# Subsets of the Real Line#

This module contains subsets of the real line that can be constructed as the union of a finite set of open and closed intervals.

EXAMPLES:

sage: RealSet(0,1)
(0, 1)
sage: RealSet((0,1), [2,3])
(0, 1) ∪ [2, 3]
sage: RealSet((1,3), (0,2))
(0, 3)
sage: RealSet(-oo, oo)
(-oo, +oo)


Brackets must be balanced in Python, so the naive notation for half-open intervals does not work:

sage: RealSet([0,1))
Traceback (most recent call last):
...
SyntaxError: ...


Instead, you can use the following construction functions:

sage: RealSet.open_closed(0,1)
(0, 1]
sage: RealSet.closed_open(0,1)
[0, 1)
sage: RealSet.point(1/2)
{1/2}
sage: RealSet.unbounded_below_open(0)
(-oo, 0)
sage: RealSet.unbounded_below_closed(0)
(-oo, 0]
sage: RealSet.unbounded_above_open(1)
(1, +oo)
sage: RealSet.unbounded_above_closed(1)
[1, +oo)


The lower and upper endpoints will be sorted if necessary:

sage: RealSet.interval(1, 0, lower_closed=True, upper_closed=False)
[0, 1)


Relations containing symbols and numeric values or constants:

sage: RealSet(x != 0)
(-oo, 0) ∪ (0, +oo)
sage: RealSet(x == pi)
{pi}
sage: RealSet(x < 1/2)
(-oo, 1/2)
sage: RealSet(1/2 < x)
(1/2, +oo)
sage: RealSet(1.5 <= x)
[1.50000000000000, +oo)


Note that multiple arguments are combined as union:

sage: RealSet(x >= 0, x < 1)
(-oo, +oo)
sage: RealSet(x >= 0, x > 1)
[0, +oo)
sage: RealSet(x >= 0, x > -1)
(-1, +oo)


AUTHORS:

class sage.sets.real_set.InternalRealInterval(lower, lower_closed, upper, upper_closed, check=True)#

A real interval.

You are not supposed to create InternalRealInterval objects yourself. Always use RealSet instead.

INPUT:

• lower – real or minus infinity; the lower bound of the interval.

• lower_closed – boolean; whether the interval is closed at the lower bound

• upper – real or (plus) infinity; the upper bound of the interval

• upper_closed – boolean; whether the interval is closed at the upper bound

• check – boolean; whether to check the other arguments for validity

boundary_points()#

Generate the boundary points of self

EXAMPLES:

sage: list(RealSet.open_closed(-oo, 1).boundary_points())

sage: list(RealSet.open(1, 2).boundary_points())
[1, 2]

closure()#

Return the closure

OUTPUT:

The closure as a new InternalRealInterval

EXAMPLES:

sage: RealSet.open(0,1).closure()
[0, 1]
sage: RealSet.open(-oo,1).closure()
(-oo, 1]
sage: RealSet.open(0, oo).closure()
[0, +oo)

contains(x)#

Return whether $$x$$ is contained in the interval

INPUT:

• x – a real number.

OUTPUT:

Boolean.

EXAMPLES:

sage: i = RealSet.open_closed(0,2); i
(0, 2]
sage: i.contains(0)
False
sage: i.contains(1)
True
sage: i.contains(2)
True

convex_hull(other)#

Return the convex hull of the two intervals

OUTPUT:

The convex hull as a new InternalRealInterval.

EXAMPLES:

sage: I1 = RealSet.open(0, 1);  I1
(0, 1)
sage: I2 = RealSet.closed(1, 2);  I2
[1, 2]
sage: I1.convex_hull(I2)
(0, 2]
sage: I2.convex_hull(I1)
(0, 2]
sage: I1.convex_hull(I2.interior())
(0, 2)
sage: I1.closure().convex_hull(I2.interior())
[0, 2)
sage: I1.closure().convex_hull(I2)
[0, 2]
sage: I3 = RealSet.closed(1/2, 3/2); I3
[1/2, 3/2]
sage: I1.convex_hull(I3)
(0, 3/2]

element_class#
interior()#

Return the interior

OUTPUT:

