Reals sets consisting of intervals and isolated points

[sagetrac-ares]

Finite unions of open/closed/semi-closed subsets of the real line, for example

    sage: RealSet(0,2) + RealSet.unbounded_above_closed(10)
    (0, 2) + [10, +Infinity)

This ticket also encountered a number of bugs in the comparison of infinity with various field elements like

sage: RLF(0) < oo
False

These are now also fixed and extensive doctests were added.

CC: @kcrisman @tscrim @robertwb

Component: calculus

Author: Volker Braun , Jordi Saludes , Ares Ribó

Branch/Commit: c818746

Reviewer: Ralf Stephan, Peter Bruin

Issue created by migration from https://trac.sagemath.org/ticket/13125

[sagetrac-ares]

Description changed:

--- 
+++ 
@@ -4,9 +4,9 @@
 
 Laurent Claessens defined a class Interval that represents an interval (can be open, closed, half open, unbounded), and implemented union() and intersection() methods, as well as the __contains__() method that tests if a number is contained in the interval. Also defined the class ContinuousSet represening finite union and intersections of intervals by a list of disjoint intervals, and for this class ContinuousSet, union() and __contain__() methods were implemented.
 
-We extend the previous work of Laurent Claessens defining the class RealSet, that describes any real set as a list of disjoint intervals and a list of isolated points.  We define the class RInterval of real intervals. A RInterval is now a RealSet, consituted as a list of disjoint intervals with a unique element and an empty list of isolated points. Our class RInterval is now always an open interval. The boundary/ies can be added as isolated point/s if necessary, constituting a RealSet.
+We extend the previous work of Laurent Claessens defining the class "RealSet", that describes any real set as a list of disjoint intervals and a list of isolated points.  We define the class 'RInterval' of real intervals. A 'RInterval' is now a 'RealSet', consituted as a list of disjoint intervals with a unique element and an empty list of isolated points. Our class 'RInterval' is now always an open interval. The boundary/ies can be added as isolated point/s if necessary, constituting a 'RealSet'.
 
-For the class RealSet, we implement the intersection(), union() and __contain__(). We implement the function 'subsets' that, given two different real sets A and B, returns if A is a (proper) subset of B, and the function 'setdiff' that returns the difference of two given real sets. We implement also the infimum and the supremum of a RealSet. 
+For the class 'RealSet', we implement the intersection(), union() and __contain__(). We implement the function 'subsets' that, given two different real sets A and B, returns if A is a (proper) subset of B, and the function 'setdiff' that returns the difference of two given real sets. We implement also the infimum and the supremum of a 'RealSet'. 
 
-Also we support definite integration over a RealSet.
+Also we support definite integration over a 'RealSet'.
     

[sagetrac-ares]

Description changed:

--- 
+++ 
@@ -4,9 +4,9 @@
 
 Laurent Claessens defined a class Interval that represents an interval (can be open, closed, half open, unbounded), and implemented union() and intersection() methods, as well as the __contains__() method that tests if a number is contained in the interval. Also defined the class ContinuousSet represening finite union and intersections of intervals by a list of disjoint intervals, and for this class ContinuousSet, union() and __contain__() methods were implemented.
 
-We extend the previous work of Laurent Claessens defining the class "RealSet", that describes any real set as a list of disjoint intervals and a list of isolated points.  We define the class 'RInterval' of real intervals. A 'RInterval' is now a 'RealSet', consituted as a list of disjoint intervals with a unique element and an empty list of isolated points. Our class 'RInterval' is now always an open interval. The boundary/ies can be added as isolated point/s if necessary, constituting a 'RealSet'.
+We extend the previous work of Laurent Claessens defining the class RealSet, that describes any real set as a list of disjoint intervals and a list of isolated points.  We define the class RInterval of real intervals. A RInterval is now a RealSet, consituted as a list of disjoint intervals with a unique element and an empty list of isolated points. Our class RInterval is now always an open interval. The boundary/ies can be added as isolated point/s if necessary, constituting a RealSet.
 
-For the class 'RealSet', we implement the intersection(), union() and __contain__(). We implement the function 'subsets' that, given two different real sets A and B, returns if A is a (proper) subset of B, and the function 'setdiff' that returns the difference of two given real sets. We implement also the infimum and the supremum of a 'RealSet'. 
+For the class RealSet, we implement the intersection(), union() and __contain__(). We implement the function subsets that, given two different real sets A and B, returns if A is a (proper) subset of B, and the function setdiff that returns the difference of two given real sets. We implement also the infimum and the supremum of a RealSet. 
 
-Also we support definite integration over a 'RealSet'.
+Also we support definite integration over a RealSet.
     

[kcrisman]

comment:3

Patch?

[sagetrac-ares]

comment:4

Replying to @kcrisman:

Patch?

uploaded

[vbraun]

Initial patch

[vbraun]

Changed author from Jordi Saludes and Ares Ribó to Volker Braun

[vbraun]

Description changed:

--- 
+++ 
@@ -1,12 +1,9 @@
-This is based of previous work available from
-     http://www.mail-archive.com/sage-support@googlegroups.com/msg21326.html
-but supporting now integration on real intervals and real sets.
+Finite unions of open/closed/semi-closed subsets of the real line
 
-Laurent Claessens defined a class Interval that represents an interval (can be open, closed, half open, unbounded), and implemented union() and intersection() methods, as well as the __contains__() method that tests if a number is contained in the interval. Also defined the class ContinuousSet represening finite union and intersections of intervals by a list of disjoint intervals, and for this class ContinuousSet, union() and __contain__() methods were implemented.
+For example
 
-We extend the previous work of Laurent Claessens defining the class RealSet, that describes any real set as a list of disjoint intervals and a list of isolated points.  We define the class RInterval of real intervals. A RInterval is now a RealSet, consituted as a list of disjoint intervals with a unique element and an empty list of isolated points. Our class RInterval is now always an open interval. The boundary/ies can be added as isolated point/s if necessary, constituting a RealSet.
+```
+    sage: RealSet(0,2) + RealSet.unbounded_above_closed(10)
+    (0, 2) + [10, +Infinity)
+```
 
-For the class RealSet, we implement the intersection(), union() and __contain__(). We implement the function subsets that, given two different real sets A and B, returns if A is a (proper) subset of B, and the function setdiff that returns the difference of two given real sets. We implement also the infimum and the supremum of a RealSet. 
-
-Also we support definite integration over a RealSet.
-    

[vbraun]

comment:5

Attachment: trac_13125_real_set.patch.gz

I'm taking over this ticket since I need this for piecewise functions. I'm not sure what happened with the originally proposed patch, but what was attached here is not the actual code.

[kcrisman]

comment:7

Is + what we're looking for here? Maybe something like u or U? Just worried about the difference between this "sum" and the Minkowski sum (which is, in fact, already in Sage).

In fact, maybe these should inherit from something related to that... probably not, but just raising the possibility.

How do I get more exotic sets like the union of 1/n for positive integer n? ;-)