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______

CourseCode

CourseTitleAssignmentNumber

Assignment Marks

WeightageLastDates forSubmission :

Thereare eight questions inmarks areforviva-voce. An

diagrams to enhancetheexp

assignments given in thePr

Question 1:

a.Maketruth tableforfollowin

~p (q

~

ii) ~p ~r

b.What areconditional connewith an example.

___ __________

: MCS-013

: DiscreteMathematics: MCA(1)/013/Assign/201

: 100

: 25%15th October,2012 (ForJuly 2012 S

15th April, 2013 (ForJanuary 2013 S

this assignment, which carries 80 marks. Rewerall thequestions. You may useillustratio

lanations. Pleasego through theguidelines re

gramme Guidefortheformat of presentation

gs: i)

r) _ p _

q

Marks (

q _ ~p _

tives? Explain useofconditional connectives

ession)

ession)

st 20s and

arding

+ 3 +4)

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______

c.Writedown suitable mathematicfollowingsymbolic properties.

i)

ans :(_x) ( _

___ __________

al statement that can berepresented bythe

y) (_ z) P

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(there exist for only x statement)(for all of Y statement)(for all of Z statement)

ii) (x) ( y) ( z) P

[there exist for only(for X)](for all of Y statement)(for all of Z statement)

Question 2: Marks (3 + 3+3)

a.What is proof? Explainmethod of direct proofwith thehelp of one example.

In mathematics,a proofis a demonstration that if some fundamental statements (axioms) are

assumed to be true, then some mathematical statementis necessarily true.[1][2] Proofs are obtained

fromdeductive reasoning, rather than from inductiveorempiricalarguments; a proof must

demonstrate that a statement is always true (occasionally by listingallpossible cases and showing

that it holds in each), rather than enumerate many confirmatory cases. An unproven proposition that

is believed to be true is known as a conjecture.Proofs employ logicbut usually include some amount

of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in

written mathematics can be considered as applications of rigorous informal logic.

Purely formal proofs, written in symbolic language instead of natural language, are considered in proof

theory. The distinction betweenformal and informal proofshas led to much examination of current and

historical mathematical practice,quasi- empiricism in mathematics, and so-calledfolk mathematics (in

both senses of that term). Thephilosophy of mathematicsis concerned with the role of language and logic

in proofs, and mathematics as a language.

Direct proof

Main article: Direct proof

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier

theorems.[11] For example, direct proof can be used to establish that the sum of two even integers i

always even:

Consider two even integersxandy. Since they are even, they can bewritten asx=2aandy=2brespectively

for integersaandb. Then thesum. From this it is clearx+yhas 2 as afactor and therefore is even, so the sumof any two even integers is even.

This proof uses definition of even integers, as well as distribution law.

b.Show whether 11is rational or irrational.

Suppose sqrt(11) were rational.

Then, sqrt(11) = p/q for some integers p and q.

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(Assume the fraction p/q has been reduced to lowest terms.)

Now, 11 = p^2/q^2. And so 11*q^2 = p^2.

This means that p^2 is divisible by 11. Hence, p is also divisible by 11. And so

p^2 is divisible by 11*11 = 121.

Since 11*q^2 = p^2, we get that q^2 = p^2/11.

Since p^2 is divisible by 121, we can divide q^2 by 11 without remainder. That is,

q^2 is divisible by 11.

And so, q is divisible by 11.

In summary, we have learned that both p and q are divisible by 11.

This is a contradiction, because p/q was assumed to be in lowest terms. Hence, sqrt(11) is irrational.

c. Provethat A -(A - B) : A B(because A-B=A-(A B))

=A-[ A-(A B)]

=A-A+(A B)

=(A B)

Question 3: Marks (4 + 4+ 4)

a. Set X has 10 members, how manymembers do P(X) has ? Howmanymembers ofP(X)areproper

subset of X?

the number of members of ~P is 2^10 or 1024. This is the number of members in

the set of subsets.

That number could be 1023 or 1022 depending on whether you count the empty set. X

is not a proper subset of itself A proper subset of X is just (by definition) a subset of X

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which is not equal to X So th

and it is a proper subset of eve

Q(X) the set of all proper s

empty set.

b. Establish the equivalencec. If p an q arestatements, s

is a tautologyornot.

