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Comments:
<0> isn't a permutation just an ordered set though?
<0> or you don't really have such things?
<1> Where (2,3,4) represents the cycle 2 -> 3 -> 4 -> 2
<0> hmm
<1> Well, if the initial set is ordered, then a permutation is a reordering of it.
<1> But you might have something that's not obviously ordered
<1> like {grape, banana, orange, apple}
<1> is orange < apple?
<2> lexicographically, no :p
<1> Hard to say, but you can still have a permutation of that set.
<0> right but for a permutation f of that set, isn't there a notation to write f as an ordered set?
<0> for example, maybe use square brackets or something
<1> You could write it as a tuple.
<0> with brackets
<0> hmm yes
<1> or sure, use square brackets if you want
<1> It doesn't really matter, mathematically.
<0> so (grape, banana, orange, apple) is a permutation of the set {grape,banana, orange, apple}
<0> of course
<1> well...
<1> Not quite, in some formal sense :)
<1> But it could be, if we have some convention for what the slots mean
<3> JohnFlux : you can read it 2 ways just as an ordered set (or as map from an defined original order to the current one)
<2> disjoint cycle notation is so much better though because you can easily see the properties of the permutation
<1> yeah
<1> Almost the first thing you should probably do when you have a specific permutation is write it in disjoint cycle notation, because that will immediately tell you a whole lot of things about it.
<1> For example, its order (the least power k such that composing the permutation with itself k times will give the identity permuation)
<3> mbot is back :)
<0> Cale: wikipedia says a permutation has to be on a finite set. why couldn't you have a permutation on an infinite but countable set?
<1> You can have a permutation on any set
<1> It's just a bijection
<3> JohnFlux :-> is just describing the mapping in general term (domain -> codomai), and to notate the specific mapping of an element of the domain you use |->
<0> kmh: yeah that describes the type
<0> kmh: ah and |-> for a element thanks
<0> so if f(1) =1
<3> so f:X->Y and y=f(x) or x|->y
<0> then f |-> 1 -> 1 ?
<1> flippo: 1 |-> 1
<3> for x in X and y in Y
<1> ugh
<0> hehe
<0> ah: f: 1 |-> 1 i see
<1> Or 1 |-> 1 with a little f over the arrow
<0> so really it's the -> bit that says they are types
<3> that would mean f(1)=1
<1> \mapsto in TeX
<0> so can you have a permutation on an uncountable set?
<0> how would that work?
<3> yes
<0> hmm
<1> It's just a bijection
<0> i guess you could have something that just doubles the number for example
<0> hmm
<1> Do you know what a bijection is?
<3> if you read permutations as maps, you can general is it as bijective map
<0> yeah it's where all the elements in one map to the something else in the other one, uniquely
<0> and vice versa
<1> Yeah, it's a function which has those properties I mentioned above
<0> oh hang on
<1> A function f is injective if whenever f(a) = f(b), you have a = b
<0> say we have a set FRUIT
<0> FRUIT= {banana, grape, etc}
<0> hmm wait
<1> A function f: X -> Y is surjective if for every y in Y, there is some x in X for which f(x) = y.
<0> FRUIT is both a set and a type?
<1> A function is bijective if it is both injective (one-to-one) and surjective (onto)
<1> There's not really any such thing as types.
<3> btw. http://en.wikipedia.org/wiki/Permutation
<0> Cale: ah okay
<0> Cale: so i can treat sets and types as the same thing?
<1> Well, I suppose.
<3> starts with permutation as you know and then the general definition (allowing an infinite amount)
<1> What you've been calling the type of a function is really just a specification of its domain and codomain
<1> which are sets.
<0> if we had a permutation f of FRUIT
<1> for example,
<0> it's type would be f : N -> FRUIT or what?
<1> no
<1> banana |-> grape
<0> hmm
<1> grape |-> orange
<1> orange |-> banana
<0> oh i see
<1> say
<1> f would be a function FRUIT -> FRUIT
<0> that's why it's not an ordered set
<3> f: {F,R,U,I,T} -> {F,R,U,I,T}
<0> I thought permutations were ordered sets. but they aren't at all
<1> kmh: FRUIT = {banana, grape, orange}
<3> oh
<0> so, what do you call a something which was of type f: N-> FRUIT ? :)
<0> so then I can an ordered set of fruit
<0> so, what do you call a something which was of type f: N+-> FRUIT ? :)
<1> I suppose you call it a fruit valued function of natural numbers, or a sequence of fruit.
<0> or do you call it Z in math?
<1> Z is the integers
<1> N is the natural numbers {0,1,2,3,...}
<1> or occasionally {1,2,3,...}
<0> I think in comp. sci. we say N+ for {1,2,3}...
<1> yeah
<3> you can do it this way Fruit -> N, banana |-> 1, orange |-> 2 , ....
<1> N = {0,1,2,3,...} is far more common in mathematics as well
<1> but there are contexts where people will define it the other way because they'd only be using N+ anyhow.
<0> offtopic, but japanese do most of their counting from 1 where we do it from 0
<3> then you get the "cl***ic" permutations of 123
<0> for example, pregnancies are for 10 months
<2> 9 months
<4> :)
<1> are you sure?
<0> and in the old days, you started at age 1
<0> whereas we start at age 0
<3> JohnFlux : actually we count from 1 to in many cases
<1> That would mess up addition, I don't think they really do that.
<4> JohnFlux, weird :)
<0> Cale: in the first month for example, even today they say the woman is one month pregnant
<4> I didn't note that when meeting japaneses :)
<0> whereas we say 0 months
<0> so they give birth at 10 months ;)
<3> well that are language remainder from times before the use of 0
<4> heh
<1> Well, at the end of 9 months, sure
<3> after all the common knowledge of 0 os _younger_ than most languages
<3> os = is
<0> in the parents generation, you would say you were 24 for example when we would say 23
<0> in the parents generation, the japanese would say you were 24 for example when we would say 23
<1> I've seen Japanese people count, and they count the same way as we do, for the most part, except that the numbers are modified based on what you're counting.
<0> Cale: i'm just saying what my jap gf is saying
<3> for instance we also speak of child's first year not his zeroth
<0> she's sitting next to me ;)
<0> kmh: ... no we don't
<3> we do
<0> kmh: oh well "first year" i suppose
<0> but we still count that as 0 years old
<1> Oh, there's a difference between saying 1st year, 2nd year, and saying one year old, two years old, and so on.
<3> yes
<0> and by the 'second year' we probably switch to saying 1 year old
<0> Cale: right, and the japanese tend to use that latter
<1> former, you mean
<1> we use the latter :)
<0> oh wait right
<3> JohnFlux : i'm just saying you partially have these things in western languages as well, depending how (or what in a grammatical sense) numbers you use
<0> bah i'm getting confused now ;)
<2> how do you write the identity permutation of S4 in disjoint cycle notation? my lecturer leaves off cycles of order 1, but that would mean there's nothing left
<3> no the japanse use ordinals as nth, where we often don't
<1> iodine: ()
<2> ok
<2> thankyou
<3> like for describing the age
<0> Cale: looking at wikipedia, it seems the meaning of permutation changes between fields
<1> It doesn't really know
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