## Monday, 13 March 2017

### 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 = Ac ⋂ Bc. (4)
(e) Define homomorphic and isomorphic graph. (4)

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) Ac ⋂ B
(v) Bc — Ac (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}.