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(-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)
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:
Laurent Claessens (2010-12-10): Interval and ContinuousSet, posted to sage-devel at http://www.mail-archive.com/sage-support@googlegroups.com/msg21326.html.
Ares Ribo (2011-10-24): Extended the previous work defining the class RealSet.
Jordi Saludes (2011-12-10): Documentation and file reorganization.
Volker Braun (2013-06-22): Rewrite
- class sage.sets.real_set.InternalRealInterval(lower, lower_closed, upper, upper_closed, check=True)¶
Bases:
sage.structure.unique_representation.UniqueRepresentation,sage.structure.parent.ParentA real interval.
You are not supposed to create
RealIntervalobjects yourself. Always useRealSetinstead.INPUT:
lower– real or minus infinity; the lower bound of the interval.lower_closed– boolean; whether the interval is closed at the lower boundupper– real or (plus) infinity; the upper bound of the intervalupper_closed– boolean; whether the interval is closed at the upper boundcheck– boolean; whether to check the other arguments for validity
- boundary_points()¶
Generate the boundary points of
selfEXAMPLES:
sage: list(RealSet.open_closed(-oo, 1)[0].boundary_points()) [1] sage: list(RealSet.open(1, 2)[0].boundary_points()) [1, 2]
- closure()¶
Return the closure
OUTPUT:
The closure as a new
RealIntervalEXAMPLES:
sage: RealSet.open(0,1)[0].closure() [0, 1] sage: RealSet.open(-oo,1)[0].closure() (-oo, 1] sage: RealSet.open(0, oo)[0].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)[0]; 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
RealInterval.EXAMPLES:
sage: I1 = RealSet.open(0, 1)[0]; I1 (0, 1) sage: I2 = RealSet.closed(1, 2)[0]; 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)[0]; 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
RealIntervalEXAMPLES:
sage: RealSet.closed(0, 1)[0].interior() (0, 1) sage: RealSet.open_closed(-oo, 1)[0].interior() (-oo, 1) sage: RealSet.closed_open(0, oo)[0].interior() (0, +oo)
- intersection(other)¶
Return the intersection of the two intervals
INPUT:
other– aRealInterval
OUTPUT:
The intersection as a new
RealIntervalEXAMPLES:
sage: I1 = RealSet.open(0, 2)[0]; I1 (0, 2) sage: I2 = RealSet.closed(1, 3)[0]; 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)[0]; 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)[0]; I1 (0, 1) sage: I2 = RealSet.closed(1, 2)[0]; 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)[0]; 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
RealSethas all intervals non-empty, so this method usually returnsFalse.OUTPUT:
Boolean.
EXAMPLES:
sage: I = RealSet(0, 1)[0] sage: I.is_empty() False
- is_point()¶
Return whether the interval consists of a single point
OUTPUT:
Boolean.
EXAMPLES:
sage: I = RealSet(0, 1)[0] 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)[0] 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)[0]; 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)[0]; 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)[0] 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)[0]; 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)[0]; 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(*intervals)¶
Bases:
sage.structure.unique_representation.UniqueRepresentation,sage.structure.parent.Parent,sage.sets.set.Set_base,sage.sets.set.Set_boolean_operators,sage.sets.set.Set_add_sub_operatorsA 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 \((a, b)\), ora 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),an
OpenInterval.
structure– (default:None) ifNone, construct the real set as an instance ofRealSet; if"differentiable", construct it as a subset of an instance ofRealLine, representing the differentiable manifold \(\RR\).ambient– (default:None) an instance ofRealLine; construct a subset of it. Using this keyword impliesstructure='differentiable'.namesorcoordinate– coordinate symbol for the canonical chart; seeRealLine. Using these keywords impliesstructure='differentiable'.name,latex_name,start_index– seeRealLine.
There are also specialized constructors for various types of intervals:
Constructor
Interval
\((a, b)\)
\([a, b]\)
\(\{a\}\)
\((a, b]\)
\([a, b)\)
\((-\infty, b]\)
\((-\infty, b)\)
\([a, +\infty)\)
\((a, +\infty)\)
\((-\infty, +\infty)\)
any
EXAMPLES:
sage: RealSet(0,1) # open set from two numbers (0, 1) sage: i = RealSet(0,1)[0] sage: RealSet(i) # interval (0, 1) sage: RealSet(i, (3,4)) # tuple of two numbers = open set (0, 1) ∪ (3, 4) sage: RealSet(i, [3,4]) # list of two numbers = closed set (0, 1) ∪ [3, 4]
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
coordinateornames: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 sets are pairwise disjoint
INPUT:
*real_set_collection– a list/tuple/iterable ofRealSet.
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.EXAMPLES:
sage: RealSet(-oo, oo).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– seeRealSet.
