# IPU BCA Semester 2 - Mathematics (End Term Paper 2015)

END TERM EXAMINATION

Second Semester [BCA]

MAY-JUNE 2015

Paper Code: BCA-102

## Subject: Mathematics

[2011 onwards]

**Time**: 3 Hours

**Maximum Marks**: 75

Note: Attempt any ﬁve questions including Q. no.1 which is compulsory. Select one question from each Unit.

**Question 1**:

(a) Let R be the relation in the natural number N deﬁned by the open sentence "(x-y) is divisible by 5", prove that R is an equivalence relation. (4)

(b) Consider the bounded lattice L (4)

Find the compliments of a & c, if they exist.

(c) If f(x) = x³, then ﬁnd f⁻ⁱ for all X ∈ R. (4)

(d) Show that (A ⋃ E)

^{c}= A

^{c}⋂ B

^{c}. (4)

(e) Define homomorphic and isomorphic graph. (4)

(f) Deﬁne Tauology and Contradiction.

__Unit-I__**Question 2:**

(a) Let R and S be the following relations on:

B = {a, b. c, d}, R = {(a,a), (a,c), (c, b), (c,d), (d, b)} and

S = {(b, a), (c, c), (c, d), (d, a)}

Find the following composition relations.

(i) ROS

(ii) SOR

(iii) ROR

(iv) SOS

(b) Let U = {a, b, c, d, e}, A = (a,b,d) and B = (b, d, e).

Find

(i) A ⋃ E

(ii) B ⋂ A

(iii) B — A

(iv) A

^{c}⋂ B

(v) B

^{c}— A

^{c}(6.5)

**Question 3:**

(a) Let R be the relation in the natural numbers N = {1, 2, 3, .... ..} deﬁned by the open sentence "2x+y=10", that is, let R = {(x,y)| (x∈ N. y∈N, 2x + y = 10}.

Find:

(i) the domain of R

(ii) the range of R

(iii) R⁻ⁱ (6)

(b) Among 50 students in a class, 26 got an A in the first examination and 21 got an A in the second examination. If 17 students did not get an A in the either examination, how many students got A in both the examination? (6.5)

__Unit-II__**Question 4:**

(a) Let B = {2, 3, 4, 5, 6, 8, 9, 10} be ordered by “x is a multiple of y”. (6)

(i) Find all maximal elements of E.

(ii) Find all minimal elements of B.

(iii) Does B have a first or a last element?

(b) State whether or not each of the following subsets of N is totally ordered: (6.5)

(i) {24, 2, 6)

(ii) {3,15, 5}

(iii) {15,5,30}

(iv) {1,2, 3,... )

**Question 5**:

(a) Let R be the relation on A. (6)

A = {2, 3, 4, 6, 8, 12, 36, 48}.

R = {(a, b) | a is divisor of b}. Draw Hasse diagram.

(b) Consider the lattice M is given below ﬁgure: (6.5)

(i) Find complements of a and b, if exist.

(ii) Is M distributive? Complemented?

__Unit-III__**Question 6:**

(a) Give an example of Isomorphic graphs. Show that the graph G₁ and G₂ are not isomorphic. (6)

(b) Deﬁne Adjacent matrix. Find the adjacency matrix of the graph G. (6.5)

**Question 7:**

(a) Deﬁne

(i) bipartite graph

(ii) Hamilton Graph

(iii) Cut-Vertical.

(b) Draw the directed graph of following incidence matrix:

e₁ e₂ e₃ e₄ e₅ e₆ V₁ │ 1 0 0 0 1 0 │ V₂ │ 1 1 0 0 0 1 │ V₃ │ -1 0 0 0 0 1 │ V₄ │ 0 0 1 1 0 1 │

Also find the degree of all vertex.

__Unit IV__**Question 8:**

(a) Construct the truth table of the following: (6)

(i) (~p ⋁ q) ⋁ ~p

(ii) (~q → ~p) → (p → q)

(b) Verify whether following are tautologies or not: (6.5)

(i) (q → p) ↔ (~q ⋁ p)

(ii) (p ⋀ (q — p)) → p

**Question 9:**

(a) Consider the following: (6)

p: Today is Monday.

q: It is hot.

r: It is not raining.

Write in simple sentence the meaning of the following:

(i) ~r ⤇(r ⋀ q)

(ii) (p ⋁ r)c ⇔ a

(b) What is the truth value of the quantification (∃ x)Q(x), if the statement Q(x) and inverse of discourse is given as follows: (6.5)

(i) Q(x): x > 32 U = {all real numbers}

(ii) Q(x): x = x + 2 U = {all real numbers}

(iii) Q (x) : x² < 12 U = {positive integer not exceeding 3}.