*8234*

8234

(Pages : 3)

Reg. No. : ..................................... Name : ..........................................

Third Semester B.Tech. Degree Examination, November 2009 (2008 Scheme) 08.303 : DISCRETE STRUCTURES (R F) Time : 3 Hours

Max. Marks : 100 PART – A

Answer all questions. 4

1. Construct the truth table for the formula

(

P ∧ Q)

2. Show that

Q ∧ ( P → Q) tautologically implies

4

P.

3. Write an equivalent formula for P ∧ (q ↔ r) ∨ r ↔ p which contains neither the biconditional nor the conditional.

4

4. Symbolize the predicate “x is the father of the mother of y”.

4

5. Let X = { 1, 2, 3, 4} and R = { (1,1), (1, 4), (4, 1), (4, 4), (2, 2), (2, 3), (3, 2), (3, 3)}. Write the matrix of R and sketch its graph.

4

6. Show that “less than or equal to “or ' ≤ ' is a partial ordering on the set of real numbers.

4

7. Show by mathematical induction that 2n < n ! for n ≥ 4.

4

8. Construct the composition tables of non-isomorphic groups of order 4.

4

9. Using Lagrange’s theorem show that an = e where e is the identity and a is any element of a finite group of order n.

4

10. Define path, elementary path, cycle and elementary cycle of a graph.

4 P.T.O.

8234

-2-

*8234*

PART – B Module – 1 10

11. a) Prove the following equivalences i) p → q ⇔ (

p ∨ q)

ii) p → (q → r) ⇔ (p ∧ q) → r b) Show that (x) (p(x) ∨ Q(x)) ⇒ (x) p(x) ∨ ( ∃ x) Q(x)

10

OR 12. a) Prove the equivalence (p ∨ q) ∧ ( p ∧ ( p ∧ q)) ⇔ (

p ∧ q)

b) Explain the procedure used in “proof by contradiction”. Also show that ( p ∧ q) follows from ( p ∧ q).

10

10

Module – 2 13. a) Define equivalence relation and partition on a set. Show by an example that every equivalence relation on a set corresponds to a partition of the set and vice versa. 10 b) Show that the set (0, 1) is uncountable.

10

OR 14. a) Let Z be the set of integers and R be the relation defined by R = {(x,y); x, y ∈ Z and (x – y) is divisible by 3}. Show that R is an equivalence relation defined on Z and determine the equivalence classes. 10 b) State and prove Cantor’s theorem of power sets.

10

*8234*

-3-

8234

Module – 3 15. a) Show that the intersection of any two congruence relations on a set is also a congruence relation. Show by an example that the composition of two congruence relations on a set is not necessarily a congruence relation. 10 b) Let S3 denote the set of all permutations on a set S = {1, 2, 3}. Determine all non-trivial subgroups and also the normal subgroups of S3.

10

OR 16. a) Define homomorphism between two semigroups. If (S, *) is a semigroup, show that there exists a homomorphism f. S → Ss where Ss is a semigroup of all functions from S to S under composition of functions. 10 b) Let G be the set of all non-zero real numbers. Define a*b=

ab for a, b, ∈ G. Show that (G, *) is a group. Is it abelian ? 2

_________________

10