DISCRETE MATHS

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Course Code : MCS-013Course Title : Discrete MathematicsAssignment Number : MCA(1)/013/Assign/2014-15Assignment Marks : 100Weightage : 25%Last Dates for Submission : 159

thOctober,

2014 (For July 2014 Session)15thApril, 2015 (For January 2015 Session)

There are eight questions in this assignment, which carries 80 marks. Rest 20marks are for viva-voce. Answer all the questions. You may use illustrations anddiagrams to enhance the explanations. Please go through the guidelinesregarding assignments given in the Programme Guide for the format ofpresentation.

(i)answer:p ~P q ~Q r ~R Q^~r ~pV~q (Q^~r)^(~pV~q) P(Q^~r)^(~pV~q)T F T F T F F F F TT F T F F T T F F TT F F T T F F T F FT F F T F T F T F FF T T F T F F T F FF T T F F T T T T TF T F T T F F T F FF T F T F T F T F F

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(ii)answer:p ~P q ~Q r ~R ~r^q ~pVr (~r^q)^(~pVr) ~p(~r^q)^(~pVr)T F T F T F F T F TT F T F F T T F F TT F F T T F F T F TT F F T F T F F F TF T T F T F F T F FF T T F F T T T T TF T F T T F F T F FF T F T F T F T F F

AB ={1,3,5,7}( AB) C =[ (AB)-C]U[C-( AB)]

={5,7}U{2,10,12,15,45,57}={2, 5,7, 10,12,15,45,57}

(I)ANSWER:{ALL OF X}{ANY ONE OF Y}{ALL OF Z}(II)ANSWER: :{ALL OF X}{ANY ONE OF Y}{ANY ONE OF OF Z}







WE HAVE P(n):2N >=N+1THEREFORE, P(1):21 >=1+1=2, WHICH IS TRUEWE ASSUME THAT P(k) IS TRUE THAT IS2K >=K+1..(1)WE WISH TO PROOVE THAT P(k+1) IS TRUE.THAT IS 2K+1 >=K+1+1NOW MULTIPLY 2 BOTH THE SIDE







2 * 2K+1 >=2(K+1+1)2K+2 >=2K+4..THERE FORE P(K+1) TRUESO IT IS TRUE FOR ANY NATURAL NUMBER N

Assume that 17 is rational. That means it can be written in the form of a/b, where a and b aretwo coprime integers (if they had any factors in common, then we could just reduce the fractionand get a new a/b where a and b are coprime). Then:

17 = a/b17 = (a/b)^217b^2 = a^2

Since a^2 is a multiple of 17, and 17 is a prime number, that means 17 is a prime factor of a^2.But since 17 is prime, this also means that it has to be a prime factor of 'a' too. So let a = 17c forsome other integer c.

17b^2 = (17c)^217b^2 = (17)^2 c^2b^2 = 17c^2

By the same reasoning as before, b^2 is a multiple of 17 and so b has to be a multiple of 17 too.But if a and b are both multiples of 17, then they can't be coprime, which contradicts our initialassumption. So 17 must be irrational.







EXAMPLES:







PURE BATSMAN =20PURE BOWLER =10ALL ROUNDER =40WICKET KEEPER AND BATSMAN = 3

i) How many players can either bat or bowl?HERE FOUR PROCESS ARE INDEPENDENT SO20+10+40+3=73

ii) How many players can bowl?=PURE BOWLER +ALL ROUNDER=10+40=50

ii) How many players can bat?=PURE BATSMAN+ALL ROUNDER+WICKETKEEPERAND BATSMAN

=20+40+3=63

TAUTOLOGY IS ONLY OCCUR WHEN FINAL RESULT IN TRUTH TABLE ARE ONLYAND ONLY IN TRUE FORM

P Q ~P ~Q PQ (PQ)^Q ~P^~Q [(PQ)^Q](~P^~Q)

T T F F T T F FT F F T F F F TF T T F T F F TF F T T T F T TIT IS NOTTAUTOLOGYBECAUSEONE F ISAVAILABLE







DUALITY OF

IS.(X+YZ) ^ (XY+Z) ^ (XY+Z) ^ (XY+Z)=(X+YZ) ^ (XY+Z) ^ (X+YZ) ^ (XY+Z)=(XXY + XZ+XYYZ)(XXY+XZ+XYYZ+YZZ)







(FOG)(X)= LOG(G(x))=LOG eX

=X(GOF)(X)=eX

=eLOG X=X

c) Explain inclusion-exclusion principle with example. (2 Marks)www.vijay-jotani.weebly.com






Now f(x1) = f(x2)

3(x1)2 + 5 = 3(x2)2 +53(x1)2 = 3(x2)2

3(x1)2 - 3(x2)2 =0 (x1)2 - (x2)2 =0 (x1) = (x2)there fore it is one-one function

Y=3(x)2 +5







X = y-5/3]1/3

It isnt onto

INVERSE FUNCTIONX = y-5/3]1/3

Question 7a) Find how many 4 digit numbers are odd? (2 Marks)Since it cannot equal 0, there are 9 possibilities for the 1st digit.The 2nd and 3rd digits can be any number from 0-9 so there are 10possibilities for them.

For the number to be odd, there are 5 possibilites for the 4th digit:1,3,5,7,9So the total probability is 9*10*10*5 = 4500

b)

How many different 10 professionals committees can be formedeach containing at least 2 Project Delivery Managers, at least 2Technical Architects and 3 Security Experts from list of 10 ProjectDelivery Managers 12 Technical Architects and 5 Security Experts?(4 Marks)

5 project delivery manager and 2 technical architect and 3 security expert =10c2*12c2*5c2www.vijay-jotani.weebly.com






+4 project delivery manager and 2 technical architect and 4 security expert =10c2*12c2*5c2+3 project delivery manager and 2 technical architect and 5 security expert =10c2*12c2*5c2+2 project delivery manager and 5 technical architect and 3 security expert =10c2*12c2*5c2+3 project delivery manager and 4 technical architect and 3 security expert =10c2*12c2*5c2+4 project delivery manager and 3 technical architect and 3 security expert =10c2*12c2*5c2c)

Explain concept of permutation with an example. How it is differentfrom combination, explain with an example?(4 Marks)

Permutation : Permutation means arrangement of things. The word arrangement is used, if the order ofthings is considered.

Combination: Combination means selection of things. The word selection is used, when the order of thingshas no importance.

Example: Suppose we have to form a number of consisting of three digits using the digits 1,2,3,4, To formthis number the digits have to be arranged. Different numbers will get formed depending upon the order inwhich we arrange the digits. This is an example of Permutation.

Now suppose that we have to make a team of 11 players out of 20 players, This is an example of combination,because the order of players in the team will not result in a change in the team. No matter in which order we listout the players the team will remain the same! For a different team to be formed at least one player will have tobe changed.

Now let us look at two fundamental principles of counting:







Addition rule : If an experiment can be performed in n ways, & another experiment can be performed in mways then either of the two experiments can be performed in (m+n) ways. This rule can be extended toany finite number of experiments.

Example: Suppose there are 3 doors in a room, 2 on one side and 1 on other side. A man want to go outfrom the room. Obviously he has 3 options for it. He can come out by door A or door B or door C.

Multiplication Rule : If a work can be done in m ways, another work can be done in n ways, then both of theoperations can be performed in m x n ways. It can be extended to any finite number of operations.

Example.: Suppose a man wants to cross-out a room, which has 2 doors on one side and 1 door on othersite. He has 2 x 1 = 2 ways for it.

Factorial n : The product of first n natural numbers is denoted by n!.

n! = n(n-1) (n-2) ..3.2.1.

Ex. 5! = 5 x 4 x 3 x 2 x 1 =120

Note 0! = 1

Proof n! =n, (n-1)!

Or (n-1)! = [n x (n-1)!]/n = n! /n

Putting n = 1, we have

O! = 1!/1

or 0 = 1







What's the Difference?

In English we use the word "combination" loosely, without thinking if the order of things isimportant. In other words:

"My fruit salad is a combination of apples, grapes and bananas" We don'tcare what order the fruits are in, they could also be "bananas, grapes and apples"or "grapes, apples and bananas", its the same fruit salad.

"The combination to the safe was 472". Now we do care about the order. "724"would not work, nor would "247". It has to be exactly 4-7-2.

So, in Mathematics we use more precise language:

If the order doesn't matter, it is a Combination.

If the order does matter it is a Permutation.







b)

How many words can be formed using letter of STUDENT using







each letter at most once:

i) If each letter must be used,ii) If some or all the letters may be omitted.

7p1*7p2*7p3*7p4*7p5*7p6*7p7=7+ 7*6+ 7*6*5+ 7*6*5*4+ 7*6*5*4*3+ 7*6*5*4*3*2+ 7*6*5*4*3*2*1=245+840+2520+5040+5040=13685(ii)answer:7p1*7p2*7p3*7p4*7p5*7p6=7+ 7*6+ 7*6*5+ 7*6*5*4+ 7*6*5*4*3+ 7*6*5*4*3*2=245+840+2520+5040=8645

p q pq qp (pq)v(qp)T T T T TT F F T TF T T F TF F T T T

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DISCRETE MATHS

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