BT0069, Discrete Mathematics

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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.

(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

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)

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,

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