The interior as a new InternalRealInterval

EXAMPLES:

sage: RealSet.closed(0, 1).interior()
(0, 1)
sage: RealSet.open_closed(-oo, 1).interior()
(-oo, 1)
sage: RealSet.closed_open(0, oo).interior()
(0, +oo)

intersection(other)#

Return the intersection of the two intervals

INPUT:

OUTPUT:

The intersection as a new InternalRealInterval

EXAMPLES:

sage: I1 = RealSet.open(0, 2);  I1
(0, 2)
sage: I2 = RealSet.closed(1, 3);  I2
[1, 3]
sage: I1.intersection(I2)
[1, 2)
sage: I2.intersection(I1)
[1, 2)
sage: I1.closure().intersection(I2.interior())
(1, 2]
sage: I2.interior().intersection(I1.closure())
(1, 2]

sage: I3 = RealSet.closed(10, 11);  I3
[10, 11]
sage: I1.intersection(I3)
(0, 0)
sage: I3.intersection(I1)
(0, 0)

is_connected(other)#

Test whether two intervals are connected

OUTPUT:

Boolean. Whether the set-theoretic union of the two intervals has a single connected component.

EXAMPLES:

sage: I1 = RealSet.open(0, 1);  I1
(0, 1)
sage: I2 = RealSet.closed(1, 2);  I2
[1, 2]
sage: I1.is_connected(I2)
True
sage: I1.is_connected(I2.interior())
False
sage: I1.closure().is_connected(I2.interior())
True
sage: I2.is_connected(I1)
True
sage: I2.interior().is_connected(I1)
False
sage: I2.closure().is_connected(I1.interior())
True
sage: I3 = RealSet.closed(1/2, 3/2); I3
[1/2, 3/2]
sage: I1.is_connected(I3)
True
sage: I3.is_connected(I1)
True

is_empty()#

Return whether the interval is empty

The normalized form of RealSet has all intervals non-empty, so this method usually returns False.

OUTPUT:

Boolean.

EXAMPLES:

sage: I = RealSet(0, 1)
sage: I.is_empty()
False

is_point()#

Return whether the interval consists of a single point

OUTPUT:

Boolean.

EXAMPLES:

sage: I = RealSet(0, 1)
sage: I.is_point()
False

lower()#

Return the lower bound

OUTPUT:

The lower bound as it was originally specified.

EXAMPLES:

sage: I = RealSet(0, 1)
sage: I.lower()
0
sage: I.upper()
1

lower_closed()#

Return whether the interval is open at the lower bound

OUTPUT:

Boolean.

EXAMPLES:

sage: I = RealSet.open_closed(0, 1);  I
(0, 1]
sage: I.lower_closed()
False
sage: I.lower_open()
True
sage: I.upper_closed()
True
sage: I.upper_open()
False

lower_open()#

Return whether the interval is closed at the upper bound

OUTPUT:

Boolean.

EXAMPLES:

sage: I = RealSet.open_closed(0, 1);  I
(0, 1]
sage: I.lower_closed()
False
sage: I.lower_open()
True
sage: I.upper_closed()
True
sage: I.upper_open()
False

upper()#

Return the upper bound

OUTPUT:

The upper bound as it was originally specified.

EXAMPLES:

sage: I = RealSet(0, 1)
sage: I.lower()
0
sage: I.upper()
1

upper_closed()#

Return whether the interval is closed at the lower bound

OUTPUT:

Boolean.

EXAMPLES:

sage: I = RealSet.open_closed(0, 1);  I
(0, 1]
sage: I.lower_closed()
False
sage: I.lower_open()
True
sage: I.upper_closed()
True
sage: I.upper_open()
False

upper_open()#

Return whether the interval is closed at the upper bound

OUTPUT:

Boolean.

EXAMPLES:

sage: I = RealSet.open_closed(0, 1);  I
(0, 1]
sage: I.lower_closed()
False
sage: I.lower_open()
True
sage: I.upper_closed()
True
sage: I.upper_open()
False

class sage.sets.real_set.RealSet(normalized, *intervals)#

A subset of the real line, a finite union of intervals

INPUT:

• *args – arguments defining a real set. Possibilities are either:

• two extended real numbers a, b, to construct the open interval $$(a, b)$$, or

• a list/tuple/iterable of (not necessarily disjoint) intervals or real sets, whose union is taken. The individual intervals can be specified by either

• a tuple (a, b) of two extended real numbers (constructing an open interval),

• a list [a, b] of two real numbers (constructing a closed interval),

• structure – (default: None) if None, construct the real set as an instance of RealSet; if "differentiable", construct it as a subset of an instance of RealLine, representing the differentiable manifold $$\RR$$.

• ambient – (default: None) an instance of RealLine; construct a subset of it. Using this keyword implies structure='differentiable'.

• names or coordinate – coordinate symbol for the canonical chart; see RealLine. Using these keywords implies structure='differentiable'.

• name, latex_name, start_index – see RealLine.

• normalized – (default: None) if True, the input is already normalized, i.e., *args are the connected components (type InternalRealInterval) of the real set in ascending order; no other keyword is provided.

There are also specialized constructors for various types of intervals:

EXAMPLES:

sage: RealSet(0, 1)    # open set from two numbers
(0, 1)
sage: RealSet(1, 0)    # the two numbers will be sorted
(0, 1)
sage: s1 = RealSet((1,2)); s1    # tuple of two numbers = open set
(1, 2)
sage: s2 = RealSet([3,4]); s2    # list of two numbers = closed set
[3, 4]
sage: i1, i2 = s1, s2
sage: RealSet(i2, i1)            # union of intervals
(1, 2) ∪ [3, 4]
sage: RealSet((-oo, 0), x > 6, i1, RealSet.point(5), RealSet.closed_open(4, 3))
(-oo, 0) ∪ (1, 2) ∪ [3, 4) ∪ {5} ∪ (6, +oo)


Initialization from manifold objects:

sage: R = manifolds.RealLine(); R
Real number line ℝ
sage: RealSet(R)
(-oo, +oo)
sage: I02 = manifolds.OpenInterval(0, 2); I
I
sage: RealSet(I02)
(0, 2)
sage: I01_of_R = manifolds.OpenInterval(0, 1, ambient_interval=R); I01_of_R
Real interval (0, 1)
sage: RealSet(I01_of_R)
(0, 1)
sage: RealSet(I01_of_R.closure())
[0, 1]
sage: I01_of_I02 = manifolds.OpenInterval(0, 1, ambient_interval=I02); I01_of_I02
Real interval (0, 1)
sage: RealSet(I01_of_I02)
(0, 1)
sage: RealSet(I01_of_I02.closure())
(0, 1]


Real sets belong to a subcategory of topological spaces:

sage: RealSet().category()
Join of
Category of finite sets and
Category of subobjects of sets and
Category of connected topological spaces
sage: RealSet.point(1).category()
Join of
Category of finite sets and
Category of subobjects of sets and
Category of connected topological spaces
sage: RealSet([1, 2]).category()
Join of
Category of infinite sets and
Category of compact topological spaces and
Category of subobjects of sets and
Category of connected topological spaces
sage: RealSet((1, 2), (3, 4)).category()
Join of
Category of infinite sets and
Category of subobjects of sets and
Category of topological spaces


Constructing real sets as manifolds or manifold subsets by passing structure='differentiable':

sage: RealSet(-oo, oo, structure='differentiable')
Real number line ℝ

sage: RealSet([0, 1], structure='differentiable')
Subset [0, 1] of the Real number line ℝ
sage: _.category()
Category of subobjects of sets

sage: RealSet.open_closed(0, 5, structure='differentiable')
Subset (0, 5] of the Real number line ℝ


This is implied when a coordinate name is given using the keywords coordinate or names:

sage: RealSet(0, 1, coordinate='λ')
Open subset (0, 1) of the Real number line ℝ
sage: _.category()
Join of
Category of smooth manifolds over Real Field with 53 bits of precision and
Category of connected manifolds over Real Field with 53 bits of precision and
Category of subobjects of sets


It is also implied by assigning a coordinate name using generator notation:

sage: R_xi.<ξ> = RealSet.real_line(); R_xi
Real number line ℝ
sage: R_xi.canonical_chart()
Chart (ℝ, (ξ,))


With the keyword ambient, we can construct a subset of a previously constructed manifold:

sage: P_xi = RealSet(0, oo, ambient=R_xi); P_xi
Open subset (0, +oo) of the Real number line ℝ
sage: P_xi.default_chart()
Chart ((0, +oo), (ξ,))
sage: B_xi = RealSet(0, 1, ambient=P_xi); B_xi
Open subset (0, 1) of the Real number line ℝ
sage: B_xi.default_chart()
Chart ((0, 1), (ξ,))
sage: R_xi.subset_family()
Set {(0, +oo), (0, 1), ℝ} of open subsets of the Real number line ℝ

sage: F = RealSet.point(0).union(RealSet.point(1)).union(RealSet.point(2)); F
{0} ∪ {1} ∪ {2}
sage: F_tau = RealSet(F, names="τ"); F_tau
Subset {0} ∪ {1} ∪ {2} of the Real number line ℝ
sage: F_tau.manifold().canonical_chart()
Chart (ℝ, (τ,))

ambient()#

Return the ambient space (the real line).

EXAMPLES:

sage: s = RealSet(RealSet.open_closed(0,1), RealSet.closed_open(2,3))
sage: s.ambient()
(-oo, +oo)

static are_pairwise_disjoint(*real_set_collection)#

Test whether the real sets are pairwise disjoint

INPUT:

• *real_set_collection – a list/tuple/iterable of RealSet or data that defines one.

OUTPUT:

Boolean.

EXAMPLES:

sage: s1 = RealSet((0, 1), (2, 3))
sage: s2 = RealSet((1, 2))
sage: s3 = RealSet.point(3)
sage: RealSet.are_pairwise_disjoint(s1, s2, s3)
True
sage: RealSet.are_pairwise_disjoint(s1, s2, s3, [10,10])
True
sage: RealSet.are_pairwise_disjoint(s1, s2, s3, [-1, 1/2])
False

boundary()#

Return the topological boundary of self as a new RealSet.

EXAMPLES:

sage: RealSet(-oo, oo).boundary()
{}
sage: RealSet().boundary()
{}
sage: RealSet.point(2).boundary()
{2}
sage: RealSet([1, 2], (3, 4)).boundary()
{1} ∪ {2} ∪ {3} ∪ {4}
sage: RealSet((1, 2), (2, 3)).boundary()
{1} ∪ {2} ∪ {3}

cardinality()#

Return the cardinality of the subset of the real line.

OUTPUT:

Integer or infinity. The size of a discrete set is the number of points; the size of a real interval is Infinity.

EXAMPLES:

sage: RealSet([0, 0], [1, 1], [3, 3]).cardinality()
3
sage: RealSet(0,3).cardinality()
+Infinity

static closed(lower, upper, **kwds)#

Construct a closed interval

INPUT:

• lower, upper – two real numbers or infinity. They will be sorted if necessary.

• **kwds – see RealSet.

OUTPUT:

A new RealSet.

EXAMPLES:

sage: RealSet.closed(1, 0)
[0, 1]

static closed_open(lower, upper, **kwds)#

Construct an half-open interval

INPUT:

• lower, upper – two real numbers or infinity. They will be sorted if necessary.

• **kwds – see RealSet.

OUTPUT:

A new RealSet that is closed at the lower bound and open an the upper bound.

EXAMPLES:

sage: RealSet.closed_open(1, 0)
[0, 1)

closure()#

Return the topological closure of self as a new RealSet.

EXAMPLES:

sage: RealSet(-oo, oo).closure()
(-oo, +oo)
sage: RealSet((1, 2), (2, 3)).closure()
[1, 3]
sage: RealSet().closure()
{}

complement()#

Return the complement

OUTPUT:

The set-theoretic complement as a new RealSet.

EXAMPLES:

sage: RealSet(0,1).complement()
(-oo, 0] ∪ [1, +oo)

sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10);  s1
(0, 2) ∪ [10, +oo)
sage: s1.complement()
(-oo, 0] ∪ [2, 10)

sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10);  s2
(-oo, -10] ∪ (1, 3)
sage: s2.complement()
(-10, 1] ∪ [3, +oo)

contains(x)#

Return whether $$x$$ is contained in the set

INPUT:

• x – a real number.

OUTPUT:

Boolean.

EXAMPLES:

sage: s = RealSet(0,2) + RealSet.unbounded_above_closed(10);  s
(0, 2) ∪ [10, +oo)
sage: s.contains(1)
True
sage: s.contains(0)
False
sage: s.contains(10.0)
True
sage: 10 in s    # syntactic sugar
True
sage: s.contains(+oo)
False
sage: RealSet().contains(1)
False

static convex_hull(*real_set_collection)#

Return the convex hull of real sets.

