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**Total No. of Questions : 13] [Total No. of Pages : 03**

**Paper ID [A0208]**

**BCA (203) (Old) / (S05) (Sem. - 2**

**nd**

**)**

**B.Sc. IT (202) (New)**

**C**

**Time : 03 Hours Maximum Marks : 75**

**Instruction to Candidates:**

**1)**Section - A is

**Compulsory**.

**2)**Attempt any

**Nine**questions from Section - B.

**Section - A**

*Q1) (15***x**

*2 = 30)*
a) Define inverse relation with example.

b) Define into and onto functions.

c) Prove A ∪ B = B ∪ A.

d) Draw venn diagram for the symmetrical
difference of sets A and B.

e) Define partition of a set with example.

f) Form conjuction of

*p*and*q*for the following:*p*: Ram is healthy,

*q*: He has blue eyes.

g) If

*p*: It is cold,*q*: It is raining, write the simple verbal sentence which describe (i)*p*∨*q*(ii)*p*∨ ~*q*.
h) Define logical equivalence.

i) Prove that proposition

*p*∨ ~*p*is tautology.
j) Define Biconditional statement.

k) Define undirected graph with example.

l) Edge of a graph that joins a node to
itself is called? And Edges joins node by more than one edges are called?

m) Define Null graph with example.

n) Does there exist a 4 - regular graph on
6-vertices, if so construct a graph.

o)
Prove V (G1 ∩ G2) = V(G1) ∩ V (G2) with
example.

**Section - B**

*(9***x**

*5 = 45)***Let R = {(1, 2), (2, 3), (3, 1)} and A = {1, 2, 3}. Find Reflexive, symmetric, and transitive closure of R using composition of relation R.**

*Q2)***If**

*Q3)**f*: A → B and

*g*: B → C be functions, then prove

(a) If

*f*and*g*are injections, then*gof*: A → C is an injection.
(b) If

*f*and*g*are surjections then so is*gof*.**Prove that A – (B ∩ C) = (A – B) ∪ (A – C).**

*Q4)***Show that set of real numbers in [0, 1] is uncountable set.**

*Q5)***A man has 7 relatives, 4 of them are ladies, and 3 are gentlemen, his wife has 7 relatives and 3 of them are ladies and 4 are gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 man’s relatives and 3 of wife relatives.**

*Q6)***Using truth table show that ~ (**

*Q7)**p*∧

*q*) ≡ (~

*p*) ∨ (~

*q*).

**Consider the following:**

*Q8)**p*: It is cold day,

*q*: the temperature is 50°C

write the simple sentences meaning of the following:

(a) ~

*p*(b)*p*∨*q*(c) ~ (*p*∨*q*) (d) ~*p*∧ ~*q*(e) ~ (~*p*∨ ~*q*)**Prove that following propositions are tautology.**

*Q9)*
(a) ~ (

*p*∧*q*) ∨*q*(b)*p*⇒(*p*∨*q*)**Show that two graphs shown in figure are isomorphic.**

*Q10)***Prove a non-empty connected graph G is Eulerian if and only if its all vertices**

*Q11)*
are of even degree.

**Define graph coloring and chromatic number with two examples of each.**

*Q12)***Prove a simple graph G has a spanning tree if and only if G is connected.**

*Q13)*
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