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


Question and Answers Forum

All Questions Topic List
Others Questions
Previous in All Question Next in All Question
Previous in Others Next in Others
Question Number 193871 by Mastermind last updated on 21/Jun/23
$Ques. 8 Find the signum (sign or sgn) of the permutation θ=(12345678). Hint : for any permutation β, take sgn β = {_(−1 if β is odd) ^(1 if β is even) Ques. 9 Prove that ∣S_n ∣ = n!. Ques. 10 Provd that for b∈S_n , sgn b = sgn b^(−1) .$
$Ques . 8$ $Find the signum (sign or sgn) of the$ $permutation θ = (12345678) .$ $Hint : for any permutation β, take$ $sgn β = {_{- 1 if β is odd}^{1 if β is even}$ $Ques . 9$ $Prove that ∣ S_{n} ∣ = n! .$ $Ques . 10$ $Provd that for b \in S_{n}, sgn b = sgn b^{- 1} .$
Commented by talminator2856792 last updated on 23/Jun/23

$what is source of these questions .$ $or book name .$
	Answered by Rajpurohith last updated on 22/Jun/23
	$(8) Clearly θ=(18)(17)(16)(15)(14)(13)(12) which shows θ is an odd permutation. hence sgn(θ)=−1. ■ (9) S_n denotes the set of all permutations of a set of n elements say A={1,2,3,...n}. [you know that S_n forms a group under function composition.] let σ∈S_n , so σ is a bijection. How many choices do you have for σ(1)? You have n choices. And for σ(2)? It must be (n−1) choices. because you cannot choose the one you choosed for σ(1). [Suppose you choose σ(2)=σ(1) , ⇒2=1 as σ is bijective.] so you cannot. similarly there are (n−2) choices for σ(3). . . . There are 2 choices for σ(n−1) And only one choice for σ(n). By fundamental principle of counting, we have n(n−1)(n−2)...2.1=n! such permutations on n letters and they are the only permutations. ⇒∣S_n ∣=n! ■ (10)Let β∈S_n . Suppose β is even . ⇒βoβ^(−1) =Id ⇒sgn(βoβ^(−1) )=sgn(Id)=1 Suppose β^(−1 ) is odd ⇒βoβ^(−1) is odd ⇒sgn(βoβ^(−1) )=−1, a contradiction. Hence β^(−1 ) is not odd ⇒β^(−1) must be even. ⇒sgn(β^(−1) )=1=sgn(β) Similar argument for odd. ■$
	$(8) C l e a r l y θ = (18) (17) (16) (15) (14) (13) (12)$ $w h i c h s h o w s θ i s a n o d d p e r m u t a t i o n .$ $h e n c e s g n (θ) = - 1 . ◼$ $(9) S_{n} d e n o t e s t h e s e t o f a l l p e r m u t a t i o n s o f$ $a s e t o f n e l e m e n t s s a y A = {1, 2, 3, . . . n} .$ $[y o u k n o w t h a t S_{n} f o r m s$ $a g r o u p u n d e r f u n c t i o n c o m p o s i t i o n .]$ $l e t σ \in S_{n}, s o σ i s a b i j e c t i o n .$ $H o w m a n y c h o i c e s d o y o u h a v e f o r σ (1) ?$ $Y o u h a v e n c h o i c e s .$ $A n d f o r σ (2) ?$ $I t m u s t b e (n - 1) c h o i c e s .$ $b e c a u s e y o u c a n n o t c h o o s e t h e o n e y o u c h o o s e d$ $f o r σ (1) .$ $[S u p p o s e y o u c h o o s e σ (2) = σ (1),$ $\Rightarrow 2 = 1 a s σ i s b i j e c t i v e .]$ $s o y o u c a n n o t .$ $s i m i l a r l y t h e r e a r e (n - 2) c h o i c e s f o r σ (3) .$ $.$ $.$ $.$ $T h e r e a r e 2 c h o i c e s f o r σ (n - 1)$ $A n d o n l y o n e c h o i c e f o r σ (n) .$ $B y f u n d a m e n t a l p r i n c i p l e o f c o u n t i n g,$ $w e h a v e n (n - 1) (n - 2) . . . 2.1 = n! s u c h p e r m u t a t i o n s$ $o n n l e t t e r s a n d t h e y a r e t h e o n l y p e r m u t a t i o n s .$ $\Rightarrow∣ S_{n} ∣= n! ◼$ $(10) L e t β \in S_{n} . S u p p o s e β i s e v e n .$ $\Rightarrow β o β^{- 1} = I d$ $\Rightarrow s g n (β o β^{- 1}) = s g n (I d) = 1$ $S u p p o s e β^{- 1} i s o d d$ $\Rightarrow β o β^{- 1} i s o d d \Rightarrow s g n (β o β^{- 1}) = - 1, a c o n t r a d i c t i o n .$ $H e n c e β^{- 1} i s n o t o d d$ $\Rightarrow β^{- 1} m u s t b e e v e n .$ $\Rightarrow s g n (β^{- 1}) = 1 = s g n (β)$ $S i m i l a r a r g u m e n t f o r o d d . ◼$
	Commented by Mastermind last updated on 22/Jun/23

	$Thank you so much$
	Commented by York12 last updated on 22/Jun/23

	$s i r I a m f a c i n g a l o t o f p r o b l e m s w i t h c o m b i n a t o r i c s$ $w h a t d o y o u r e c o m m e n d t o r e a d$
	Commented by Rajpurohith last updated on 22/Jun/23

	$I w o u l d b e t t e r s u g g e s t y o u t o s t u d y$ $o n e b o o k p r o p e r l y a s a c o u r s e .$
	Commented by talminator2856792 last updated on 23/Jun/23

	$what book do you recommend .$
	Commented by York12 last updated on 23/Jun/23

	$w h a t b o o k d o y o u r e c o m m e n d s i r$
	Commented by Rajpurohith last updated on 24/Jun/23

	$A c t u a l l y I d i d n o t h a v e a c o u r s e i n i t,$ $B u t I^{'} d s u g g e s t t h e b o o k ‘ ‘ I n t r o d u c t o r y$ ${C o m b i n a t o r i c s}^{″} b y R i c h a r d . A B r u a l d i .$
	Commented by York12 last updated on 23/Jul/23

	$t h a n k s$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum