**Roll No......................**

**Total No. Of Questtions:13**

**Paper ID [ A0208]**

**BCA (Sem.-2**

^{nd})**B.Sc. IT(202)**

**MATH- I (Discrete)**

**Time : 03 Hrs.**

**Instruction to Candidates:**

1. Section
– A is Compulsory

2. Attempt
any Nine questions from Section – B

**SECTION – A**

1.

a. Define
inverse relation with example.

b. Define
into and onto functions.

c. Prove
AB=
BA.

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

e. Define
partition of a set with example.

f. From
conjunction 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 v q ii) P v ~ q.

h. Prove
that proposition p v ~ p is tautology.

i. Define
logical equivalence.

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 edge 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 ( G

_{1}G_{2}) = V(G_{1}) V(G_{2}) with example .**Section –B**

2. 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.

3. If Æ’ : A→B and g : B→C be functions , then prove

a. If Æ’ and g are injections, then gof: A → C is an injection.

b. If
Æ’ and g are surjection then so is gof.

4. Prove
that A- (BC)=(A-B) (A-C)

5. Show
that set of real number in [0,-1] is uncountable set.

6. 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.

7. Using
truth table show that ~ (p q = (~p) v(~q).

8. Consider
the following :

P: it is cold day, q:
the temperature is 50 c writer the
simple sentences meaning of the following:

(a)~p(b) p v q (c)~ (p
v q)(d)~ p ~
q (e)~(~p v~q)

9. Prove
that following propositions are tautology.

a. ~(p
q ) v q b.
P →(p v q)

10.
Show that two graphs shown in figure are
isomorphic .

11.
Prove a non – empty
connected graph G is Eulerian if and only if all vertices are of even degree.

12.
Define graph coloring
and chromatic number with two examples of each.

13.
Prove a simple graph G
has a spanning tree if and only if G is connected.

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