*7914*
7914
(Pages : 3)
Reg. No. : ..................................... Name : ..........................................
Third Semester B.Tech. Degree Examination, November 2009 (2003 Scheme) 03.303 : Discrete Structures (RF) Time : 3 Hours
Max. Marks : 100 PART – A
Answer all questions. Each question carries 4 marks. 1. Define the terms tautology and absurdity stating examples. 2. State the converse and contrapositive of the implication “If I have enough money, then I will buy a car and I will buy a house”. 3. Let A, B and C be finite sets with |A| = 6, |B| = 8, |C| = 6, | A « B « C | 11, | A ª B | 3, | A ª C | 2 and | B ª C | 5 . Find | A ª B ª C | . 4. Let R be a relation on a set A. Show that R is symmetric if and only if R = R–1 . 5. Define partial order relation on a set R. Is ‘<’ or ‘less than’ a partial order on Z+ ? Why ? 6. Find the chromatic polynomial of L4. 7. Define a cyclic group. Also give examples of finite and infinite cyclic groups. 8. Let x and y be words in Bm . Define the Hamming distance between x and y. Also find the distance between x = 110110 and y = 000101. 9. Define a commutative ring with identity. Also give an example. 10. Define Boolean homomorphism between two Boolean algebras.
P.T.O.
7914
*7914*
-2-
PART â&#x20AC;&#x201C; B Each question carries 20 marks. 11. a) Show that i) p Â&#x2018; q Â? p ½ q
ii) p Âź q Âź r
½ q Ÿ r ½ p Ÿ r � r .
b) If R1 and R2 are equivalence relations on a set X, then show that R 1 ª R 2 is also an equivalence relation on X. Give an example to show that R 1  R 2 need not be an equivalence relation on X. OR 12. a) Show that (p ½ q ) Ÿ
( p Ÿ ( q ½ r )) ½ ( p Ÿ q ) ½ (
p Âź r ) is a tautology.
b) Let A and B be finite sets with the same number of elements and f : A Â&#x2018; B be a function. Show that i) If is one to one then f is onto ii) If f is onto, then f is one to one. 13. a) Let G be graph with m edges and n vertices. Show that G has a Hamiltonian 1 2
2 circuit if m Â&#x2013; (n 3n 6) .
b) Let f : R Â&#x2018; R be defined by f ( x ) 2 x 3 1 where R is the set of real numbers. Show that f is a bijection. Find the inverse of f. OR 14. a) Define Euler path, Euler circuit, Hamiltonian path, Hamiltonian circuit with examples. b) Find an explicit formula for the sequence defined by C n the initial conditions C1 = 5 and C2 = 3.
3 C n 1 2 C n 2 with
*7914*
-3-
15. a) Let G be the set of all non-zero real numbers and let a * b
7914 ab for a , b ± G . 2
Show that (G, *) is an abelian group. b) Consider the semigroup (Z, +) and R be defined on Z by a R b if and only if a b (mod 2) . Show that R is a congruence relation. OR 16. a) Let N be a normal subgroup of a group G and R be defined by a R b if and only if ab 1 ± N . Show that R is a congruence relation on G. b) If K is a finite subgroup of a group G, then show that every left coset of K in G has exactly as many elements as in K.
__________________________