from __future__ import print_function, division
from itertools import product
from sympy.core.sympify import (_sympify, sympify, converter,
SympifyError)
from sympy.core.basic import Basic
from sympy.core.expr import Expr
from sympy.core.singleton import Singleton, S
from sympy.core.evalf import EvalfMixin
from sympy.core.numbers import Float
from sympy.core.compatibility import (iterable, with_metaclass,
ordered, range, PY3)
from sympy.core.evaluate import global_evaluate
from sympy.core.function import FunctionClass
from sympy.core.mul import Mul
from sympy.core.relational import Eq
from sympy.core.symbol import Symbol, Dummy
from sympy.sets.contains import Contains
from sympy.utilities.misc import func_name, filldedent
from mpmath import mpi, mpf
from sympy.logic.boolalg import And, Or, Not, true, false
from sympy.utilities import subsets
[docs]class Set(Basic):
"""
The base class for any kind of set.
This is not meant to be used directly as a container of items. It does not
behave like the builtin ``set``; see :class:`FiniteSet` for that.
Real intervals are represented by the :class:`Interval` class and unions of
sets by the :class:`Union` class. The empty set is represented by the
:class:`EmptySet` class and available as a singleton as ``S.EmptySet``.
"""
is_number = False
is_iterable = False
is_interval = False
is_FiniteSet = False
is_Interval = False
is_ProductSet = False
is_Union = False
is_Intersection = None
is_EmptySet = None
is_UniversalSet = None
is_Complement = None
is_ComplexRegion = False
@staticmethod
def _infimum_key(expr):
"""
Return infimum (if possible) else S.Infinity.
"""
try:
infimum = expr.inf
assert infimum.is_comparable
except (NotImplementedError,
AttributeError, AssertionError, ValueError):
infimum = S.Infinity
return infimum
[docs] def union(self, other):
"""
Returns the union of 'self' and 'other'.
Examples
========
As a shortcut it is possible to use the '+' operator:
>>> from sympy import Interval, FiniteSet
>>> Interval(0, 1).union(Interval(2, 3))
[0, 1] U [2, 3]
>>> Interval(0, 1) + Interval(2, 3)
[0, 1] U [2, 3]
>>> Interval(1, 2, True, True) + FiniteSet(2, 3)
(1, 2] U {3}
Similarly it is possible to use the '-' operator for set differences:
>>> Interval(0, 2) - Interval(0, 1)
(1, 2]
>>> Interval(1, 3) - FiniteSet(2)
[1, 2) U (2, 3]
"""
return Union(self, other)
[docs] def intersect(self, other):
"""
Returns the intersection of 'self' and 'other'.
>>> from sympy import Interval
>>> Interval(1, 3).intersect(Interval(1, 2))
[1, 2]
>>> from sympy import imageset, Lambda, symbols, S
>>> n, m = symbols('n m')
>>> a = imageset(Lambda(n, 2*n), S.Integers)
>>> a.intersect(imageset(Lambda(m, 2*m + 1), S.Integers))
EmptySet()
"""
return Intersection(self, other)
[docs] def intersection(self, other):
"""
Alias for :meth:`intersect()`
"""
return self.intersect(other)
def _intersect(self, other):
"""
This function should only be used internally
self._intersect(other) returns a new, intersected set if self knows how
to intersect itself with other, otherwise it returns ``None``
When making a new set class you can be assured that other will not
be a :class:`Union`, :class:`FiniteSet`, or :class:`EmptySet`
Used within the :class:`Intersection` class
"""
return None
[docs] def is_disjoint(self, other):
"""
Returns True if 'self' and 'other' are disjoint
Examples
========
>>> from sympy import Interval
>>> Interval(0, 2).is_disjoint(Interval(1, 2))
False
>>> Interval(0, 2).is_disjoint(Interval(3, 4))
True
References
==========
.. [1] http://en.wikipedia.org/wiki/Disjoint_sets
"""
return self.intersect(other) == S.EmptySet
[docs] def isdisjoint(self, other):
"""
Alias for :meth:`is_disjoint()`
"""
return self.is_disjoint(other)
def _union(self, other):
"""
This function should only be used internally
self._union(other) returns a new, joined set if self knows how
to join itself with other, otherwise it returns ``None``.
It may also return a python set of SymPy Sets if they are somehow
simpler. If it does this it must be idempotent i.e. the sets returned
must return ``None`` with _union'ed with each other
Used within the :class:`Union` class
"""
return None
[docs] def complement(self, universe):
"""
The complement of 'self' w.r.t the given the universe.
