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MathGroup Archive 1998

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Re: logical inconsistency in Union

  • To: mathgroup at smc.vnet.net
  • Subject: [mg13126] Re: [mg13069] logical inconsistency in Union
  • From: Daniel Lichtblau <danl>
  • Date: Tue, 7 Jul 1998 03:44:30 -0400
  • References: <199807042045.QAA02915@smc.vnet.net.>
  • Sender: owner-wri-mathgroup at wolfram.com

Veit Elser wrote:
> 
> The Union function gives logically inconsistent results when used with
> certain SameTest rules. The problem is that the argument of Union is
> first sorted into Mathematica's canonical order, which may be
> inconsistent with the supplied SameTest. Here's an innocent example of
> the problem:
> 
> 
> In[1]:= (* 3 points in the plane *)
> a={1,1};
> b={1.2,1};
> c={1.1,2};
> 
> In[2]:= (* rule which says two points are the same if each coordinate
> differs by at most 0.5 *)
> approxSame[x_,y_]:=Round[x-y]=={0,0}
> 
> In[3]:=
> approxSame[a,b]
> 
> Out[3]=
> True
> 
> In[4]:=
> approxSame[a,c]
> 
> Out[4]=
> False
> 
> In[5]:=
> Union[{a,b,c},SameTest->approxSame]
> 
> Out[5]=
> {{1,1},{1.1,2},{1.2,1}}     (* incorrect result ! *)
> 
> In[7]:=
> Union[{a,b},SameTest->approxSame]
> 
> Out[7]=
> {{1,1}}    (* correct result *)
> 
> 
> To fix this problem we have to be allowed to modify the ordering
> function used by Union in order to be consistent with SameTest. An even
> better solution would be to be able to overide Mathematica's Order
> function (which returns -1,0,1). This would accomplish both: a
> consistent rule for sorting and a criterion to decide if two elements
> are the same.
> 
> This problem affects other set theoretic functions as well (e.g.
> Complement).
> 
> 
> 
> -Veit


I do not think overriding the ordering would help in cases where the
user-provided SameTest does not give a canonical ordering. About all
that can be done for Union et al is a brute-force comparison of every
candidate element with all elements already in the set under
construction.

This problem was noted by several people at the recent Worldwide
Mathematica Conference. I fixed it in our development version.

In[6]:= Union[{a,b,c},SameTest->approxSame] Out[6]= {{1, 1}, {1.1, 2}}


Daniel Lichtblau
Wolfram Research


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