Paper ID [A0208]
BCA (203) (Old) / (S05) (Sem. - 2nd)
B.Sc. IT (202) (New)
MATH - I (Discrete)
(Please fill this Paper ID in OMR Sheet)
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)
Q2) 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.
Q3) If 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.
Q4) Prove that A – (B ∩ C) = (A – B) ∪ (A – C).
Q5) Show that set of real numbers in [0, 1] is uncountable set.
Q6) 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.
Q7) Using truth table show that ~ p ∧ q ( ) ≡ ~ p ( )∨ ~ q ( ).
Q8) Consider the following: 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 ( )
Q9) Prove that following propositions are tautology.
(a) ~ (p∧ q) ∨ q
(b) p⇒ p ∨ q ( )
Q10)Show that two graphs shown in figure are isomorphic.
Q11)Prove a non-empty connected graph G is Eulerian if and only if its all vertices are of even degree.
Q12)Define graph coloring and chromatic number with two examples of each.
Q13)Prove a simple graph G has a spanning tree if and only if G is connected.
BCA (203) (Old) / (S05) (Sem. - 2nd)
B.Sc. IT (202) (New)
MATH - I (Discrete)
(Please fill this Paper ID in OMR Sheet)
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)
Q2) 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.
Q3) If 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.
Q4) Prove that A – (B ∩ C) = (A – B) ∪ (A – C).
Q5) Show that set of real numbers in [0, 1] is uncountable set.
Q6) 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.
Q7) Using truth table show that ~ p ∧ q ( ) ≡ ~ p ( )∨ ~ q ( ).
Q8) Consider the following: 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 ( )
Q9) Prove that following propositions are tautology.
(a) ~ (p∧ q) ∨ q
(b) p⇒ p ∨ q ( )
Q10)Show that two graphs shown in figure are isomorphic.
Q11)Prove a non-empty connected graph G is Eulerian if and only if its all vertices are of even degree.
Q12)Define graph coloring and chromatic number with two examples of each.
Q13)Prove a simple graph G has a spanning tree if and only if G is connected.