Examples
========
>>> from sympy import Interval, S
>>> Interval(0, 1).complement(S.Reals)
(-oo, 0) U (1, oo)
>>> Interval(0, 1).complement(S.UniversalSet)
UniversalSet() \ [0, 1]
"""
return Complement(universe, self)
def _complement(self, other):
# this behaves as other - self
if isinstance(other, ProductSet):
# For each set consider it or it's complement
# We need at least one of the sets to be complemented
# Consider all 2^n combinations.
# We can conveniently represent these options easily using a
# ProductSet
# XXX: this doesn't work if the dimentions of the sets isn't same.
# A - B is essentially same as A if B has a different
# dimentionality than A
switch_sets = ProductSet(FiniteSet(o, o - s) for s, o in
zip(self.sets, other.sets))
product_sets = (ProductSet(*set) for set in switch_sets)
# Union of all combinations but this one
return Union(p for p in product_sets if p != other)
elif isinstance(other, Interval):
if isinstance(self, Interval) or isinstance(self, FiniteSet):
return Intersection(other, self.complement(S.Reals))
elif isinstance(other, Union):
return Union(o - self for o in other.args)
elif isinstance(other, Complement):
return Complement(other.args[0], Union(other.args[1], self), evaluate=False)
elif isinstance(other, EmptySet):
return S.EmptySet
elif isinstance(other, FiniteSet):
return FiniteSet(*[el for el in other if self.contains(el) != True])
def symmetric_difference(self, other):
return SymmetricDifference(self, other)
def _symmetric_difference(self, other):
return Union(Complement(self, other), Complement(other, self))
@property
def inf(self):
"""
The infimum of 'self'
Examples
========
>>> from sympy import Interval, Union
>>> Interval(0, 1).inf
0
>>> Union(Interval(0, 1), Interval(2, 3)).inf
0
"""
return self._inf
@property
def _inf(self):
raise NotImplementedError("(%s)._inf" % self)
@property
def sup(self):
"""
The supremum of 'self'
Examples
========
>>> from sympy import Interval, Union
>>> Interval(0, 1).sup
1
>>> Union(Interval(0, 1), Interval(2, 3)).sup
3
"""
return self._sup
@property
def _sup(self):
raise NotImplementedError("(%s)._sup" % self)
[docs] def contains(self, other):
"""
Returns True if 'other' is contained in 'self' as an element.
As a shortcut it is possible to use the 'in' operator:
Examples
========
>>> from sympy import Interval
>>> Interval(0, 1).contains(0.5)
True
>>> 0.5 in Interval(0, 1)
True
"""
other = sympify(other, strict=True)
ret = sympify(self._contains(other))
if ret is None:
ret = Contains(other, self, evaluate=False)
return ret
def _contains(self, other):
raise NotImplementedError("(%s)._contains(%s)" % (self, other))
[docs] def is_subset(self, other):
"""
Returns True if 'self' is a subset of 'other'.
Examples
========
>>> from sympy import Interval
>>> Interval(0, 0.5).is_subset(Interval(0, 1))
True
>>> Interval(0, 1).is_subset(Interval(0, 1, left_open=True))
False
"""
if isinstance(other, Set):
return self.intersect(other) == self
else:
raise ValueError("Unknown argument '%s'" % other)
[docs] def issubset(self, other):
"""
Alias for :meth:`is_subset()`
"""
return self.is_subset(other)
[docs] def is_proper_subset(self, other):
"""
Returns True if 'self' is a proper subset of 'other'.
Examples
========
>>> from sympy import Interval
>>> Interval(0, 0.5).is_proper_subset(Interval(0, 1))
True
>>> Interval(0, 1).is_proper_subset(Interval(0, 1))
False
"""
if isinstance(other, Set):
return self != other and self.is_subset(other)
else:
raise ValueError("Unknown argument '%s'" % other)
[docs] def is_superset(self, other):
"""
Returns True if 'self' is a superset of 'other'.
Examples
========
>>> from sympy import Interval
>>> Interval(0, 0.5).is_superset(Interval(0, 1))
False
>>> Interval(0, 1).is_superset(Interval(0, 1, left_open=True))
True
"""
if isinstance(other, Set):
return other.is_subset(self)
else:
raise ValueError("Unknown argument '%s'" % other)
[docs] def issuperset(self, other):
"""
Alias for :meth:`is_superset()`
"""
return self.is_superset(other)
[docs] def is_proper_superset(self, other):
"""
Returns True if 'self' is a proper superset of 'other'.