INPUT:

• *real_set_collection – a list/tuple/iterable of RealSet or data that defines one.

OUTPUT:

The convex hull as a new RealSet.

EXAMPLES:

sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10);  s1 # unbounded set
(0, 2) ∪ [10, +oo)
sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10);  s2
(-oo, -10] ∪ (1, 3)
sage: s3 = RealSet((0,2), RealSet.point(8)); s3
(0, 2) ∪ {8}
sage: s4 = RealSet(); s4  # empty set
{}
sage: RealSet.convex_hull(s1)
(0, +oo)
sage: RealSet.convex_hull(s2)
(-oo, 3)
sage: RealSet.convex_hull(s3)
(0, 8]
sage: RealSet.convex_hull(s4)
{}
sage: RealSet.convex_hull(s1, s2)
(-oo, +oo)
sage: RealSet.convex_hull(s2, s3)
(-oo, 8]
sage: RealSet.convex_hull(s2, s3, s4)
(-oo, 8]

difference(*other)#

Return self with other subtracted

INPUT:

OUTPUT:

The set-theoretic difference of self with other removed as a new RealSet.

EXAMPLES:

sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10);  s1
(0, 2) ∪ [10, +oo)
sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10);  s2
(-oo, -10] ∪ (1, 3)
sage: s1.difference(s2)
(0, 1] ∪ [10, +oo)
sage: s1 - s2    # syntactic sugar
(0, 1] ∪ [10, +oo)
sage: s2.difference(s1)
(-oo, -10] ∪ [2, 3)
sage: s2 - s1    # syntactic sugar
(-oo, -10] ∪ [2, 3)
sage: s1.difference(1,11)
(0, 1] ∪ [11, +oo)

get_interval(i)#

Return the i-th connected component.

Note that the intervals representing the real set are always normalized, i.e., they are sorted, disjoint and not connected.

INPUT:

• i – integer.

OUTPUT:

The $$i$$-th connected component as a InternalRealInterval.

EXAMPLES:

sage: s = RealSet(RealSet.open_closed(0,1), RealSet.closed_open(2,3))
sage: s.get_interval(0)
(0, 1]
sage: s    # shorthand
(0, 1]
sage: s.get_interval(1)
[2, 3)
sage: s == s.get_interval(0)
True

inf()#

Return the infimum

OUTPUT:

A real number or infinity.

EXAMPLES:

sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10);  s1
(0, 2) ∪ [10, +oo)
sage: s1.inf()
0

sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10);  s2
(-oo, -10] ∪ (1, 3)
sage: s2.inf()
-Infinity

interior()#

Return the topological interior of self as a new RealSet.

EXAMPLES:

sage: RealSet(-oo, oo).interior()
(-oo, +oo)
sage: RealSet().interior()
{}
sage: RealSet.point(2).interior()
{}
sage: RealSet([1, 2], (3, 4)).interior()
(1, 2) ∪ (3, 4)

intersection(*real_set_collection)#

Return the intersection of real sets

INPUT:

• *real_set_collection – a list/tuple/iterable of RealSet or data that defines one.

OUTPUT:

The set-theoretic intersection as a new RealSet.

EXAMPLES:

sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10);  s1
(0, 2) ∪ [10, +oo)
sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10);  s2
(-oo, -10] ∪ (1, 3)
sage: s1.intersection(s2)
(1, 2)
sage: s1 & s2    # syntactic sugar
(1, 2)
sage: s3 = RealSet((0, 1), (2, 3));  s3
(0, 1) ∪ (2, 3)
sage: s4 = RealSet([0, 1], [2, 3]);  s4
[0, 1] ∪ [2, 3]
sage: s3.intersection(s4)
(0, 1) ∪ (2, 3)
sage: s3.intersection([1, 2])
{}
sage: s4.intersection([1, 2])
{1} ∪ {2}
sage: s4.intersection(1, 2)
{}
sage: s5 = RealSet.closed_open(1, 10);  s5
[1, 10)
sage: s5.intersection(-oo, +oo)
[1, 10)
sage: s5.intersection(x != 2, (-oo, 3), RealSet.real_line())
[1, 2) ∪ (2, 3)

static interval(lower, upper, lower_closed, upper_closed, **kwds)#

Construct an interval

INPUT:

• lower, upper – two real numbers or infinity. They will be sorted if necessary.

