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Intervals. Was Re: More /.{I->-1} craziness. Schools are conservative.

• To: mathgroup at smc.vnet.net
• Subject: [mg106926] Intervals. Was Re: More /.{I->-1} craziness. Schools are conservative.
• From: Richard Fateman <fateman at cs.berkeley.edu>
• Date: Thu, 28 Jan 2010 02:45:44 -0500 (EST)
• References: <hjbvc0$2tp$1@smc.vnet.net> <hjeqh1$g3c$1@smc.vnet.net> <hjh877$r4r$1@smc.vnet.net> <201001261133.GAA00712@smc.vnet.net> <hjonfa$4m2$1@smc.vnet.net>

Daniel Lichtblau wrote:
....

  Let us further define two intervals to be equal
> whenever they have nonempty intersection.
> 
> With these definitions, which I think are sensible, ..

I disagree.
The general meaning of a binary operation f on intervals A,B
is

f(A,B) = {f(a,b) |   a in A   and b in B}.

This works for f  defined as +,-,  *, /, >, <, min, max, ...

Now for f being "Equal" we get

Equal(A,B) = { Equal(a,b) | a in A and b in B},  so the only way
that two intervals can be equal is if they each contain only one point,
and it is the same point.

Even Mathematica disagrees with DanL's definition of equality of intervals:

Interval[{1,4}]==Interval[{2,4}] is False, but DanL's def would say True.

unfortunately, Mathematica (7.0) also does this:

Interval[{-1,1}] == Interval[{-1,1}]    comes out True

which would make most people who work with Reliable Computation 
(intervals) unhappy.

On the plus side,
Interval[{-1,1}] - Interval[{-1,1}]    is not zero, but Interval[{-2,2}]

which is OK, assuming, as we probably should, that the two intervals are 
independent.

• References:
  • Re: More /.{I->-1} craziness. Schools are conservative. So are [people]
    • From: Richard Fateman <fateman@cs.berkeley.edu>