Examples
========
>>> from sympy import Interval
>>> Interval(0, 1).is_proper_superset(Interval(0, 0.5))
True
>>> Interval(0, 1).is_proper_superset(Interval(0, 1))
False
"""
if isinstance(other, Set):
return self != other and self.is_superset(other)
else:
raise ValueError("Unknown argument '%s'" % other)
def _eval_powerset(self):
raise NotImplementedError('Power set not defined for: %s' % self.func)
[docs] def powerset(self):
"""
Find the Power set of 'self'.
Examples
========
>>> from sympy import FiniteSet, EmptySet
>>> A = EmptySet()
>>> A.powerset()
{EmptySet()}
>>> A = FiniteSet(1, 2)
>>> a, b, c = FiniteSet(1), FiniteSet(2), FiniteSet(1, 2)
>>> A.powerset() == FiniteSet(a, b, c, EmptySet())
True
References
==========
.. [1] http://en.wikipedia.org/wiki/Power_set
"""
return self._eval_powerset()
@property
def measure(self):
"""
The (Lebesgue) measure of 'self'
Examples
========
>>> from sympy import Interval, Union
>>> Interval(0, 1).measure
1
>>> Union(Interval(0, 1), Interval(2, 3)).measure
2
"""
return self._measure
@property
def boundary(self):
"""
The boundary or frontier of a set
A point x is on the boundary of a set S if
1. x is in the closure of S.
I.e. Every neighborhood of x contains a point in S.
2. x is not in the interior of S.
I.e. There does not exist an open set centered on x contained
entirely within S.
There are the points on the outer rim of S. If S is open then these
points need not actually be contained within S.
For example, the boundary of an interval is its start and end points.
This is true regardless of whether or not the interval is open.
Examples
========
>>> from sympy import Interval
>>> Interval(0, 1).boundary
{0, 1}
>>> Interval(0, 1, True, False).boundary
{0, 1}
"""
return self._boundary
@property
def is_open(self):
if not Intersection(self, self.boundary):
return True
# We can't confidently claim that an intersection exists
return None
@property
def is_closed(self):
return self.boundary.is_subset(self)
@property
def closure(self):
return self + self.boundary
@property
def interior(self):
return self - self.boundary
@property
def _boundary(self):
raise NotImplementedError()
def _eval_imageset(self, f):
from sympy.sets.fancysets import ImageSet
return ImageSet(f, self)
@property
def _measure(self):
raise NotImplementedError("(%s)._measure" % self)
def __add__(self, other):
return self.union(other)
def __or__(self, other):
return self.union(other)
def __and__(self, other):
return self.intersect(other)
def __mul__(self, other):
return ProductSet(self, other)
def __xor__(self, other):
return SymmetricDifference(self, other)
def __pow__(self, exp):
if not sympify(exp).is_Integer and exp >= 0:
raise ValueError("%s: Exponent must be a positive Integer" % exp)
return ProductSet([self]*exp)
def __sub__(self, other):
return Complement(self, other)
def __contains__(self, other):
symb = sympify(self.contains(other))
if not (symb is S.true or symb is S.false):
raise TypeError('contains did not evaluate to a bool: %r' % symb)
return bool(symb)
[docs]class ProductSet(Set):
"""
Represents a Cartesian Product of Sets.
Returns a Cartesian product given several sets as either an iterable
or individual arguments.
Can use '*' operator on any sets for convenient shorthand.
