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Posted Date: 27 Dec 2007      Posted By:: ashish    Member Level: Gold    Points: 5 (Rs. 1)

2007 Mangalore University B.C.A Computer Application DISCRETE MATHEMATICS Question paper

 Course: B.C.A Computer Application University/board: Mangalore University

III SEMISTER BCA DEGREE EXAMINATION
MODEL QUESTION PAPER 2007
DISCRETE MATHEMATICS
Time : 3 Hours Max. Marks : 80

Note : Answer any ten questions from PART A and one full questions from each unit of PART B

Part A
1. Answer any ten of the following questions : 2 x 10 = 20
(i) Draw a Ven Diagram (1) A - B (2) (AnB)nC)
(ii) If A={1,2,3,4} B={2,3,7} C={1,2,5,7} Find (A - B) x (AnC)
(iii) Define Partial order relations
(iv) Let A={1,2,3} If f : A->A defined by f(1)=2, f(2)=1 and f(3)=3 Find f -1.
(v) Define Characteristic function of a Set
(vi) Define invertible function
(vii) Write the following Statement in symbolic forms
“If either Jerry takes Calculus or Ken takes Sociology, then Larry takes English”
(viii) Write the Truth table for Disjunction Statement
(ix) State true or false ‘P ^ ~P is always Tantology’
(x) Define loop with an example
(xi) Define isomorphic graph
(xii) Define in-degree and out-degree

Part B
Note: Answer one full question from each unit

Unit I
2. (a) State and prove DeMorgan’s laws for two sets A and B (05)
(b) Define Cartesian product of two sets A and B
If A={1} and B={a, b}C= {2,3} Find B2, B2xA, AxB and AxBxC (05)
(c) Let A be a given finite set and P(A) is its Power Set. Let C be the Inclusion
relation on elements of P(A). Draw Hasse diagram of ( P(A),C )
for (i) A={a, b} (ii) A={a, b, c} (iii) A={a, b, c, d} (1+2+2)
OR
3. (a) Prove that Ax (BUC)=(AxB)U(AxC)
Ax (BnC)=(AxB)n(AxC) (05)
(b) Let X={1,2,3…… 7} R={ (x, y)/x-y is divisible by 3 } Show that R is an
Equivalence relation and draw the graph of R and find its matrix. (05)
(c) Let R={<1,2>, <3,4>, <2,2>} S={<4,2>, <2,5>, <3,1>, <1,3>}Find R0S,
R0R, S0S, S0R, R0(S0R) (05)

Unit II
4. (a) Let * be binary operation on X which is Associative and which has the identity
e €X. If an element a€X is invertible then prove that both its left and right inverses
are equal (05)
(b) Let f(x)=x+2, g(x)=x-2 and h(x)=3x for x €R, R is a set of real numbers .
Find f0g, f0f, g0h, (f0h)0g. (04)
(c) Define Surjective , Injective and Bijective functions along with an example. (06)
OR
5. (a) Let g: RxR->R, where R is the set of integer and g(x,y)=x*y=x+y-xy. Show
that the binary operation * is commutative and Associative. Find the Identity
element and indicate the Inverse of each element especially when x?1. (05)
(b) Define binary operation with an example .State the general properties of binary
operations. (05)
(c) If A={1,2,3…….n} Show that any function from A to A which is one to one must
also be onto and conversely. (05)

Unit III
6. (a) Define the following with an example
(i) Logical statement (ii) Conditional statement
(iii) Tautological statement (06)
(b) Construct the Truth table for the following formulas
( 7(P/\Q)V7R)V((7P/\Q)/\7R) (04)
(c) Prove that the statement (((PVQ)->R)/\(~P))->(Q->R) is a tautology (05)
OR
7. (a) Prove that ~(P/\Q)->(~PV(~PVQ)?(~PVQ) (05) (b) Prove that (Q->(P/\~P))->(R->(P/\~P)?(R->Q) (05)
(c) Define Conjunction and Disjunction and give their Truth tables (05)

Unit IV

8. (a) Explain the terms reachability and connectedness (04)
(b) Let A= {a,b,c,d} and R be the relation on A that has the matrix

1 0 1 0
MR = 0 1 0 1
1 1 1 0
0 1 0 1
Construct the diagraph of R and list in-degree and out-degrees of all nodes (05)
(c) Define i) null graph ii) weighted graph iii) multi graph (06)

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