1. ## Permutations project help

I have to do a few questions regarding 'permutations' and i don't know the answers!!!! now i know you people aren't here to do my homework for me but hopefully some better mathematicians than myself can give me some guidance.

let X = {x1, x2, x3 all the way up to xn} be a finite set with |X|=n>1 (duno what this means and dunno what a set is????) and let Sx denote the set of all bijective mappings f:X->X often we shall suppose that X=(1,2,3..n) so thats Sn for Sx???? (what?)
each f E Sx is convenitently represented by a standaed array of the form

so i dont even get what the queestion IS let alone how to solve it, please help!

2. Originally Posted by mathcore
I have to do a few questions regarding 'permutations' and i don't know the answers!!!! now i know you people aren't here to do my homework for me but hopefully some better mathematicians than myself can give me some guidance.

let X = {x1, x2, x3 all the way up to xn} be a finite set with |X|=n>1 (duno what this means and dunno what a set is????) and let Sx denote the set of all bijective mappings f:X->X often we shall suppose that X=(1,2,3..n) so thats Sn for Sx???? (what?)
each f E Sx is convenitently represented by a standaed array of the form

so i dont even get what the queestion IS let alone how to solve it, please help!

What are you studying and what's your maths level? Your questions make it clear that you don't have the required level to understand all the words of the problem, leave alone to understand the problem and a solution to it, so I doubt you'd understand any hints given to you ...

Tonio

3. its part of the math shield but its uni level

4. Originally Posted by mathcore
its part of the math shield but its uni level

I suggest one of three things:

1. if you have notes, read them.

2. if you don't have notes, look up permutations on wikipedia, and see if you can find any of the references in a library.

3. if you can't find any of those books in your library, find any book which has the words group' and theory' in the title. Permutation groups are fundamental examples of groups, so any basic book on groups will cover them.

Don't try to answer the problem yet, just try to understand it!

Also, what is `the math shield'?

5. i have the question sorry
i understand part of the math
math shield is a local qualification, dont worry about that, its just we have uni level questions to do, its part of the company i work fors policies

anyway i think what i wrote first was just the explanation part, i dont think its a question, because the question is:

find a standard array for the following:

gf
g^-1
(gf)^-1
f^-1g^-1

f:
(1 2 3)
(2 3 1)

g:
(1 2 3)
(2 1 3)

the gf means composite functions.

the answer i get for gf is:

gf:
(1 2 3)
(1 3 2)

these are bijective mappings i am denoting the mapped value beneath the original value. so is my gf correct, can anyone tell me please?

but for the inverse ones, i dont know what to do and i have read a lot and i am struggling to find out so can anybody please tell me what to do? i hope you follow my notations...

EDIT wait since g is a transposition does that mean g=g^-1? is this a general rule? i worked it out as such but i dont want to make a mistake in the exam.

6. Originally Posted by mathcore
anyone? please? this is kind of an emergency...
To say that $f = \left( {\begin{array}{*{20}c}
1 & 2 & 3 & 4 & 5 \\ 3 & 4 & 1 & 2 & 5 \\ \end{array} } \right)$
is a permutation means it is an onto mapping from a finite set to itself.
In this case a set of five where $f(1)=3,~ f(2)=4,~ f(3)=1,~ f(4)=2,~ f(5)=5$.
In this example we have four active elements and one inactive, $5$.

Here is the part you are not going to like: different authors define multiplication/composition of permutations in different ways.

Here what I mean: if $g = \left( {\begin{array}{*{20}c}
1 & 2 & 3 & 4 & 5 \\ 5 & 3 & 4 & 2 & 1 \\ \end{array} } \right)$

Some authors, partially older authors, say that $fg = \left( {\begin{array}{*{20}c} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 5 & 3 & 1 \\ \end{array} } \right)$, working left to right as with reading.

Whereas others use function composition so that $f \circ g
= \left( {\begin{array}{*{20}c} 1 & 2 & 3 & 4 & 5 \\ 5 & 1 & 2 & 4 & 3 \\ \end{array} } \right)$
, working right to left as with functions.

In any case the inverse works like this $g^{-1} = \left( {\begin{array}{*{20}c} 1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 2 & 3 & 1 \\ \end{array} } \right)$, turn it on its head.

7. ok thanks but basically youve just told me everything that i DO know and not what i asked...

8. Originally Posted by mathcore
ok thanks but basically youve just told me everything that i DO know and not what i asked...
You understand there are different way of defining the operations?
Which are you using?

9. sorry didnt realise thats what you were getting at, i am using right to left aka fg is apply g then apply f just like a composite function f(g(x))
i got these resultsbased on what i posted above:

gf=
(1 2 3)
(1 3 2)

g^-1=
(1 2 3)
(2 1 3)

(gf)^-1=
(1 2 3)
(3 2 1)

and f^-1g^-1 i don't know how to do at all...

also i asked is it a general rule that if you have a transposition lets call g then g=g^-1 (g = inverse g)???? since the two cancel each other out youre just mapping each part of the transposition back

10. Using $f = \left( {\begin{array}{*{20}c} 1 & 2 & 3 \\ 2 & 3 & 1 \\\end{array} } \right)~\&~ g=\left( {\begin{array}{*{20}c} 1 & 2 & 3 \\ 2 & 1 & 3 \\\end{array} } \right)$
Then $f\circ g= \left( {\begin{array}{*{20}c} 1 & 2 & 3 \\ 3 & 2 & 1 \\\end{array} } \right)$

And $g^{-1}= \left( {\begin{array}{*{20}c} 1 & 2 & 3 \\ 2 & 1 & 3 \\\end{array} } \right)=g$ its own inverse.

While $f^{-1}= \left( {\begin{array}{*{20}c} 1 & 2 & 3 \\ 3 & 1 & 2 \\\end{array} } \right)$

So is this true $\left( {f \circ g} \right) ^{-1}= g^{ - 1} \circ f^{ - 1} = g \circ f^{ - 1}?$

11. i dont know if its true its what im asking

i have done my questions tried to solve answers but i need to know if they're correct:

i get: gf = (gf)^-1 = f^-1g^-1, all three of those the same (1,3,2) but i dont know if its right

edit: yes i believe those 3 are equal sorry i dont know why youre asking me though im the one who need help

12. Look this is a standard theorem.

So is this true $\left( {f \circ g} \right) ^{-1}= g^{ - 1} \circ f^{ - 1}$.

That is true for all permutations regardless of the type of operational definition,

13. WHY ARE YOU ASKING ME? why do you keep saying "is this true?"... I DNT KNOW! im the one asking the question... stop asking me is it true... i dont like it that way.

14. Originally Posted by mathcore
WHY ARE YOU ASKING ME? why do you keep saying "is this true?"... I DNT KNOW! im the one asking the question... stop asking me is it true... i dont like it that way.
I have kept this thread open, against my better judgement for some time now, to see how things would go. I was particularly concerned with the following remark you made:
math shield is a local qualification, dont worry about that, its just we have uni level questions to do, its part of the company i work fors policies
Well, I do worry about things like this. It gives me the impression that this question is part of some sort of assessment. Clearly you are out of your depth with this question. It is not our job to give explicit solutions, especially to questions that count towards an assessment.

Several members have been more than patient in providing help with this question. You should be grateful, not resentful. If the help is not to your liking, then that only reinforces my view that you are out of your depth. You will qualify or not, on your own merit, not the merits of others.

15. Originally Posted by mathcore
WHY ARE YOU ASKING ME? why do you keep saying "is this true?"... I DNT KNOW! im the one asking the question... stop asking me is it true... i dont like it that way.
You are replying to someone with the user name plato. That makes me think of Socrates, so perhaps it's part of a Socratic dialog (this is a rhetorical question)? You are meant to play your part by trying to answer or other wise finding the answer.

CB