Examples
========
>>> from sympy import Interval, FiniteSet, ProductSet
>>> I = Interval(0, 5); S = FiniteSet(1, 2, 3)
>>> ProductSet(I, S)
[0, 5] x {1, 2, 3}
>>> (2, 2) in ProductSet(I, S)
True
>>> Interval(0, 1) * Interval(0, 1) # The unit square
[0, 1] x [0, 1]
>>> coin = FiniteSet('H', 'T')
>>> set(coin**2)
set([(H, H), (H, T), (T, H), (T, T)])
Notes
=====
- Passes most operations down to the argument sets
- Flattens Products of ProductSets
References
==========
.. [1] http://en.wikipedia.org/wiki/Cartesian_product
"""
is_ProductSet = True
def __new__(cls, *sets, **assumptions):
def flatten(arg):
if isinstance(arg, Set):
if arg.is_ProductSet:
return sum(map(flatten, arg.args), [])
else:
return [arg]
elif iterable(arg):
return sum(map(flatten, arg), [])
raise TypeError("Input must be Sets or iterables of Sets")
sets = flatten(list(sets))
if EmptySet() in sets or len(sets) == 0:
return EmptySet()
if len(sets) == 1:
return sets[0]
return Basic.__new__(cls, *sets, **assumptions)
def _eval_Eq(self, other):
if not other.is_ProductSet:
return
if len(self.args) != len(other.args):
return false
return And(*(Eq(x, y) for x, y in zip(self.args, other.args)))
def _contains(self, element):
"""
'in' operator for ProductSets
Examples
========
>>> from sympy import Interval
>>> (2, 3) in Interval(0, 5) * Interval(0, 5)
True
>>> (10, 10) in Interval(0, 5) * Interval(0, 5)
False
Passes operation on to constituent sets
"""
try:
if len(element) != len(self.args):
return false
except TypeError: # maybe element isn't an iterable
return false
return And(*
[set.contains(item) for set, item in zip(self.sets, element)])
def _intersect(self, other):
"""
This function should only be used internally
See Set._intersect for docstring
"""
if not other.is_ProductSet:
return None
if len(other.args) != len(self.args):
return S.EmptySet
return ProductSet(a.intersect(b)
for a, b in zip(self.sets, other.sets))
def _union(self, other):
if not other.is_ProductSet:
return None
if len(other.args) != len(self.args):
return None
if self.args[0] == other.args[0]:
return self.args[0] * Union(ProductSet(self.args[1:]),
ProductSet(other.args[1:]))
if self.args[-1] == other.args[-1]:
return Union(ProductSet(self.args[:-1]),
ProductSet(other.args[:-1])) * self.args[-1]
return None
@property
def sets(self):
return self.args
@property
def _boundary(self):
return Union(ProductSet(b + b.boundary if i != j else b.boundary
for j, b in enumerate(self.sets))
for i, a in enumerate(self.sets))
@property
def is_iterable(self):
return all(set.is_iterable for set in self.sets)
def __iter__(self):
if self.is_iterable:
return product(*self.sets)
else:
raise TypeError("Not all constituent sets are iterable")
@property
def _measure(self):
measure = 1
for set in self.sets:
measure *= set.measure
return measure
def __len__(self):
return Mul(*[len(s) for s in self.args])
[docs]class Interval(Set, EvalfMixin):
"""
Represents a real interval as a Set.
Usage:
Returns an interval with end points "start" and "end".
For left_open=True (default left_open is False) the interval
will be open on the left. Similarly, for right_open=True the interval
will be open on the right.
Examples
========
>>> from sympy import Symbol, Interval
>>> Interval(0, 1)
[0, 1]
>>> Interval(0, 1, False, True)
[0, 1)
>>> Interval.Ropen(0, 1)
[0, 1)
>>> Interval.Lopen(0, 1)
(0, 1]
>>> Interval.open(0, 1)
(0, 1)
>>> a = Symbol('a', real=True)
>>> Interval(0, a)
[0, a]
Notes
=====
- Only real end points are supported
- Interval(a, b) with a > b will return the empty set
- Use the evalf() method to turn an Interval into an mpmath
'mpi' interval instance
References
==========
.. [1] http://en.wikipedia.org/wiki/Interval_%28mathematics%29
"""
is_Interval = True
def __new__(cls, start, end, left_open=False, right_open=False):
start = _sympify(start)
end = _sympify(end)
left_open = _sympify(left_open)
right_open = _sympify(right_open)
if not all(isinstance(a, (type(true), type(false)))
for a in [left_open, right_open]):
raise NotImplementedError(
"left_open and right_open can have only true/false values, "
"got %s and %s" % (left_open, right_open))
inftys = [S.Infinity, S.NegativeInfinity]
# Only allow real intervals (use symbols with 'is_real=True').
if not all(i.is_real is not False or i in inftys for i in (start, end)):
raise ValueError("Non-real intervals are not supported")
# evaluate if possible
if (end < start) == True:
return S.EmptySet
elif (end - start).is_negative:
return S.EmptySet
if end == start and (left_open or right_open):
return S.EmptySet
if end == start and not (left_open or right_open):
return FiniteSet(end)
# Make sure infinite interval end points are open.
if start == S.NegativeInfinity:
left_open = true
if end == S.Infinity:
right_open = true
return Basic.__new__(cls, start, end, left_open, right_open)
@property
def start(self):
"""
The left end point of 'self'.