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– seeRealSet.
OUTPUT:
A new
RealSetthat 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.EXAMPLES:
sage: RealSet(-oo, oo).closure() (-oo, +oo) sage: RealSet((1, 2), (2, 3)).closure() [1, 3]
- 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: 10 in s # syntactic sugar True
- difference(*other)¶
Return
selfwithothersubtractedINPUT:
other– aRealSetor data that defines one.
OUTPUT:
The set-theoretic difference of
selfwithotherremoved as a newRealSet.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, see
normalize().INPUT:
i– integer.
OUTPUT:
The \(i\)-th connected component as a
RealInterval.EXAMPLES:
sage: s = RealSet(RealSet.open_closed(0,1), RealSet.closed_open(2,3)) sage: s.get_interval(0) (0, 1] sage: s[0] # shorthand (0, 1] sage: s.get_interval(1) [2, 3) sage: s[0] == 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.EXAMPLES:
sage: RealSet(-oo, oo).interior() (-oo, +oo) sage: RealSet.point(2).interior() {} sage: RealSet([1, 2], (3, 4)).interior() (1, 2) ∪ (3, 4)
- intersection(*other)¶
Return the intersection of the two sets
INPUT:
other– aRealSetor 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: s1 = RealSet((0, 1), (2, 3)); s1 (0, 1) ∪ (2, 3) sage: s2 = RealSet([0, 1], [2, 3]); s2 [0, 1] ∪ [2, 3] sage: s3 = RealSet([1, 2]); s3 [1, 2] sage: s1.intersection(s2) (0, 1) ∪ (2, 3) sage: s1.intersection(s3) {} sage: s2.intersection(s3) {1} ∪ {2}
- 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– seeRealSet.
OUTPUT:
A new
RealSet.EXAMPLES:
sage: RealSet.interval(1, 0, lower_closed=True, upper_closed=False) [0, 1)
- is_closed()¶
Return whether
selfis 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
- is_disjoint(*other)¶
Test whether the two sets are disjoint
INPUT:
other– aRealSetor data defining one.
OUTPUT:
Boolean.
EXAMPLES:
sage: s1 = RealSet((0, 1), (2, 3)); s1 (0, 1) ∪ (2, 3) sage: s2 = RealSet([1, 2]); s2 [1, 2] sage: s1.is_disjoint(s2) True sage: s1.is_disjoint([1, 2]) True
- 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
- is_included_in(*args, **kwds)¶
Deprecated: Use
is_subset()instead. See trac ticket #31927 for details.
- is_open()¶
Return whether
selfis 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
- is_subset(*other)¶
Return whether
selfis a subset ofother.INPUT:
*other– aRealSetor something that defines one.
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
xto the ambient space forself.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
- static 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))[0] sage: i2 = RealSet([1, 2])[0] sage: i3 = RealSet((2, 3))[0] sage: RealSet.normalize([i1, i2, i3]) ((0, 3),) sage: RealSet((0, 1), [1, 2], (2, 3)) (0, 3) sage: RealSet((0, 1), (1, 2), (2, 3)) (0, 1) ∪ (1, 2) ∪ (2, 3) sage: RealSet([0, 1], [2, 3]) [0, 1] ∪ [2, 3] sage: RealSet((0, 2), (1, 3)) (0, 3) sage: RealSet(0,0) {}
- static open(lower, upper, **kwds)¶
Construct an open interval
INPUT:
lower,upper– two real numbers or infinity. They will be sorted if necessary.**kwds– seeRealSet.
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– seeRealSet.
OUTPUT:
A new
RealSetthat 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:
p– a real number.**kwds– seeRealSet.
OUTPUT:
A new
RealSet.EXAMPLES:
sage: RealSet.open(1, 0) (0, 1)
- static real_line(**kwds)¶
Construct the real line
INPUT:
**kwds– seeRealSet.
EXAMPLES:
sage: RealSet.real_line() (-oo, +oo)
- retract(x)¶
Retract
xtoself.It raises an error if
xdoes not lie in the setself.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
selfandother.INPUT:
other– aRealSetor data that defines one.
OUTPUT:
The set-theoretic symmetric difference of
selfandotheras a newRealSet.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:
bound– a real number.**kwds– seeRealSet.
OUTPUT:
A new
RealSetfrom 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:
bound– a real number.**kwds– seeRealSet.
OUTPUT:
A new
RealSetfrom 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
RealSetfrom minus infinity to the bound (including).**kwds– seeRealSet.
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
RealSetfrom minus infinity to the bound (excluding).**kwds– seeRealSet.
EXAMPLES:
sage: RealSet.unbounded_below_open(1) (-oo, 1)
- union(*other)¶
Return the union of the two sets
INPUT:
other– aRealSetor 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)