This document provides information about an assignment for the subject Discrete Mathematics in the BSc IT program for the second semester. It gives the subject code, credit hours, maximum marks, and includes the first question asking students to calculate the number of bytes formed from sequences of 0s and 1s with certain patterns, explain the principle of inclusion and exclusion, and define terms related to groups. It provides contact information to obtain fully solved assignments via email or phone.

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ASSIGNMENT
PROGRAM BSc IT
SEMESTER SECOND
SUBJECT CODE & NAME BT0069, Discrete Mathematics
CREDIT 4
BK ID B0953
MAX.MARKS 60
Q. No. 1 A bit is either 0 or 1: a byte is a sequence of 8 bits. Find the number of
bytes that, (a) can be formed (b)begin with 11 and end with 11 (c)begin with 11 and
do not end with 11 (d) begin with 11 or end with 11. 4x2.5 10
Answer: (a) Since the bits 0 or 1 can repeat, the eight positions can be filled up either by 0 or 1
in 28 ways. Hence the number of bytes that can be formed is 256.
(b) Keeping two positions at the beginning by 11 and the two positions at the end by 11, there are
four open positions, which can be filled up
2 (i) State the principle of inclusion and exclusion.

2. (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) |P ∩ Q| ≤ min (|P|, |Q|)
iii) |P O Q| = |P| + |Q| – 2|P ∩ Q| where O is the symmetric difference.
ii) 3X8! – 6!
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)
4 (i) Define valid argument
(ii) Show that ~(P ^Q) follows from ~ P ^ ~Q. 5+5= 10
Answer: i)
Definition

3. Any conclusion, which is arrived at by following the rules is called a valid conclusion and
argument is called a valid argument.
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. 5+5 10
Answer: i)
Let T = {a, b} and N = {S, A, B},
S is a starting symbol.
ii)
1. Here, S  A0, A  B0 and B  012 are of type 2, while A  1 and A 2 are type 3.
Therefore, the highest type number is 2.
2. Here, S  ASB is
6 (i) Define tree with example
(ii) Any connected graph with ‘n’ vertices and n -1 edges is a tree. 5+5 10
Answer: i)

4. 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}, {b, c}, {c, d}, {c, e}, {e, g}, {f,
Dear students get fully solved assignments
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