Question 4:

a.Makelogiccircuit forthe fo

i.(x.y+z) +(x+y+z

ii) (x'+y).(y+ z).(

answer is 1023. The empty set is a subset of e

ry set except of itself. If P(X) is the power set o

bsets, then |Q(X)|=|P(X)-1|. one member, na

: (P _Q) _ (P _ Q) _ (~P +Q) _ (Q _ P)

ow whetherthestatement [(~pq) _(q)] (p

llowing Boolean expressions:

) +(x+y.z)

.z+x)

1

Marks (

ery set,

X, and

ely the

_ q)

+ 3 +3)

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b. What is dual of aBooleanfollowinglogiccircuit

expression? Find dual of boolean expression o the output of the

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c. Set A,Band C are: A ={1thefollowings

i.A-(A-B)

Ans:

{1, 2, 3, 4

={1, 2, 3,

={1, 2,5}

ii. A B C

iii. A\B

Question 5:

a.Draw aVenn diagram to re

i. (A _B)

, 2, 3, 4, 5,6,9,19,15}, B ={ 1,2,5,22,33,99 } an

, 5,6,9,19,15}-({1, 2, 3, 4, 5,6,9,19,15}-{ 1,2,5,

, 5,6,9,19,15}-(3, 4, 6,9,19,15)

resent followings:

(CB)

1

Marks (

C { 2, 5,11,19,15}. Find

2,33,99 })

+4 +2)

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c.Explain the concept of cou

Question 6:a.What is inclusion-exclusio

terexamplewith thehelp of an example.

principle? Explain inclusion-exclusion princip

1

arks (5+4)

le with an example.

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Find inverseof thefollowingfunctionsi)

f(x) =

ii)f(x) =

x3

__

_

5

x_

3x3 ___

7

x2 _

4

x _ 3

x __ 2

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Question 7:

a. Find how many3 digit nu______&_!'

__!%_)____(__&"_#

#____%_______

_&_&___#"%_&_"!_

___!"&___

"&__$)_%___&_)__

_&_&___#"%_&_________*__ )_

_&_&___#"%_&_

______"$______%"__&"&___!"__"

_"&___!"__"__&_

Howmanydifferent 20 persons c

Associate Professorfrom aset of

Ans.2 professors and 18 associat3professors and 17 associate

4professors and 16 associate5professors and 15 associate

6professors and 14 associate

7professors and 13 associate8professors and 12 associate

9professors and 11 associate

10 professors and 10 associa

TOTAL==============

c. Provethat foreverypositiv

Marks (

mbers areeven? How many3 digit numbers arec

__$%_!'

'&_________&__&__

_______$%&_#______)____!_

_!"&_________&_!' __$__%"__

!_"_____%__"!__#______)____!%___

!_"_____&__$__#______)____!

_*__)_*%_______&_____ _______

_______&__(_!_!' __$%____

mmittees can beformed eachcontainingat least

10 Professors and 42 AssociateProfessors.

professor = C(10,2)*C(42*18)=45*

professor = C(10,3)*C(42*17)=

professor = C(10,4)*C(42*16)=professor = C(10,5)*C(42*15)=

professor = C(10,6)*C(42*14)=

professor = C(10,7)*C(42*13)=professor = C(10,8)*C(42*12)=

professor = C(10,9)*C(42*11)=

e professor = C(10,10)*C(42*10)=

======================_______way

integer n, n3 +n is even.

+ 3 + 3)

omposed ofodd digits.

__$_

'&__!*_____&_

___)_*%

_#'&__

#'&__

Professors and at least 3

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Question 8:

a.What is Demorgan_s Law?

b. Howmanyways arethereti.At least two em

ii) No emptybox.

c.In afiftyquestion true falsee

student answerrandomlywha

Marks (

Explain the use ofDemorgen_s lawwith an exa

distribute10 district object into 4 distinct boxtybox.

xamination astudent must achieve twentyfivec

is theprobabilitythat student will fail.

+4 +2)

ple?

s with

rrect answers to pass. If