Clases de equivalencia

7 Comentarios

1. Usando que dos clases de equivalencia siempre son iguales o disjuntas y que, por tanto, no puede haber un elemento perteneciente a dos clases distintas:

   Haskell

   import Data.List clasesEquivalencia :: Eq a => [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia [] _ = [] clasesEquivalencia ys@(x:xs) r = zs : clasesEquivalencia (xs \\ zs) r where zs = filter (`r` x) ys

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   import Data.List clasesEquivalencia :: Eq a => [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia [] _ = [] clasesEquivalencia ys@(x:xs) r = zs : clasesEquivalencia (xs \\ zs) r      where zs = filter (`r` x) ys

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2. Haskell

   --He necesitado añadir (Eq a) al tipo de la función. import Data.List clasesEquivalencia :: Eq a => [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia [] r = [] clasesEquivalencia (x:xs) r = (x:lista):clasesEquivalencia (xs \\ lista) r where lista = [z | z <- xs, r x z]

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   --He necesitado añadir (Eq a) al tipo de la función.   import Data.List   clasesEquivalencia :: Eq a => [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia [] r = [] clasesEquivalencia (x:xs) r = (x:lista):clasesEquivalencia (xs \\ lista) r   where lista = [z | z <- xs, r x z]

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3. Haskell

   clasesEquivalencia :: [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia [] _ = [] clasesEquivalencia (x:xs) p = (x : [y | y <- xs, p x y]) : clasesEquivalencia [z | z <- xs, not $ p x z] p

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   clasesEquivalencia :: [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia []     _ = [] clasesEquivalencia (x:xs) p =   (x : [y | y <- xs, p x y]) : clasesEquivalencia [z | z <- xs, not $ p x z] p

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4. Haskell

   clasesEquivalencia :: Eq a => [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia xs r = clasesEquivalenciaAux xs r xs [] clasesEquivalenciaAux :: Eq a => [t] -> (a -> t -> Bool) -> [a] -> [a] -> [[a]] clasesEquivalenciaAux [] _ _ _ = [] clasesEquivalenciaAux (y:xs) r ys zs | notElem h zs = rs : clasesEquivalenciaAux xs r ys (h:zs) | otherwise = clasesEquivalenciaAux xs r ys zs where rs@(h:_) = filter (`r` y) ys

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   clasesEquivalencia :: Eq a => [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia xs r = clasesEquivalenciaAux xs r xs []   clasesEquivalenciaAux :: Eq a => [t] -> (a -> t -> Bool)                          -> [a] -> [a] -> [[a]] clasesEquivalenciaAux [] _ _ _ = [] clasesEquivalenciaAux (y:xs) r ys zs   | notElem h zs = rs : clasesEquivalenciaAux xs r ys (h:zs)   | otherwise    = clasesEquivalenciaAux xs r ys zs   where rs@(h:_) = filter (`r` y) ys

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5. Haskell

   import Data.list clasesEquivalencia :: Ord a => [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia xs r = nub [ sort $ partes x xs r | x <- xs] where partes n xs r = nub $ concat [ n:[y] | y <- xs, r n y]

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   import Data.list   clasesEquivalencia :: Ord a => [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia xs r = nub [ sort $ partes x xs r | x <- xs]        where partes n xs r = nub $ concat [ n:[y] | y <- xs, r n y]

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6. Haskell

   import Data.List (partition) clasesEquivalencia :: [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia [] _ = [] clasesEquivalencia (x:xs) r = (x:ys) : clasesEquivalencia zs r where (ys, zs) = partition (r x) xs

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   import Data.List (partition)   clasesEquivalencia :: [a] -> (a -> a -> Bool) -> [[a]] clasesEquivalencia [] _     = [] clasesEquivalencia (x:xs) r = (x:ys) : clasesEquivalencia zs r     where (ys, zs) = partition (r x) xs

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7. En Maxima,

   clasesEquivalencia (xs,r) := equiv_classes (setify (xs),r)$

   1 2	clasesEquivalencia (xs,r) :=   equiv_classes (setify (xs),r)$

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