PU-4 BCA-2
BCA (IV- SEMESTER)
END SEMESTER EXAMINATION, JUNE 2022
MATHEMATICS - II
(DISCRETE MATHEMATICS)
Time: 3 Hours Maximum Marks: 70
Note: Candidates are to give answers in their own words as far as practicable. Answer any five Questions. All Questions are of equal value.
GROUP - A
Q.1
(a) (i) Find the truth of the proposition
(: (P ∧ Q): R) v (((: P ∧ Q) v : P) ∧ S)
(ii) Using Truth table, show that P → Q ⇒ P → (P ∧ Q)
(b) construct circuits to produce the following output:
(i) (A'+Β').B'
(II)((A.B)'+B)).(A+B)'
Q.2
(a) Obtain the disjunctive normal from and conjunctive normal form of
\(p\land \left(p\implies q\right) and,\left(p\lor q\right)\iff \left(p\land q\right)\)
(b) (i) Define Quantities and its type with suitable examples.
(ii) Verify the validity of the following argument: All men are mortal. Socrates is a man. Therefore, Socrates is mortal
Q.3
(a) Using Boolean Algebra Identities prove that
(ABC)' (A+B+C)'=A'B'C'
(b) Define trees and draw all the spanning trees of the given graphs.
Q.4. If R and S are relations on A=\ a, b, c ) represented by the matrices.
$$ M_R = \begin{bmatrix} a & 0 & a \\ 0 & a & 0 \\ 0 & 0 & 0 \end{bmatrix} $$ and,
$$ M_S = \begin{bmatrix} 0 & a & a \\ a & a & 0 \\ 0 & 0 & a \end{bmatrix} $$
Find the matrices that represent
(i) R ∪ S (ii) R ∩ S
(iii) R ∘ S (iv) S ∘ R
Q.5
(a) Define an equivalence relation on a set and prove that the relation congruence modulo 5 is an equivalence on the set I to integers.
(b) For any partially order set ( X ,<=) Prove that every non-empty subset of X which has an upper bound has a supermum.
Q.6
(a) Write the Boolean expression for the logic diagram given below and simplify it as much as possible and draw that logic diagram that implements the simplified expression.
(b) Reduce the expression
f= ((AB)' + A'+ AB)'
GROUP - B
Q.7
a)
i) Prove that for any graph (G, *)
(a* b) = ba¹ where a,be G
ii) Prove that if group (G, *) is an Abelian group then
(a* b)²= a²* b²
b)
i) Prove that the set \{1, w, w ^ 2\} where w is a cube root of unity is a cyclic group with respect to multiplication.
ii) Every subgroup <H, *> of a Cyclic Group G <= a > is also cyclic group, prove it.
Q.8
a) Express the following permutation as product of disjoint cycles
f =(13 2 5)(1 4 3)(2 5 1)
b) If a,b,c are elements of a Ring R then prove that
(i) - (a + b) = - a - b
(ii) (a - b) - c = a - (b + c)
Q.9
a) Prove that the set \ a+ ib |a,b in z\ is an integral domain but not a field.
b) Show that if H_{1} and H_{2} be two subgroups of a Group G then H_{1} cap H 2 is also a subgroup of G.
Q.10
á) Find all the subspaces of
(i) V2(I/(2))
(ii) V3 (I/(2))
b) Determine whether the three vectors (1,0,0), (0,1,0),
(0,0,1) of V3(R) are linearly dependent.