This property takes the same value as the 'inf' property.
Examples
========
>>> from sympy import Interval
>>> Interval(0, 1).start
0
"""
return self._args[0]
_inf = left = start
@classmethod
[docs] def open(cls, a, b):
"""Return an interval including neither boundary."""
return cls(a, b, True, True)
@classmethod
[docs] def Lopen(cls, a, b):
"""Return an interval not including the left boundary."""
return cls(a, b, True, False)
@classmethod
[docs] def Ropen(cls, a, b):
"""Return an interval not including the right boundary."""
return cls(a, b, False, True)
@property
def end(self):
"""
The right end point of 'self'.
This property takes the same value as the 'sup' property.
Examples
========
>>> from sympy import Interval
>>> Interval(0, 1).end
1
"""
return self._args[1]
_sup = right = end
@property
def left_open(self):
"""
True if 'self' is left-open.
Examples
========
>>> from sympy import Interval
>>> Interval(0, 1, left_open=True).left_open
True
>>> Interval(0, 1, left_open=False).left_open
False
"""
return self._args[2]
@property
def right_open(self):
"""
True if 'self' is right-open.
Examples
========
>>> from sympy import Interval
>>> Interval(0, 1, right_open=True).right_open
True
>>> Interval(0, 1, right_open=False).right_open
False
"""
return self._args[3]
def _intersect(self, other):
"""
This function should only be used internally
See Set._intersect for docstring
"""
# We only know how to intersect with other intervals
if not other.is_Interval:
return None
# handle (-oo, oo)
infty = S.NegativeInfinity, S.Infinity
if self == Interval(*infty):
l, r = self.left, self.right
if l.is_real or l in infty or r.is_real or r in infty:
return other
# We can't intersect [0,3] with [x,6] -- we don't know if x>0 or x<0
if not self._is_comparable(other):
return None
empty = False
if self.start <= other.end and other.start <= self.end:
# Get topology right.
if self.start < other.start:
start = other.start
left_open = other.left_open
elif self.start > other.start:
start = self.start
left_open = self.left_open
else:
start = self.start
left_open = self.left_open or other.left_open
if self.end < other.end:
end = self.end
right_open = self.right_open
elif self.end > other.end:
end = other.end
right_open = other.right_open
else:
end = self.end
right_open = self.right_open or other.right_open
if end - start == 0 and (left_open or right_open):
empty = True
else:
empty = True
if empty:
return S.EmptySet
return Interval(start, end, left_open, right_open)
def _complement(self, other):
if other == S.Reals:
a = Interval(S.NegativeInfinity, self.start,
True, not self.left_open)
b = Interval(self.end, S.Infinity, not self.right_open, True)
return Union(a, b)
if isinstance(other, FiniteSet):
nums = [m for m in other.args if m.is_number]
if nums == []:
return None
return Set._complement(self, other)
def _union(self, other):
"""
This function should only be used internally
See Set._union for docstring
"""
if other.is_UniversalSet:
return S.UniversalSet
if other.is_Interval and self._is_comparable(other):
from sympy.functions.elementary.miscellaneous import Min, Max
# Non-overlapping intervals
end = Min(self.end, other.end)
start = Max(self.start, other.start)
if (end < start or
(end == start and (end not in self and end not in other))):
return None
else:
start = Min(self.start, other.start)
end = Max(self.end, other.end)
left_open = ((self.start != start or self.left_open) and
(other.start != start or other.left_open))
right_open = ((self.end != end or self.right_open) and
(other.end != end or other.right_open))
return Interval(start, end, left_open, right_open)
# If I have open end points and these endpoints are contained in other
if ((self.left_open and sympify(other.contains(self.start)) is S.true) or
(self.right_open and sympify(other.contains(self.end)) is S.true)):
# Fill in my end points and return
open_left = self.left_open and self.start not in other
open_right = self.right_open and self.end not in other
new_self = Interval(self.start, self.end, open_left, open_right)
return set((new_self, other))
return None
@property
def _boundary(self):
finite_points = [p for p in (self.start, self.end)
if abs(p) != S.Infinity]
return FiniteSet(*finite_points)
def _contains(self, other):
if not isinstance(other, Expr) or (
