Re: IntervalComplement

• To: mathgroup at smc.vnet.net
• Subject: [mg70361] Re: IntervalComplement
• From: "Philpp" <piotr at bigpond.net.au>
• Date: Sat, 14 Oct 2006 03:07:07 -0400 (EDT)
• References: <egl468\$53j\$1@smc.vnet.net>

```I thought I'd share a bit of an insight into why the
IntervalComplement was never included in Mathematica.

The reason is rather obvious; the implementation of Interval, as it
stands now, does NOT allow for a consistent definition of
IntervalComplement.

Consider,

In[1]:=  a = Interval[{5, 5}];
IntervalMemberQ[a, 5]
Out[2]=  True

Thus, 5 belongs to the interval a.

Let's assume that a complement of this Interval, with respect to
(say) Real number set, could be expressed using Mathematica Interval
type as,

In[3]:=  c = Interval[{-Infinity, 5}, {5, Infinity}];
IntervalMemberQ[c, 5]
Out[4]=  True

Thus, 5 also belongs to the interval c.

This leads to a contradiction; a number (5) cannot belong to an
interval and its complement simultaneously.

Thus, c = Interval[{-Infinity, 5}, {5, Infinity}] is not a complement
of a = Interval[{5, 5}].

Philipp

Chris Chiasson wrote:

> Has anyone implemented this function before? I need something that can do this.
>
> (IntervalComplement is to IntervalUnion as Complement is to Union)
>
> --
> http://chris.chiasson.name/

```

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