Realsets

realsets.py is a Sage module to support subsets of the real field consisting of intervals and isolated points and was developed to demonstrate set operations of the MGL Set1 module.

It is based of previous work from Interval1Sage adding integration on real sets and real intervals.

An object in this module consists of a list of disjoint open intervals plus a list of isolated points (not belonging to these intervals). Notice that Infinite is acceptable as interval bound. Therefore, one can define:

• All sort of real intervals: open, close and half-open
• Finite sets
• Unbounded intervals
• And combinations of these by union, intersection and taking complements.

Represent a set that can be the union of some intervals and isolated points. It consists of:

• A list of disjoint open non-empty intervals.
• A list of points. Each of these points belongs at most to one interval.

Examples

A closed interval:

? RealSet.cc_interval(1,4); 
[ 1 :: 4 ]

A single point:

? RealSet.singleton(1)
{1}

Union

Union is supported with intervals and can be nested :

? I = RealSet.co_interval(1, 4)
? J = RealSet.co_interval(4, 5)
? M = RealSet.oc_interval(7, 8)
? I.union(J).union(M)
[ 1 :: 5 [ ∪ ] 7 :: 8 ]

Intersection

? I.intersection(J)
()
? I.intersection(RealSet.cc_interval(2,5))
[ 2 :: 4 [

Queries

Is a point in the set?

? I = RealSet.oo_interval(1, 3)
? 2 in I
True
? 3 in I
False

Is a set discrete (i.e: does not contain intervals)?

? RealSet.oo_interval(0,1).discrete
False
? RealSet(points=(1,2,3)).discrete
True

Size of a discrete is the number of points:

? RealSet(points=range(5)).size
5
? RealSet.oo_interval(0,3).size
+Infinity

A is subset of B

? A = RealSet.oo_interval(0,1)
? B = RealSet.cc_interval(0,1)
? RealSet().subset(A)
True
? B.subset(A)
False
? A.subset(B)
True
? A.subset(A)
True
? A.subset(A, proper=True)
False

Return the infimum (greatest lower bound)

? RealSet(points=range(3)).infimum()
0
? RealSet.oo_interval(1,3).infimum()
1

The opposite of a set: –A = {-x | x ∈ A}

? -RealSet.oo_interval(1,2)
] -2 :: -1 [

Return the supremum (least upper bound)

? RealSet(points=range(3)).supremum()
2
? RealSet.oo_interval(1,3).supremum()
3

The complementary of a set:

? RealSet.oo_interval(2,3).complement()
] -Infinity :: 2 ] ∪ [ 3 :: +Infinity [
? RealSet(points=range(3)).complement()
] 0 :: 1 [ ∪ ] 1 :: 2 [ ∪ ] 2 :: +Infinity [ ∪ ] -Infinity :: 0 [

The set difference of A and B: \{x \in A, x\notin B\}

? I = RealSet.oo_interval(2,+Infinity)
? J = RealSet.oo_interval(-Infinity, 5)
? I.setdiff(J)
[ 5 :: +Infinity [
? J.setdiff(I)
] -Infinity :: 2 ]