other is S.Infinity or
other is S.NegativeInfinity or
other is S.NaN or
other is S.ComplexInfinity) or other.is_real is False:
return false
if self.start is S.NegativeInfinity and self.end is S.Infinity:
if not other.is_real is None:
return other.is_real
if self.left_open:
expr = other > self.start
else:
expr = other >= self.start
if self.right_open:
expr = And(expr, other < self.end)
else:
expr = And(expr, other <= self.end)
return _sympify(expr)
def _eval_imageset(self, f):
from sympy.functions.elementary.miscellaneous import Min, Max
from sympy.solvers.solveset import solveset
from sympy.core.function import diff, Lambda
from sympy.series import limit
from sympy.calculus.singularities import singularities
# TODO: handle functions with infinitely many solutions (eg, sin, tan)
# TODO: handle multivariate functions
expr = f.expr
if len(expr.free_symbols) > 1 or len(f.variables) != 1:
return
var = f.variables[0]
if expr.is_Piecewise:
result = S.EmptySet
domain_set = self
for (p_expr, p_cond) in expr.args:
if p_cond is true:
intrvl = domain_set
else:
intrvl = p_cond.as_set()
intrvl = Intersection(domain_set, intrvl)
if p_expr.is_Number:
image = FiniteSet(p_expr)
else:
image = imageset(Lambda(var, p_expr), intrvl)
result = Union(result, image)
# remove the part which has been `imaged`
domain_set = Complement(domain_set, intrvl)
if domain_set.is_EmptySet:
break
return result
if not self.start.is_comparable or not self.end.is_comparable:
return
try:
sing = [x for x in singularities(expr, var)
if x.is_real and x in self]
except NotImplementedError:
return
if self.left_open:
_start = limit(expr, var, self.start, dir="+")
elif self.start not in sing:
_start = f(self.start)
if self.right_open:
_end = limit(expr, var, self.end, dir="-")
elif self.end not in sing:
_end = f(self.end)
if len(sing) == 0:
solns = list(solveset(diff(expr, var), var))
extr = [_start, _end] + [f(x) for x in solns
if x.is_real and x in self]
start, end = Min(*extr), Max(*extr)
left_open, right_open = False, False
if _start <= _end:
# the minimum or maximum value can occur simultaneously
# on both the edge of the interval and in some interior
# point
if start == _start and start not in solns:
left_open = self.left_open
if end == _end and end not in solns:
right_open = self.right_open
else:
if start == _end and start not in solns:
left_open = self.right_open
if end == _start and end not in solns:
right_open = self.left_open
return Interval(start, end, left_open, right_open)
else:
return imageset(f, Interval(self.start, sing[0],
self.left_open, True)) + \
Union(*[imageset(f, Interval(sing[i], sing[i + 1], True, True))
for i in range(0, len(sing) - 1)]) + \
imageset(f, Interval(sing[-1], self.end, True, self.right_open))
@property
def _measure(self):
return self.end - self.start
def to_mpi(self, prec=53):
return mpi(mpf(self.start._eval_evalf(prec)),
mpf(self.end._eval_evalf(prec)))
def _eval_evalf(self, prec):
return Interval(self.left._eval_evalf(prec),
self.right._eval_evalf(prec),
left_open=self.left_open, right_open=self.right_open)
def _is_comparable(self, other):
is_comparable = self.start.is_comparable
is_comparable &= self.end.is_comparable
is_comparable &= other.start.is_comparable
is_comparable &= other.end.is_comparable
return is_comparable
@property
def is_left_unbounded(self):
"""Return ``True`` if the left endpoint is negative infinity. """
return self.left is S.NegativeInfinity or self.left == Float("-inf")
@property
def is_right_unbounded(self):
"""Return ``True`` if the right endpoint is positive infinity. """
return self.right is S.Infinity or self.right == Float("+inf")
[docs] def as_relational(self, x):
"""Rewrite an interval in terms of inequalities and logic operators."""
x = sympify(x)
if self.right_open:
right = x < self.end
else:
right = x <= self.end
if self.left_open:
left = self.start < x
else:
left = self.start <= x
return And(left, right)
def _eval_Eq(self, other):
if not other.is_Interval:
if (other.is_Union or other.is_Complement or
other.is_Intersection or other.is_ProductSet):
return
return false
return And(Eq(self.left, other.left),
Eq(self.right, other.right),
self.left_open == other.left_open,
self.right_open == other.right_open)
[docs]class Union(Set, EvalfMixin):
"""
Represents a union of sets as a :class:`Set`.
