Sets : Relation And Function Assignment

3.1Question1

Letusconsiderthefollowingquestionwithdifferentsolutions.Choosethebestsolutionandexplainwhyyouchooseit.

Problem:Provethattheproductoftwooddnumbersisalsoanoddnumber.

• Anintegerisanoddnumberifitistwotimesanotherintegerplus1.Thereforethe product of two odd numbers is four times the product of two other integers plus twotimes the sum of the two other integers plus one. This shows that the product of two oddnumbersisodd
• m=2k+1,n=2l +1

mn=4kl+2(k+l)+1=2(2kl+k+l)+1.

• Supposenandmaretwooddnumbers,thenm= 2k+1andn=2l+1forsome

k,l?Z.Wehave

mn=(2k+1)(2l+1)

=4kl+2k+2l+1

=2(2kl+k+l)+1

=2q+1,

whereq=2kl +k+l.

Wecanseethatq=2kl+k+l Zask,l Z.Therefore,mn=2q+1isanoddnumber.Inotherwords,theproductoftwooddnumbersisalsoodd.

• Supposenandmaretwooddnumbers,thenm=2k+1andn=2l+1forsomeintegerskandl.Wehave

mn=(2k+1)(2l+1)=4kl+2k+2l+1=2(2kl+k+l)+1=2q+1, where q=2kl+k+l.

Wecanseethatqisanintegerask,l Z.Therefore,mn=2q+1isanoddnumber.Inotherwords,theproductoftwooddnumbersisalsoodd.

3.2Question2

Considerthefollowingproblemanditssolution.

• Problem:CalculateA¹⁰⁰where

A= 0 1 .

?1 ?1

• Solution:Byastraightforwardcalculation,weobtain

A100 = 0 1 .

?1 ?1

ThissolutiondoesnotexplainhowwecalculateA¹⁰⁰.Therefore,thissolutionwillbemarkedasinsufficientexplanation.

Canyouprovideanexplanationtothissolutiontomakeitcomplete?

3.3Question3

Beforeusingatheoremorlemma,makesureyou(carefully)checkandstateifallconditionsofthetheoremorlemmaareactuallysatisfied.

Letf(x)=tanx.Inthefollowingstatements,choosethecorrectstatement.Brieflyexplainyouranswer.

• Sincef(0)=f(?)=0,byRollestheoremthereexistsc?(0,?)suchthatf^?(c)=
• Calculatingthederivativeoff,wehave

f^?(x)= 1cos2(x)

=1+tan²x.

Wenotethatovertheinterval[0,?],fisnotdefinedatx=?/2. As1 +tan²x>0forall

x?[0,?]andx/=?/2,theredoesnotexistc?(0,?)suchthatf^?(c)=0.

• Both(i)and(ii)are
• Both(i)and(ii)are

3.4Question4

Considerthefollowingstatementandexplanation.Canyouconcludethatthestatementistrue?

• Statement:Ifpisprimethen2^p?1isalso
• Explanation:Thestatementholdsforp=2,3,5, Therefore,thestatementistrue,i.e.,ifpisprimethen2^p?1isalsoprime.

3.5Question5

Considerthefollowingproblemandsolution.Isthesolutiongivenbelowcorrect?Ifnot,canyougiveacorrectsolutionfortheproblem?

• Problem:ShowthatifAisasquarematrix,thenA+A^Tis
• Solution:ForthecasewhereAisa11matrix,letA=[a]. Wehave

A+A^T=[a]+[a]=[2a],

whichissymmetric.

ForthecasewhereAisa22matrix,let

A= a b.

c d

Wehave

A+AT = a b+ a c= 2a b+c,

whichissymmetric.

c d b d

b+c 2d

Therefore,ifAisasquarematrix,thenA+A^Tsymmetric.

3.6Question6

Considerthefollowingproblemandchooseallthevalidsolutionsfromtheoptionsgivenbelow.

• Problem:Provethatifab=acanda/=0thenb=cfora,b,c?R.