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Re: confused about == vs === in this equality

  • To: mathgroup at smc.vnet.net
  • Subject: [mg103761] Re: [mg103732] confused about == vs === in this equality
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 5 Oct 2009 07:38:26 -0400 (EDT)
  • References: <20091003104738.LCJ3I.416659.imail@eastrmwml34> <200910040935.FAA07794@smc.vnet.net>

I amy be taking a bit of a risk here, but I would guess that  
ComplexInfinity and Indeterminate are the only symbols in Mathematica  
with this property, that is we get:

a=ComplexInfinity

TrueQ[Unevaluated[x == x] /. x -> a]

False

  a = Indeterminate;

TrueQ[Unevaluated[x == x] /. x -> a]

  False

I believe that there are no other symbols for which this happens (?)
(If I am right and it is the only one that there is no need to be  
seriously concerned or, as you say, "careful" about this issue.)

Why does and Indeterminate and ComplexInfinity behave in this way? Of  
course this is a matter of design and not (for example) mathematics so  
the question really is, is this a reasonable and useful thing rather  
than if it is right. I guess it is pretty clear that since  
Indeterminate refers to a magnitude that cannot be determined, you  
would not really want to assert that two expressions, both of which  
evaluate to Indeterminate, are in any sense equal. For example it  
would seem very strange if

Infinity - Infinity == Infinity/Infinity

returned True (as would have to be the case if  
Indeterminate==Indeterminate returned True). Similar considerations  
perhaps apply to ComplexInfinity, which refers to a complex quantity  
with infinite magnitude but with an indeterminate argument. (However,  
I am less convinced of that in the case of ComplexInfinty than in the  
case of Indeterminate, because ComplexInfinity has a natural  
interpretation as a unique point on the Riemann sphere).

(Of course === asks quite a different question and there is no doubt  
that when you have identical expressions on both sides of === the  
answer should always be True.)

Andrzej Kozlowski






On 4 Oct 2009, at 18:35, Nasser Abbasi wrote:

> ?===
> lhs===rhs yields True if the expression lhs is identical to rhs, and  
> yields
> False otherwise.
>
> ?==
> lhs==rhs returns True if lhs and rhs are identical.
>
> But looking at this example:
>
> a = ComplexInfinity;
> If[a == ComplexInfinity, Print["YES"]]
>
> Expecting it would print "YES", but it does not. it just returns the  
> whole
> thing unevaluated? But
>
> If[a === ComplexInfinity, Print["YES"]]
>
> does return YES.
>
> I guess I am a little confused about the "expression" bit in the  
> definition.
>
> So, when using the 3"=", it is looking at the _value_ of the  
> expression, but
> when using the 2"=", it is looking at the expression _as it is_, i.e.
> without evaluating it?  Is this the difference?  I've always used  
> the 2"="
> for equality, now I have to be more careful which to use.
>
> --Nasser
>
>
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