Examples
========
>>> from sympy import Union, Interval
>>> Union(Interval(1, 2), Interval(3, 4))
[1, 2] U [3, 4]
The Union constructor will always try to merge overlapping intervals,
if possible. For example:
>>> Union(Interval(1, 2), Interval(2, 3))
[1, 3]
See Also
========
Intersection
References
==========
.. [1] http://en.wikipedia.org/wiki/Union_%28set_theory%29
"""
is_Union = True
def __new__(cls, *args, **kwargs):
evaluate = kwargs.get('evaluate', global_evaluate[0])
# flatten inputs to merge intersections and iterables
args = list(args)
def flatten(arg):
if isinstance(arg, Set):
if arg.is_Union:
return sum(map(flatten, arg.args), [])
else:
return [arg]
if iterable(arg): # and not isinstance(arg, Set) (implicit)
return sum(map(flatten, arg), [])
raise TypeError("Input must be Sets or iterables of Sets")
args = flatten(args)
# Union of no sets is EmptySet
if len(args) == 0:
return S.EmptySet
# Reduce sets using known rules
if evaluate:
return Union.reduce(args)
args = list(ordered(args, Set._infimum_key))
return Basic.__new__(cls, *args)
@staticmethod
[docs] def reduce(args):
"""
Simplify a :class:`Union` using known rules
We first start with global rules like
'Merge all FiniteSets'
Then we iterate through all pairs and ask the constituent sets if they
can simplify themselves with any other constituent
"""
# ===== Global Rules =====
# Merge all finite sets
finite_sets = [x for x in args if x.is_FiniteSet]
if len(finite_sets) > 1:
a = (x for set in finite_sets for x in set)
finite_set = FiniteSet(*a)
args = [finite_set] + [x for x in args if not x.is_FiniteSet]
# ===== Pair-wise Rules =====
# Here we depend on rules built into the constituent sets
args = set(args)
new_args = True
while(new_args):
for s in args:
new_args = False
for t in args - set((s,)):
new_set = s._union(t)
# This returns None if s does not know how to intersect
# with t. Returns the newly intersected set otherwise
if new_set is not None:
if not isinstance(new_set, set):
new_set = set((new_set, ))
new_args = (args - set((s, t))).union(new_set)
break
if new_args:
args = new_args
break
if len(args) == 1:
return args.pop()
else:
return Union(args, evaluate=False)
def _complement(self, universe):
# DeMorgan's Law
return Intersection(s.complement(universe) for s in self.args)
@property
def _inf(self):
# We use Min so that sup is meaningful in combination with symbolic
# interval end points.
from sympy.functions.elementary.miscellaneous import Min
return Min(*[set.inf for set in self.args])
@property
def _sup(self):
# We use Max so that sup is meaningful in combination with symbolic
# end points.
from sympy.functions.elementary.miscellaneous import Max
return Max(*[set.sup for set in self.args])
def _contains(self, other):
return Or(*[set.contains(other) for set in self.args])
@property
def _measure(self):
# Measure of a union is the sum of the measures of the sets minus
# the sum of their pairwise intersections plus the sum of their
# triple-wise intersections minus ... etc...
# Sets is a collection of intersections and a set of elementary
# sets which made up those intersections (called "sos" for set of sets)
# An example element might of this list might be:
# ( {A,B,C}, A.intersect(B).intersect(C) )
# Start with just elementary sets ( ({A}, A), ({B}, B), ... )
# Then get and subtract ( ({A,B}, (A int B), ... ) while non-zero
sets = [(FiniteSet(s), s) for s in self.args]
measure = 0
parity = 1
while sets:
# Add up the measure of these sets and add or subtract it to total
measure += parity * sum(inter.measure for sos, inter in sets)