• lower_closed, upper_closed – boolean; whether the interval is closed at the lower and upper bound of the interval, respectively.

• **kwds – see RealSet.

OUTPUT:

A new RealSet.

EXAMPLES:

sage: RealSet.interval(1, 0, lower_closed=True, upper_closed=False)
[0, 1)

is_closed()#

Return whether self is a closed set.

EXAMPLES:

sage: RealSet().is_closed()
True
sage: RealSet.point(1).is_closed()
True
sage: RealSet([1, 2]).is_closed()
True
sage: RealSet([1, 2], (3, 4)).is_closed()
False
sage: RealSet(-oo, +oo).is_closed()
True

is_connected()#

Return whether self is a connected set.

OUTPUT:

Boolean.

EXAMPLES:

sage: s1 = RealSet((1, 2), (2, 4));  s1
(1, 2) ∪ (2, 4)
sage: s1.is_connected()
False
sage: s2 = RealSet((1, 2), (2, 4), RealSet.point(2));  s2
(1, 4)
sage: s2.is_connected()
True
sage: s3 = RealSet(1,3) + RealSet.unbounded_below_closed(-10);  s3
(-oo, -10] ∪ (1, 3)
sage: s3.is_connected()
False
sage: RealSet(x != 0).is_connected()
False
sage: RealSet(-oo, oo).is_connected()
True
sage: RealSet().is_connected()
False

is_disjoint(*other)#

Test whether the two sets are disjoint

INPUT:

OUTPUT:

Boolean.

EXAMPLES:

sage: s = RealSet((0, 1), (2, 3));  s
(0, 1) ∪ (2, 3)
sage: s.is_disjoint(RealSet([1, 2]))
True
sage: s.is_disjoint([3/2, 5/2])
False
sage: s.is_disjoint(RealSet())
True
sage: s.is_disjoint(RealSet().real_line())
False

is_disjoint_from(*args, **kwds)#

Deprecated: Use is_disjoint() instead. See trac ticket #31927 for details.

is_empty()#

Return whether the set is empty

EXAMPLES:

sage: RealSet(0, 1).is_empty()
False
sage: RealSet(0, 0).is_empty()
True
sage: RealSet.interval(1, 1, lower_closed=False, upper_closed=True).is_empty()
True
sage: RealSet.interval(1, -1, lower_closed=False, upper_closed=True).is_empty()
False

is_included_in(*args, **kwds)#

Deprecated: Use is_subset() instead. See trac ticket #31927 for details.

is_open()#

Return whether self is an open set.

EXAMPLES:

sage: RealSet().is_open()
True
sage: RealSet.point(1).is_open()
False
sage: RealSet((1, 2)).is_open()
True
sage: RealSet([1, 2], (3, 4)).is_open()
False
sage: RealSet(-oo, +oo).is_open()
True

is_subset(*other)#

Return whether self is a subset of other.

INPUT:

OUTPUT:

Boolean.

EXAMPLES:

sage: I = RealSet((1,2))
sage: J = RealSet((1,3))
sage: K = RealSet((2,3))
sage: I.is_subset(J)
True
sage: J.is_subset(K)
False

is_universe()#

Return whether the set is the ambient space (the real line).

EXAMPLES:

sage: RealSet().ambient().is_universe()
True

lift(x)#

Lift x to the ambient space for self.

This version of the method just returns x.

EXAMPLES:

sage: s = RealSet(0, 2); s
(0, 2)
sage: s.lift(1)
1

n_components()#

Return the number of connected components

See also get_interval()

EXAMPLES:

sage: s = RealSet(RealSet.open_closed(0,1), RealSet.closed_open(2,3))
sage: s.n_components()
2

normalize(intervals)#

Bring a collection of intervals into canonical form

INPUT:

• intervals – a list/tuple/iterable of intervals.

OUTPUT:

A tuple of intervals such that

• they are sorted in ascending order (by lower bound)

• there is a gap between each interval

• all intervals are non-empty

EXAMPLES:

sage: i1 = RealSet((0, 1))
sage: i2 = RealSet([1, 2])
sage: i3 = RealSet((2, 3))
sage: RealSet.normalize([i1, i2, i3])
((0, 3),)

static open(lower, upper, **kwds)#

Construct an open interval

INPUT:

• lower, upper – two real numbers or infinity. They will be sorted if necessary.

