First question is compulsory.

Answer any FOUR from the remaining questions.

All questions carry equal marks.

Answer all parts any question at one place.

  1.  Answer the following.

(a)   If  A={1,2} and B={a,b} then write ρ (AXB) where ρ (AXB)denotes the power set of A X B.

(b)   Write the statement in predicate calculus:

There is a shop in wvery street.

(c)    How many three digit numbers can be formed from the digits 0,1,2,3,4?

(d)   Find the number of terms in the expansinon of (x+y+z)5.

(e)   Define binary tree.

(f)     Give an example of a digraph with four nodes each having in-degree 2.

(g)   Degfine partial ordering.

 

2. (a) Verify whether the folowing is a tautology or not.

((P->Q)->R)->((P->Q)->(P->R)).

(b) Using mathematical induction,prove that the product of any 3 consecutive integers is divisible by 6.

3.  (a) If A and B are two sets then define AB=(A-B)(B-A) with A={1,2,3} and B={1,3,5} then find the set ((AB)B)-(A(BB)).

(b) Ifρ(φ) denotes the power set of the empty set φ then write explicitly the elements of ρ(

4. (a) How many integral solutions are ther to x1+x2+x3+x4+x5+=20 where each xi>=2?

(b) Represent the relatin R={(1,2),(1,3),(2,3(3,1)} on the set A={ 1,2,3} as a digraph and find it’s transitive closure.

5. (a) If A is a non- empty finite set then prove that any function f:A->A is one-one if and only if it is onto.

(b) Show that the value of A(2,2)=7 where the A(m,n) is recursively fefined as follows:

A(0,n)=n+1

A(m,0)=A(m-1,1)if m>0

A(m,n)=A(m-1, A(m,n-1)) if m>0 and n>0.

6. (a) Prove that the sum of all the in-degree of the nodes of a graph is equal to the sum of all the out-degree of the nodes of al graph.

(b)  Check whether the following graphs are isomorphic or not ?

(GRAPH)

7.  (a) Find all solutions of the recurrence relation an = 5 an-1-6 an-2+7n.

(b) Writ e the Kruskal’s algorithm for finding the minimun spanning tree of a graph.

8.  (a) Define the reaversals of a binary tree and illustrate with an example.

(b) Construct the binary tree whose preorder sequence is A B C D E F G H I  and with the in-order sequnece is B C A E D G H F I.

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First question is compulsory. Answer any FOUR from the remaining questions. All questions carry equal marks. Answer all parts any question at one place.  Answer the following. (a)   If  A={1,2} and B={a,b} then write ρ (AXB) where ρ (AXB)denotes the power set of A X B. (b)   Write the statement in predicate calculus: There is a...