Important questions in MA2265 Discrete Mathematics For Anna University, Chennai Nov/Dec 2012 Exam - V Sem CSE

Anna University, Chennai Nov/Dec 2012 Examinations

MA2265 Discrete Mathematics

V Sem CSE

Unit I-V

1. Without constructing truth table obtain PCNF of ( p®(q Ù r)) Ù (Øp ®(Øq ÙØr)) and hence find pdnf.

2. Using CP or otherwise obtain the following implication.

"x( p(x)®Q(x);"x(R(x)®ØQ(x))Þ"x(R(x)®ØP(x)

3. Show that (Øp Ù (Øq Ù r))Ú (q Ù r)Ú ( p Ù r)Ûr

4. Find PCNF and PDNF for ( p Ù q)Ú (Øp Ù q)Ú (q Ù r)

5. Prove that ( ) ( ) ( ) éë p Ú q Ù p ® r Ù q ® r ùû ® r is a tautology

6. Show that ( ) ( ( ) ( ) ) ( ) ( ( ) ( ) ) ( ) ( ( ) ) x P x ® Q x Ù x Q x ® R x Þ x P x ® R(x)

7. Prove that 8

n

– 3

n

is a multiple of 5 using mathematical induction

8. Using mathematical induction show that 2n+2 + 32n+1 is divisible by 7, n³ 0 .

9. Solve s(k) – 10 s(k-1) + 9 s(k-2) = 0 with s(0) = 3 , s(1) = 11.

10. Show that

1.2.3 2.3.4 3.4.5 . . . ( 1)( 2) ( 1)( 2)( 3) , 1.

4

+ + + + n n + n + = n n + n + n + n³

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

15. Check 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

2

(n - k) (n - k +1)

16. State and prove Lagrange’s theorem

17. Let G be a group and a Î G.Show that the map f : G G defined by f(x) = a x a-1 for every x Î G is an

isomorphism.

18. If H is a group of G such that x2 ÎH"xÎG , 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

25. In a Boolean Algebra, show that (ab) + (bc) + (ca) = (ab) + (bc) + (ca)

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