Anna University, Chennai Nov/Dec 2012 Examinations
Important Questions
MA2265 Discrete Mathematics
V Sem CSE
1. Without constructing truth table obtain PCNF of ( p → ( q ∧ r )) ∧ (¬p → (¬q ∧ ¬r )) and hence find pdnf.
Using CP or otherwise obtain the following implication.
∀x ( p ( x) → Q( x ); ∀x ( R ( x ) → ¬Q( x )) ⇒ ∀x( R( x ) → ¬P( x )
3. Show that (¬p ∧ (¬q ∧ r )) ∨ (q ∧ r ) ∨ ( p ∧ r ) ⇔ r
4. Find PCNF and PDNF for ( p ∧ q ) ∨ (¬p ∧ q ) ∨ (q ∧ r )
( p ∨ q ) ∧ ( p → r ) ∧ ( q → r ) → r
5. Prove that
is a tautology
( x ) ( P ( x ) → Q ( x ) ) ∧ ( x ) ( Q ( x ) → R ( x ) ) ⇒ ( x ) ( P ( x ) → R( x ) )
Prove that 8 – 3 is a multiple of 5 using mathematical induction
n+ 2
2 n +1
8. Using mathematical induction show that 2 + 3
is divisible by 7, n ≥ 0 .
Solve s(k) – 10 s(k-1) + 9 s(k-2) = 0 with s(0) = 3 , s(1) = 11.
10. Show that
11. Using generating function method to solve the Fibonacci series
12. If G is a simple graph with n vertices and k components, then the number of edges is at most
(n − k )( n − k + 1) / 2
13. A connected graph G is Eulerian if and only if every vertex of G is of even degree
14. Prove that if a graph G has not more than two vertices of odd degree, then there can be Euler path in G
the given graph is strongly connected, weakly connected and unilaterally
connected or not. If
G is a simple graph with n- vertices and k- components then the no.of
edges is atmost
1.2.3 + 2.3.4 + 3.4.5 + . . . + n( n + 1)( n + 2) =
n(n + 1)(n + 2)(n + 3)
, n ≥1.
4
16. State and prove Lagrange’s theorem
17. Let G be a group and a ∈ G.Show that
isomorphism.
2
18. If H is a group of G such that x ∈ H ∀x ∈ G , Prove that H is normal subgroup of G
19. State and prove Fundamental theorem on homomorphism of groups
20. Prove that every finite group of order n is isomorphic to a permutation group of degree n.
21. Establish De.Morgan’s laws in a Boolean Algebra
22. State and prove distributive inequalities of a Lattice
23. Show that every distributive lattice is modular. Whether the converse is true?
24. In a distributive lattice prove that a *b = a * c and a b = a c implies that b = c
a Boolean Algebra, show that (ab) + (bc) + (ca) = (ab) + (bc) + (ca)
the map f : G G defined by f(x) = a x a-1 for every x ∈ G is an
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