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Ques-8-Find-the-signum-sign-or-sgn-of-the-permutation-12345678-Hint-for-any-permutation-take-sgn-1-if-is-odd-1-if-is-even-Ques-9-Prove-that




Question Number 193871 by Mastermind last updated on 21/Jun/23
Ques. 8        Find the signum (sign or sgn) of the  permutation θ=(12345678).  Hint : for any permutation β, take  sgn β = {_(−1       if β is odd) ^(1            if β is even)       Ques. 9       Prove that ∣S_n ∣ = n!.    Ques. 10        Provd that for b∈S_n , sgn b = sgn b^(−1)  .
Ques.8Findthesignum(signorsgn)ofthepermutationθ=(12345678).Hint:foranypermutationβ,takesgnβ={1ifβisodd1ifβisevenQues.9ProvethatSn=n!.Ques.10ProvdthatforbSn,sgnb=sgnb1.
Commented by talminator2856792 last updated on 23/Jun/23
  what is source of these questions.      or book name.
whatissourceofthesequestions.orbookname.
Answered by Rajpurohith last updated on 22/Jun/23
(8) Clearly θ=(18)(17)(16)(15)(14)(13)(12)  which shows θ is an odd permutation.  hence sgn(θ)=−1.   ■  (9) S_n  denotes the set of all permutations of  a set of n elements say A={1,2,3,...n}.  [you know that S_n  forms  a group under function composition.]  let σ∈S_n  , so σ is a bijection.  How many choices do you have for σ(1)?  You have n choices.  And for σ(2)?  It must be (n−1) choices.  because you cannot choose the one you choosed  for σ(1).  [Suppose you choose σ(2)=σ(1) ,  ⇒2=1 as σ is bijective.]  so you cannot.  similarly there are (n−2) choices for σ(3).  .  .  .  There are 2 choices for σ(n−1)  And only one choice for σ(n).  By fundamental principle of counting,  we have n(n−1)(n−2)...2.1=n! such permutations  on n letters and they are the only permutations.  ⇒∣S_n ∣=n!        ■    (10)Let β∈S_n . Suppose β is even .  ⇒βoβ^(−1) =Id  ⇒sgn(βoβ^(−1) )=sgn(Id)=1  Suppose β^(−1 ) is odd   ⇒βoβ^(−1)  is odd ⇒sgn(βoβ^(−1) )=−1, a contradiction.  Hence β^(−1 )  is not odd  ⇒β^(−1)  must be even.  ⇒sgn(β^(−1) )=1=sgn(β)     Similar argument for odd.      ■
(8)Clearlyθ=(18)(17)(16)(15)(14)(13)(12)whichshowsθisanoddpermutation.hencesgn(θ)=1.◼(9)SndenotesthesetofallpermutationsofasetofnelementssayA={1,2,3,n}.[youknowthatSnformsagroupunderfunctioncomposition.]letσSn,soσisabijection.Howmanychoicesdoyouhaveforσ(1)?Youhavenchoices.Andforσ(2)?Itmustbe(n1)choices.becauseyoucannotchoosetheoneyouchoosedforσ(1).[Supposeyouchooseσ(2)=σ(1),2=1asσisbijective.]soyoucannot.similarlythereare(n2)choicesforσ(3)....Thereare2choicesforσ(n1)Andonlyonechoiceforσ(n).Byfundamentalprincipleofcounting,wehaven(n1)(n2)2.1=n!suchpermutationsonnlettersandtheyaretheonlypermutations.⇒∣Sn∣=n!◼(10)LetβSn.Supposeβiseven.βoβ1=Idsgn(βoβ1)=sgn(Id)=1Supposeβ1isoddβoβ1isoddsgn(βoβ1)=1,acontradiction.Henceβ1isnotoddβ1mustbeeven.sgn(β1)=1=sgn(β)Similarargumentforodd.◼
Commented by Mastermind last updated on 22/Jun/23
Thank you so much
Thankyousomuch
Commented by York12 last updated on 22/Jun/23
sir I am facing a lot of problems with combinatorics  what do you recommend to read
sirIamfacingalotofproblemswithcombinatoricswhatdoyourecommendtoread
Commented by Rajpurohith last updated on 22/Jun/23
I would better suggest you to study  one book properly as a course.
Iwouldbettersuggestyoutostudyonebookproperlyasacourse.
Commented by talminator2856792 last updated on 23/Jun/23
  what book do you recommend.
whatbookdoyourecommend.
Commented by York12 last updated on 23/Jun/23
what book do you recommend sir
whatbookdoyourecommendsir
Commented by Rajpurohith last updated on 24/Jun/23
Actually I did not have a course in it,  But I ′d suggest the book “Introductory  Combinatorics” by Richard .A Brualdi.
ActuallyIdidnothaveacourseinit,ButIdsuggestthebookIntroductoryCombinatoricsbyRichard.ABrualdi.
Commented by York12 last updated on 23/Jul/23
thanks
thanksthanks

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