Re: how can I do this functionally?

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg939] Re: how can I do this functionally?
• From: tomi (Tom Issaevitch)
• Date: Wed, 3 May 1995 00:18:23 -0400
• Organization: Wolfram Research, Inc.

```drc at gate.net (David Cabana)  writes:

>Below I define a function, Classify[S_, f_], in an imperative (and
>not particularly efficient) style.  Can anyone tell me how to define
>it either functionally or via pattern matching?
>
>The function Classify takes 2 arguments, f and S.  f is a function,
>and S is a subset of the domain of S.  In particular, S is a list
>of elements of the domain of f, and contains no repeated elements.
>
>The function f induces an equivalence relation on S as follows:
>a is equivalent to b modulo f if and only if f[a] == f[b]. Classify
>returns the equivalence classes of S modulo f.
>
>In[1]:=
>Classify[S_, f_]:=
>Module[
>   {values, partition, len, index},
>   len = Length[S];
>   partition = Table[{}, {len}];
>   values = Map[f,S];
>
>   Do[
>      index = First[Flatten[Position[values, values[[i]]]]];
>      partition[[index]] = Prepend[partition[[index]], S[[i]]],
>      {i, 1, len}
>   ];
>
>   (* remove any empty lists from partition *)
>   Select[partition, (#!={})&]
>]
>
>Here are some examples:
>
>In[2]:= Classify[{-1,-2,1,2,3,4,5}, Positive]
>Out[2]= {{-2, -1}, {5, 4, 3, 2, 1}}
>
>In[3]:= Classify[{-1,-2,1,2,3,4,5}, EvenQ]
>Out[3]= {{5, 3, 1, -1}, {4, 2, -2}}
>
>In[4]:= square[x_]:= x x
>
>In[5]:=Classify[{-1,-2,1,2,3,4,5}, square]
>Out[5]={{1, -1}, {2, -2}, {3}, {4}, {5}}
>
>In[6]:= cube[x_] := x x x
>
>In[7]:= Classify[{-1,-2,1,2,3,4,5}, cube]
>Out[7]= {{-1}, {-2}, {1}, {2}, {3}, {4}, {5}}
>
>David Cabana  drc at gate.net

You can use Union and select:

In[195]:= Classify[s_, f_]:=
With[{v = Union[Map[f, s]]}, Map[Select[s, Function[{z}, f[z] == #]]&, v]]

In[196]:= Classify[{-1,-2,1,2,3,4,5}, Positive]

Out[196]= {{-1, -2}, {1, 2, 3, 4, 5}}

In[197]:= Classify[{-1,-2,1,2,3,4,5}, #^2&]

Out[197]= {{-1, 1}, {-2, 2}, {3}, {4}, {5}}

Tom
Wolfram Research

```

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