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Spring 2017] ASSIGNMENT
PROGRAM BSc IT
SEMESTER SECOND
SUBJECT CODE & NAME BT0069, Discrete
Mathematics
CREDIT 4 BK ID B0953 MAX.
MARKS 60
Q1. Find the sum of all the
four digit number that can be obtained by using the digits 1, 2, 3, 4 once in
each.
Answer:
There
are 4! or 24 such numbers. So there are 24 digits in each column.
There
are the same number of 1's in each column as there are 2's, 3's and 4's. so there are 24÷4 or 6 of each digit
in each column. So each column sums to
6*1+6*2+6*3+6*4 = 6(1+2+3+4) = 6(10) = 60, so the
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
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.
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
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 І bc.
Answer: i)
Let T = {a, b} and N = {S, A, B},
S is a
starting symbol.
The
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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