MCS 013 Discrete Mathematics

MCS-013 

Discrete Mathematics

2016-17 

(last session )

 

 

(a) Make logic circuit for the following Boolean expressions:

.

.

.

(b) Find Boolean Expression of Q in the figure given below.
 .

Figure 1: Boolean Circuit.

sol: 

(A.B)  +  (B+C).(C.B)

 

 

(c) Find Boolean Expression of Q in the figure given below.
Figure 2: Boolean Circuit
 

sol.

(X.Y)'  +  Z' +  Y

 

(d) What is integer partition? Write down all partitions of 8.
sol .

check on Google or Wikipedia... 

 

4. (a) How many different committees can be formed of 10
professionals, each containing at least 4 Professors, at
least 3 General Managers and 3 Finance Advisors from list
of 10 Professors, 12 General Managers and 5 Finance
Advisors?
 

PROFESSOR        GENERAL MANAGER        FINANCE ADVISORS

                4                              3                                              3

           SO

SOLUTION IS

C(10,4)  + C(12,3)  +  C(5,3)

( 10X9X8X7)/(4X3X2X1)+(12X11X10)/(3X2X1)+(5X2)/(2X1)

(10X3X7) +(11X10X2) + 5

210+220+5

435

 

(b) There are two mutually exclusive events A and B with
P(A) =0.5 and P(B) = 0.4. Find the probability of
followings:

(i)                 A does not occur

P(A’)=1-P(A)=1-0.5=0.5

(ii)               Both A and B does not occur

P(A’)=1-P(A)=1-0.5=0.5

P(B’)=1-P(B)=1-0.4=0.6

P(A’ AND B’) = P(A’) X P(B’) =0.5 X 0.6 = 0.3

 (iii) Either A or B does not occur

P(A’)=1-P(A)=1-0.5=0.5

P(B’)=1-P(B)=1-0.4=0.6

P(A’ OR B’) = P(A’) + P(B’) - P(A’ AND B’) =0.5 + 0.6 - 0.3 = 0.9

 

(c) What is set? Explain the basic properties of sets. 

 

sol .

check on Google or Wikipedia... 

 

5. (a) How many words can be formed using letter of
UMBRELLA using each letter at most once?

(i)                 If each letter must be used,

{U,M,B,R,E,L,A}

n=7

SO

NO’S OF WORD IS = 7!

7!= 7X6X5X4X3X2X1= 5040 WORD ( in the umbrella letter L two time be we count once time)  

(ii) If some or all the letters may be omitted

=P(7,0) + p(7,1)+P(7,2) + p(7,3)+P(7,4) + p(7,5)+P(7,6) + p(7,7)

= 1 + 7 + (7x6) + (7x6x5) + (7x6x5x4) +(7x6x5x4x3) +(7x6x5x4x3x2) +(7x6x5x4x3x2x1)

=1+7+42 +210+840 +2520 + 5040+5040

= 13700

(b) Show using truth that:

(p -> q) -> q ⇒ p ⋁ q

P	q	p -> q	(p->q)->q	p or q
TRUE	FALSE	FALSE	TRUE	TRUE
TRUE	TRUE	TRUE	TRUE	TRUE
FALSE	FALSE	TRUE	FALSE	FALSE
FALSE	TRUE	TRUE	TRUE	TRUE

(c) Explain whether (p -> q) -> (q -> r) is a tautology or not. 

p	q	r	p -> q	q -> r	(p -> q) - > (q -> r)
TRUE	TRUE	FALSE	TRUE	FALSE	FALSE
TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
TRUE	FALSE	FALSE	FALSE	TRUE	TRUE
TRUE	FALSE	TRUE	FALSE	TRUE	TRUE
FALSE	TRUE	FALSE	TRUE	FALSE	FALSE
FALSE	TRUE	TRUE	TRUE	TRUE	TRUE
FALSE	FALSE	FALSE	TRUE	TRUE	TRUE
FALSE	FALSE	TRUE	TRUE	TRUE	TRUE

Not a Tautology 2 value is false

(d) Prove that: 1 + 2 + 3 + . . . + n = ½n(n + 1) using mathematical induction.

6. (a) How many ways are there to distribute 15 district objects
into 5 distinct boxes with:

(i) At least three empty box.

            C(5,3) X P(15,2)X2¹³  +  C(5,4) X P(15,1)X1¹⁴  + C(5,5) X P(15,0)X0¹⁵ 

(ii) No empty box.

                C(15,5) x 5⁵ x5!

 

(b) Explain principle of multiplication with an example. 

 

    check on google....
 

 

(c)

Set A,B and C are:

A = {1, 2, 3,5, 8, 11 12,13},

B = { 1,2, 3 ,4, 5,6 } and

C { 7,8,12, 13}.

Find A∩ B U C , A U B U C, A U B ∩ C and (B~C)

A∩ B U C ={1,2,3,5,7,8,12,13}

 A U B U C={ 1,2,3,4,5,6,7,8,11,12,13}

A U B ∩ C= {8,12,13}

(B~C)= { 1,2, 3 ,4, 5,6 }

 

(d) Out of 30 students in college 15 takes art courses, 8 takes
biology courses and 6 takes chemistry. It is also known
that 3 students take all the three courses. Show that 7 or
more students taken none of the course.

7. (a) Explain principle of duality with example?

sol

check on google....
(b) What is power set? Write power set of set
A={1,2,3,4,5,6}.
 

In mathematics, the power set (or powerset) of any set S is the set of all subsets of S, including the empty set and S itself.

 power set of A = {{123456}{12345}{23456}{34561}...................................{ }} 

2 ^6 subset 

= 64 subset (dear friend make a posibal 64 set in your answer sheat)

 (c) What is a function? Explain domain and range in context
of function with example.
        A function is a rule which relates the values of one variable quantity to the values of another variable quantity, and does so in such a way that the value of the second variable quantity is uniquely determined by (i.e. is a function of) the value of the first variable quantity.

 

 (d) State and prove the Pigeonhole principle. 

 

check on google...

 

8. (a) Find inverse of the following functions

 

(b) Explain circular permutation with the help of an example. 

check on google.......

 

(c) What is indirect proof? Explain with an example. 

check on google....

 

(d) What is Boolean algebra?

check on google.....

thanks