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

III SEMISTER BCA DEGREE exam
MODEL ques. PAPER 2007
DiscRETE MATHEMATICS
Time : three Hours Max. Marks : 80

Note : ans any ten ques. from PART A and 1 full ques. from every unit of PART B

Part A
1. ans any ten of the subsequent ques. : 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} obtain (A - B) x (AnC)
(iii) describe Partial order relations
(iv) Let A={1,2,3} If f : A->A described by f(1)=2, f(2)=1 and f(3)=3 obtain f -1.
(v) describe Characteristic function of a Set
(vi) describe invertible function
(vii) Write the subsequent 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 actual or false ‘P ^ ~P is always Tantology’
(x) describe loop with an example
(xi) describe isomorphic graph
(xii) describe in-degree and out-degree

Part B
Note: ans 1 full ques. from every unit

Unit I
2. (a) State and prove DeMorgan’s laws for 2 sets A and B (05)
(b) describe Cartesian product of 2 sets A and B
If A={1} and B={a, b}C= {2,3} obtain B2, B2xA, AxB and AxBxC (05)
(c) Let A be a provided 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 three } Show that R is an
Equivalence relation and draw the graph of R and obtain 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 .
obtain f0g, f0f, g0h, (f0h)0g. (04)
(c) describe 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. obtain the Identity
element and indicate the Inverse of every element especially when x?1. (05)
(b) describe 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 1 to 1 must
also be onto and conversely. (05)

Unit III
6. (a) describe the subsequent with an example
(i) Logical statement (ii) Conditional statement
(iii) Tautological statement (06)
(b) Construct the Truth table for the subsequent 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) describe Conjunction and Disjunction and provide their Truth tables (05)

Unit IV



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

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

Earning:   Approval pending.