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[Fall 2015] ASSIGNMENT
PROGRAM BSc IT
SEMESTER SECOND
SUBJECT CODE & NAME BT0069, Discrete
Mathematics
CREDIT 4 BK ID B0953 MAX.
MARKS 60
Q1. if U={a,b,c,d,e},
A={a,c,d}, B={d,e}, C={b,c,e} Evaluate
the following:
(a) A¢ ´ (B - C)
(b) (AU
B)’ ´ (B
ÇC)
(c) (A
-B) ´ (B -C)
(d) (B
UC)’ ´ A
(e) (B
- A) ´ C¢
Answer:
2 (i) State the principle
of inclusion and exclusion.
(ii) How many arrangements
of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 contain at least one of the patterns
289, 234 or 487? 4+6 10
Answer:
I)
Principle of Inclusion and
Exclusion
For any two
sets P and Q, we have;
i)
|P ﮟ Q| ≤ |P| + |Q| where |P| is the number of
elements in P, and |Q| is the number elements in Q.
ii)
3 If G is a group, then
i) The identity element of
G is unique.
ii) Every element in G has
unique inverse in G.
iii)
|
For any a єG, we have (a-1)-1 = a.
|
iv) For all a, b є G, we have (a.b)-1 = b-1.a-1.4x
2.5 10
Answer:i) Let e, f be two identity
elements in G. Since e is the identity, we have e.f= f.
Since f is the identity, we have e.f = e.
Therefore, e = e.f = f.
Hence the identity element is unique.
ii)Let a
4 (i)Define valid argument
(ii) Show that ~(P ^Q)
follows from ~ P ^ ~Q.5+5= 10
Answer: i)
Definition
Any
conclusion, which is arrived at by following the rules is called a valid
conclusion and argument is called a valid argument.
ii)
5
(i)Construct a grammar for the language.
|
'L⁼{x/ xє{ab} the number of as in x is a
multiple of 3.
|
(ii)Find the highest type
number that can be applied to the following productions:
1. S→A0, A → 1 І 2 І B0,
B → 012.
2. S →ASB І b, A →bA І c ,
3.
S →bS Іbc.5+5
10
Answer: i)
Let T = {a, b} and N = {S, A, B},
S is a
6 (i) Define tree with
example
(ii) Prove that any
connected graph with ‘n’ vertices and n -1 edges is a tree.
Answer:i)
Definition
A
connected graph without circuits is called a tree.
Example
Consider the two trees G1 = (V, E1) and
G2 = (V,
E2) where V = {a, b, c, d, e, f, g, h, i, j}
E1 =
{{a, c
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