• **kwds – see RealSet.

OUTPUT:

A new RealSet.

EXAMPLES:

sage: RealSet.open(1, 0)
(0, 1)

static open_closed(lower, upper, **kwds)#

Construct a half-open interval

INPUT:

• lower, upper – two real numbers or infinity. They will be sorted if necessary.

• **kwds – see RealSet.

OUTPUT:

A new RealSet that is open at the lower bound and closed at the upper bound.

EXAMPLES:

sage: RealSet.open_closed(1, 0)
(0, 1]

static point(p, **kwds)#

Construct an interval containing a single point

INPUT:

OUTPUT:

A new RealSet.

EXAMPLES:

sage: RealSet.open(1, 0)
(0, 1)

static real_line(**kwds)#

Construct the real line

INPUT:

EXAMPLES:

sage: RealSet.real_line()
(-oo, +oo)

retract(x)#

Retract x to self.

It raises an error if x does not lie in the set self.

EXAMPLES:

sage: s = RealSet(0, 2); s
(0, 2)
sage: s.retract(1)
1
sage: s.retract(2)
Traceback (most recent call last):
...
ValueError: 2 is not an element of (0, 2)

sup()#

Return the supremum

OUTPUT:

A real number or infinity.

EXAMPLES:

sage: s1 = RealSet(0,2) + RealSet.unbounded_above_closed(10);  s1
(0, 2) ∪ [10, +oo)
sage: s1.sup()
+Infinity

sage: s2 = RealSet(1,3) + RealSet.unbounded_below_closed(-10);  s2
(-oo, -10] ∪ (1, 3)
sage: s2.sup()
3

symmetric_difference(*other)#

Returns the symmetric difference of self and other.

INPUT:

OUTPUT:

The set-theoretic symmetric difference of self and other as a new RealSet.

EXAMPLES:

sage: s1 = RealSet(0,2); s1
(0, 2)
sage: s2 = RealSet.unbounded_above_open(1); s2
(1, +oo)
sage: s1.symmetric_difference(s2)
(0, 1] ∪ [2, +oo)

static unbounded_above_closed(bound, **kwds)#

Construct a semi-infinite interval

INPUT:

OUTPUT:

A new RealSet from the bound (including) to plus infinity.

EXAMPLES:

sage: RealSet.unbounded_above_closed(1)
[1, +oo)

static unbounded_above_open(bound, **kwds)#

Construct a semi-infinite interval

INPUT:

OUTPUT:

A new RealSet from the bound (excluding) to plus infinity.

EXAMPLES:

sage: RealSet.unbounded_above_open(1)
(1, +oo)

static unbounded_below_closed(bound, **kwds)#

Construct a semi-infinite interval

INPUT:

• bound – a real number.

OUTPUT:

A new RealSet from minus infinity to the bound (including).

EXAMPLES:

sage: RealSet.unbounded_below_closed(1)
(-oo, 1]

static unbounded_below_open(bound, **kwds)#

Construct a semi-infinite interval

INPUT:

• bound – a real number.

OUTPUT:

A new RealSet from minus infinity to the bound (excluding).

EXAMPLES:

sage: RealSet.unbounded_below_open(1)
(-oo, 1)

union(*real_set_collection)#

Return the union of real sets

INPUT:

• *real_set_collection – a list/tuple/iterable of RealSet or data that defines one.

OUTPUT:

The set-theoretic union as a new RealSet.

EXAMPLES:

sage: s1 = RealSet(0,2)
sage: s2 = RealSet(1,3)
sage: s1.union(s2)
(0, 3)
sage: s1.union(1,3)
(0, 3)
sage: s1 | s2    # syntactic sugar
(0, 3)
sage: s1 + s2    # syntactic sugar
(0, 3)
sage: RealSet().union(RealSet.real_line())
(-oo, +oo)
sage: s = RealSet().union([1, 2], (2, 3)); s
[1, 3)
sage: RealSet().union((-oo, 0), x > 6, s, RealSet.point(5.0), RealSet.closed_open(2, 4))
(-oo, 0) ∪ [1, 4) ∪ {5} ∪ (6, +oo)