Give an example.
ii) State and prove Hand shaking theorem.

2. iii)If G is not connected, then G is connected.
14.a) i)State and prove Legranges theorem
ii) The intersection of any two normal subgroups of a group is its normal subgroups
(OR)
b) i) The identity of a group G is unique
ii)A group G is abelian iff (a*b)2
= a2
*b2
iii) Let (G,*) and(H,  ) be groups and g:G →H be a homomorphism. Then the kernel of g is normal
subgroup
15.a)i) State and prove Demargans law
ii)In a distributive lattice (L, 
, ) if an element aL , a complement then it is unique
(OR)
b) i) In a Boolean algebra for any a,b B, a=b iff (a b
 ) )
( b
a 

 =0
ii) Show that the following i)a +a =a ii)a.a =a iii) a +1=1 iv)a +(a. b) =a
Staff incharge HOD PRINCIPAL

3. Immanuel Arasar J J College of Engineering
Model Exam
Sub: Discrete Mathematics Marks: 50
Branch: III- CSE Date:21/09/2015
PART-A (10*2=20)
1.Construct the Truth Table for the compound proposition (p →q) ( q →p)
2. Prove without using truth table (p →q)  qp
3. Prove by mathematical induction 1+2+3+….+n=
2
)
1
( +
n
n
4.Define Recurrence relation
5. Define Hamiton graph
6.Weakly connected graph
7. Define groups
8.The inverse of any element of group G is unique
9.Define POSET
10. Definedistributive lattice
PART-B (5*16=80)
11.a.i) Show that the statements (p →q)  (p →r) and p → (q  r) are logically equivalent.
(ii).Using indirect method show that p r
p
r
p
r
q
q 


→
→ ),
(
,
, r

(iii)If there was a cricket, then travelling was difficult. If they arrived on time , then travelling was not
difficult. They arrived on time therefore there was no cricket.Show that these proposition form a valid
argument.
(OR)
b)i) Show that the premises are inconsistent p r
p
s
r
s
q
q 

→
→
→ ,
,
,
ii)Show that t is a valid conclusion from the premises p s
s
r
r
q
q 
→
→
→ ,
,
, and p t

12.a)(i)Use mathematical induction prove that, 
1
=
i
Ai
n =
n
i
Ai
1
=
, where A1,A2,…..An are subsets of
U and n 2
(ii)Solve yn+2-yn+1-2yn =n2
,with the initial conditions y0=0 & y1=1
(OR)
b)(i)Solve an+2-4an+1+4an =2n
(ii) Solve s(k) + 3s(k-1) -4s(k-2) =0 ,k 2 and s(0) = 3 ,s(1) =-2
13.a)(i) Necessary and sufficient conditions for Euler circuit and path
ii) A non trivial connected graph has atmost two vertices of odd degree
b)i) DefineBi partite graph, Regular graph,Isomorphism of graph,connected graph,Eulerian graph.
Give an example.
ii) State and prove Hand shaking theorem.

4. iii)If G is not connected, then G is connected.
14.a)(i) State and prove Lagranges theorem
(ii)Fundamental theorem on homomorphism of groups
(OR)
b)(i)Any two right cosets of H in G are either identical or disjoint
ii)A group G is abelian iff (a*b)2
= a2
*b2
iii) Let (G,*) and(H,  ) be groups and g:G →H be a homomorphism. Then the kernel of g is normal
subgroup
15.a)(i) State and prove Demargans law
(ii) Every chain is a distributive lattice
(OR)
b)(i)Show that the following i)a +a =a ii)a.a =a iii) a +1=1 iv)a +(a. b) =a
(ii) In a Boolean algebra for any a,b B, a=b iff (a b
 ) )
( b
a 

 =0
Staff incharge HOD PRINCIPAL