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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