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# OpenMath Content Dictionary: interval1

Canonical URL:
http://www.openmath.org/cd/interval1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
interval1.ocd
CD as XML Encoded OpenMath:
interval1.omcd
Defines:
integer_interval, interval, interval_cc, interval_co, interval_oc, interval_oo, oriented_interval
Date:
2009-04-01
Version:
4
Review Date:
2014-04-01
Status:
official

```
This document is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

The copyright holder grants you permission to redistribute this
document freely as a verbatim copy. Furthermore, the copyright
holder permits you to develop any derived work from this document
provided that the following conditions are met.
a) The derived work acknowledges the fact that it is derived from
this document, and maintains a prominent reference in the
work to the original source.
b) The fact that the derived work is not the original OpenMath
document is stated prominently in the derived work.  Moreover if
both this document and the derived work are Content Dictionaries
then the derived work must include a different CDName element,
chosen so that it cannot be confused with any works adopted by
the OpenMath Society.  In particular, if there is a Content
Dictionary Group whose name is, for example, `math' containing
Content Dictionaries named `math1', `math2' etc., then you should
not name a derived Content Dictionary `mathN' where N is an integer.
However you are free to name it `private_mathN' or some such.  This
is because the names `mathN' may be used by the OpenMath Society
for future extensions.
c) The derived work is distributed under terms that allow the
compilation of derived works, but keep paragraphs a) and b)
intact.  The simplest way to do this is to distribute the derived
work under the OpenMath license, but this is not a requirement.
If you have questions about this license please contact the OpenMath
society at http://www.openmath.org.
```

This CD holds symbols which describe both discrete and continuous 1-dimensional intervals (with open/closed end points). There is also an oriented_interval, for use in integration

## integer_interval

Role:
application
Description:

A symbol to denote a discrete 1 dimensional interval from the first argument to the second (inclusive), where the discretisation occurs at unit intervals. The arguments are the start and the end points of the interval in that order.

Example:
The integer interval 1, 2, ..., 10.
$\left[1,10\right]$
Formal Mathematical property (FMP):
$n\in \left[a,b\right]\equiv \left(n\in \mathbb{Z}\wedge \mathrm{le}\left(a,n\right)\wedge \mathrm{le}\left(n,b\right)\right)$
Signatures:
sts

 [Next: interval] [Last: interval_co] [Top]

## interval

Role:
application
Description:

A symbol to denote a continuous 1-dimensional interval without any information about the character of the end points (used in definite integration). The arguments are the start and the end points of the interval in that order.

Example:
The interval 1.0, ..., 10.0.
$\left(1.0,10.0\right)$
Formal Mathematical property (FMP):
$\left(a,b\right)\subset \left[a,b\right]$
Formal Mathematical property (FMP):
$\left(a,b\right)\subset \left(a,b\right)$
Signatures:
sts

 [Next: oriented_interval] [Previous: integer_interval] [Top]

## oriented_interval

Role:
application
Description:

A symbol to denote a continuous 1-dimensional interval without any information about the character of the end points (used in definite integration). The arguments are the start and the end points of the integration, in either order.

Example:
$\mathrm{defintint}\left(\mathrm{oriented_interval}\left(a,b\right),f\right)=\mathrm{defintint}\left(\mathrm{oriented_interval}\left(b,a\right),f\right)$
Signatures:
sts

 [Next: interval_oo] [Previous: interval] [Top]

## interval_oo

Role:
application
Description:

A symbol to denote a continuous 1-dimensional interval with both end points excluded from the interval. The arguments are the start and the end points of the interval in that order.

Example:
The continuous open interval (1,10).
$\left(1,10\right)$
Formal Mathematical property (FMP):
$x\in \left(a,b\right)\equiv \left(x\in \mathbb{R}\wedge a
Signatures:
sts

 [Next: interval_cc] [Previous: oriented_interval] [Top]

## interval_cc

Role:
application
Description:

A symbol to denote a continuous 1-dimensional interval with both end points included in the interval. The arguments are the start and the end points of the interval in that order.

Example:
The continuous closed interval [1,10].
$\left[1,10\right]$
Formal Mathematical property (FMP):
$x\in \left[a,b\right]\equiv \left(x\in \mathbb{R}\wedge \mathrm{le}\left(a,x\right)\wedge \mathrm{le}\left(x,b\right)\right)$
Signatures:
sts

 [Next: interval_oc] [Previous: interval_oo] [Top]

## interval_oc

Role:
application
Description:

A symbol to denote a continuous 1-dimensional interval with the first point excluded from the interval, but the last included. The arguments are the start and the end points of the interval in that order.

Example:
The continuous interval open at the lower bound and closed at the higher bound (1,10].
$\left(1,10\right]$
Formal Mathematical property (FMP):
$x\in \left(a,b\right]\equiv \left(x\in \mathbb{R}\wedge a
Signatures:
sts

 [Next: interval_co] [Previous: interval_cc] [Top]

## interval_co

Role:
application
Description:

A symbol to denote a continuous 1-dimensional interval with the first point included in the interval, but the last excluded. The arguments are the start and the end points of the interval in that order.

Example:
The continuous interval closed at the lower bound and open at the higher bound [1,10).
$\left[1,10\right)$
Formal Mathematical property (FMP):
$x\in \left[a,b\right)\equiv \left(x\in \mathbb{R}\wedge \mathrm{le}\left(a,x\right)\wedge x
Signatures:
sts

 [First: integer_interval] [Previous: interval_oc] [Top]

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