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Anna University Exams Nov / Dec 2018 – Regulation 2013
Important Questions – 5th Semester BE/BTECH MA6566 Discrete
Mathematics
UNIT I - V
1. Prove that the following argument is valid, p q, r  q, r p
2. Show that (x)(P(x)  Q(x)),(y)P( y)  (x)Q(x) .
3. Prove that the premises a  (b  c), d  (b  c) and (a  d ) are inconsistent
4. Without using truth table find the PCNF and PDNF of P  (Q  P)  (P  (Q  R))
5. Prove that 2 is irrational by giving a proof using contradiction
6. Show that the hypothesis, “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 go 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. Prove (( p  q)  (p  (q  r)))  p  q (p  r) is a tautology
8. Show that (x)(P(x)  Q(x))  (x)P(x)  (x)Q(x) . Is the converse true?
9. Establish this logical equivalences, where A is a proposition not involving any quantifiers. Show that
(xp(x))  A  x( p(x)  A) and (xp(x))  A  x( p(x)  A) .
10. Prove that the premises P  Q, Q  R, R  S, S ~ R and P  S are inconsistent
11. Show that R  S is logically derived from the premises P  (Q  S ), R  P and Q .
12. Show that R˄(P˅Q) is a valid conclusion from the premises P˅Q,Q→R,P→M, ̴ M
13. Show that (x)[P(X)→Q(x)]˄(x)[Q(x)→R(x)] => )x)[P(x)→R(x
14. Find the number of integers between 1 and 250 both inclusive that are divisible by any of the integers
15.
16.
17.
18.
19.
20.
2, 3,5,7
Prove by mathematical induction that for all n  1, n3  2n is a multiple of 3
n
n(n 1)(2n 1)
.
Using Mathematical induction show that  k 2 
6
k 1
1 1
1
1
n, n  2.
Using mathematical induction show that

  
1
2
3
n
If n Pigeonholes are occupied by  kn 1 pigeons, where k is positive integer, prove that at least one
Pigeonhole is occupied by k+1 or more Pigeons. Hence, find the minimum number of m integers to be
selected from S= {1,2,3,…,9} so that the sum of two of the m integers are even.
Solve the recurrence relation S  n   S  n 1  2  n 1 , with S  0   3, S 1  1, finding its
generating function
Using the generating function , solve the recurrence relation an  5an1  6an2  0 where n  2 ,
a0  3 and a1  1
21. Solve the recurrence relation an  3an1  3an2  an3 given that a0  5,a1  9,a2 15 .
22. Use generating function to solve the recurrence relation
(𝐾) − 7𝐺(𝑘 − 1) + 10𝐺(𝑘 − 2) = 8𝐾 + 6, 𝑓𝑜𝑟 𝑘 ≥ 2
23. Find the number of positive integers ≤ 1000 and not divisible by any of 3, 5, 7 and 22.
24. Examine whether the following pair of graphs are isomorphic. If not isomorphic, give the reasons
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25. Draw the graph with 5 vertices A, B, C, D and E such that the deg(A)= 3, B is an odd vertex, deg(c)=2
and D and E are adjacent
26. The adjacency matrices of 2 pairs of graphs are given below. Examine the isomorphism of G and H by
finding a permutation matrix
0 0 1
0 1 1
𝐴𝐺 = [0 0 1] , 𝐴𝐻 = [1 0 0]
1 1 0
1 0 0
27. Show that the complete graph with n vertices 𝑲𝒏 has a Hamiltonian circuit whenever n ≥ 3.
28. Prove that number of odd degree vertices in any graph is even
29. Prove that an undirected graph has an even number of vertices of odd degree
30. Using circuits, examine whether the following pairs of graphs𝑮𝟏, 𝑮𝟐 given below are isomorphic or not
31. Prove that the maximum the number of edges in a simple disconnected graph G with n vertices and k components is
(n  k)(n  k 1)
.
2
32. Prove that a connected graph G is Eulerian(Euler graph) if and only if all the vertices of G are of even degree
33. Show that if a graph with n vertices is self-complementary then 𝑛 ≡ 0 𝑜𝑟 1(𝑚𝑜𝑑 4).
34. State and prove Lagrange’s theorem
35. If (G, *) is an abelian group, show that (𝑎 ∗ 𝑏)2 =𝑎2 ∗ 𝑏2.
36. Prove that every finite group of order n is isomorphic to a permutation group of order n.
37. Prove the theorem : Let ⟨ 𝐺,∗⟩ be a finite cyclic group generated by an element a ϵ G. If G is of order n, then
𝑎𝑛= e, so that G ={𝑎, 𝑎2, 𝑎3, … , 𝑎𝑛 = 𝑒}. Further more n is a least positive integer for which 𝑎𝑛 = 𝑒.
38. Let (𝐺,∗) and (𝐻,△) be group homomorphism. Then prove that the Kernel of g is normal subgroup of (𝐺,∗).
39. Prove that the set Z4  0,1,2,3 is a commutative ring with respect to the binary operation addition
modulo and multiplication modulo +4, ×4
40. If f : G  G is a group homomorphism from {G,*} to {G, } then prove that for any a G, f (a1)  [ f (a)]1
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and f(e) is an identity of G’,when e is an identity of G
41. Show that every cyclic group is abelian group
42. Prove that intersection of two subgroups of a group (G,*) is a subgroup of a group (G,*).
43. Prove that every subgroup of a cyclic group is cyclic.
44. Show that the direct product of any two distributive lattices is a distributive lattice
45. Prove that Demorgan’s laws hold good for a complemented distributive lattice ⟨𝐿, ∧, ∨⟩, viz (a∨ 𝑏) = 𝑎′ ∧ 𝑏′
and (a∧ 𝑏)′ = 𝑎′ ∨ 𝑏′
46. Show that every chain is a lattice
47. Draw the Hasse diagram representing the partial ordering {(A,B) : A ⊆ B} on the power set P(S) where S={a, b,
c}.Find the maximal, minimal, greatest and least elements of the poset
48. Show that every non-empty subset of a lattice has a least upper bound and a greatest lower bound
49. Show that the De Morgan’s laws are valid in a Boolean Algebra.
50. If P(S) is power set of a non—empty S, prove that {P(S), , , \, , S}is a Boolean algebra
51. In a distributive complemented lattice. Show that the following are equivalent:
1) a  b
2) a  b  0
3) a  b 1
4) b  a .
52. Show that every chain is a distributive lattice
53. Show that in a lattice if a  b and c  d , then a  c  b  d and a  c  b  d .
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