B.C.A. (Sem-IV)
END SEMESTER EXAMINATION - 2020
MATHEMATICS-II (DISCRETE MATHEMATICS)
4BCA-2
TIME: 3 HOURS MAXIMUM MARKS : 70
NOTE: Candidates are required to give answer in their own words as far as practicable.
The questions are of equal value.
Answer any five questions.
Group-A
Q-1 (a) State and prove that 'D' Morgans Law.
(b) Discuss the Contradiction and Tautology with suitable example.
Q-2 (a) What is the Boolean function? Find the dual Boolean Function F = xy + xy
(b) Draw the truth table of (i) A + BC (ii) (A+B) (A+C)
Q-3 (a) Show that $\neg\big(p\to(q\land p)\big)$ is a well formed propositional formula.
(b) Define the quantifiers. Discuss different types of quantifiers with syntax and example.
Q-4 For the given set and relations below, determine which define equivalence relations.
(a) S is the set of all people in the world today, a~b if a and b have an ancestor in common.
(b) S is the set of all people in the world today, a~b if a and b have the same father.
(c) S is the set of real numbers a~b if $a = \pm b$ .
(d) S is the set of all straight lines in the plane, a~b if a is parallel to b
Q-5 (a) If A is a finite set having n elements, prove that A has exactly 2 distinct subsets
(b) Let's consider a propositional language where p = "paola is happy", q = "Pola paints a picture" and r = "Renzo is happy". Formalize the following sentences:
(i) "if Paola is happy and paints a picture then Renzo isn't happy"
(ii) "If Paola is happy, the she paints a picture"
(iii) "Paola is happy only if the paints a picture"
Group -B
Q-6 Define the Group. If G is a group in which (ab)' = a' * b' for three consecutive integers i for all a, b \in G show that G is abelian.
Q-7 (a) If G is a group and H is a subgroup of index 2 in G, then prove that H is a normal subgroup of G.
(b) (i) Show that the set \{5, 15, 25, 35\} is a group under multiplication modulo 40.
Q-8 (a) Prove that any field is an integral domain.
(b) Let R be the field of real number and Q the field of rational numbers, In R, sqrt(2) and sqrt(3) are both algebraic over Q. Exhibit a polynomial of degree 4 over Q satisfied by sqrt(2) + sqrt(3)
Q-9 Prove that a linear transformation$T : \mathbb{R}^3 \rightarrow \mathbb{R}^2$ cannot be one-to-one and that a linear transformations $S : \mathbb{R}^2 \rightarrow \mathbb{R}^3$ cannot be onto. Generalize these assertions.
Q-10 Let V be a vector space.
(a) Let M be a family of subspaces of V. Prove that the intersection TM of this family is itself a subspace of V.
(b) Let A be a set of vectors in V. Explain carefully why it makes sense to say that the intersection of the family of all subspaces containing A is "the smallest subspace of V which contains A".
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