# For each intersection in sets, compute the intersection with every
# other set not already part of the intersection.
sets = ((sos + FiniteSet(newset), newset.intersect(intersection))
for sos, intersection in sets for newset in self.args
if newset not in sos)
# Clear out sets with no measure
sets = [(sos, inter) for sos, inter in sets if inter.measure != 0]
# Clear out duplicates
sos_list = []
sets_list = []
for set in sets:
if set[0] in sos_list:
continue
else:
sos_list.append(set[0])
sets_list.append(set)
sets = sets_list
# Flip Parity - next time subtract/add if we added/subtracted here
parity *= -1
return measure
@property
def _boundary(self):
def boundary_of_set(i):
""" The boundary of set i minus interior of all other sets """
b = self.args[i].boundary
for j, a in enumerate(self.args):
if j != i:
b = b - a.interior
return b
return Union(map(boundary_of_set, range(len(self.args))))
def _eval_imageset(self, f):
return Union(imageset(f, arg) for arg in self.args)
[docs] def as_relational(self, symbol):
"""Rewrite a Union in terms of equalities and logic operators. """
return Or(*[set.as_relational(symbol) for set in self.args])
@property
def is_iterable(self):
return all(arg.is_iterable for arg in self.args)
def _eval_evalf(self, prec):
try:
return Union(set._eval_evalf(prec) for set in self.args)
except Exception:
raise TypeError("Not all sets are evalf-able")
def __iter__(self):
import itertools
# roundrobin recipe taken from itertools documentation:
# https://docs.python.org/2/library/itertools.html#recipes
def roundrobin(*iterables):
"roundrobin('ABC', 'D', 'EF') --> A D E B F C"
# Recipe credited to George Sakkis
pending = len(iterables)
if PY3:
nexts = itertools.cycle(iter(it).__next__ for it in iterables)
else:
nexts = itertools.cycle(iter(it).next for it in iterables)
while pending:
try:
for next in nexts:
yield next()
except StopIteration:
pending -= 1
nexts = itertools.cycle(itertools.islice(nexts, pending))
if all(set.is_iterable for set in self.args):
return roundrobin(*(iter(arg) for arg in self.args))
else:
raise TypeError("Not all constituent sets are iterable")
[docs]class Intersection(Set):
"""
Represents an intersection of sets as a :class:`Set`.
Examples
========
>>> from sympy import Intersection, Interval
>>> Intersection(Interval(1, 3), Interval(2, 4))
[2, 3]
We often use the .intersect method
>>> Interval(1,3).intersect(Interval(2,4))
[2, 3]
See Also
========
Union
References
==========
.. [1] http://en.wikipedia.org/wiki/Intersection_%28set_theory%29
"""
is_Intersection = True
def __new__(cls, *args, **kwargs):
evaluate = kwargs.get('evaluate', global_evaluate[0])
# flatten inputs to merge intersections and iterables
args = list(args)
def flatten(arg):
if isinstance(arg, Set):
if arg.is_Intersection:
return sum(map(flatten, arg.args), [])
else:
return [arg]
if iterable(arg): # and not isinstance(arg, Set) (implicit)
return sum(map(flatten, arg), [])
raise TypeError("Input must be Sets or iterables of Sets")
args = flatten(args)
if len(args) == 0:
return S.EmptySet
# args can't be ordered for Partition see issue #9608
if 'Partition' not in [type(a).__name__ for a in args]:
args = list(ordered(args, Set._infimum_key))
# Reduce sets using known rules
if evaluate:
return Intersection.reduce(args)
return Basic.__new__(cls, *args)
@property
def is_iterable(self):
return any(arg.is_iterable for arg in self.args)
@property
def _inf(self):
raise NotImplementedError()
@property
def _sup(self):
raise NotImplementedError()
def _eval_imageset(self, f):
return Intersection(imageset(f, arg) for arg in self.args)
def _contains(self, other):
return And(*[set.contains(other) for set in self.args])
def __iter__(self):
no_iter = True
for s in self.args:
if s.is_iterable:
no_iter = False
other_sets = set(self.args) - set((s,))
other = Intersection(other_sets, evaluate=False)
for x in s:
c = sympify(other.contains(x))
if c is S.true:
yield x
elif c is S.false:
pass
else:
yield c
if no_iter:
raise ValueError("None of the constituent sets are iterable")
@staticmethod
def _handle_finite_sets(args):
from sympy.core.logic import fuzzy_and, fuzzy_bool
from sympy.core.compatibility import zip_longest
from sympy.utilities.iterables import sift
sifted = sift(args, lambda x: x.is_FiniteSet)
fs_args = sifted.pop(True, [])
if not fs_args:
return
s = fs_args[0]
fs_args = fs_args[1:]
other = sifted.pop(False, [])
res = []
unk = []
for x in s:
c = fuzzy_and(fuzzy_bool(o.contains(x))
for o in fs_args + other)
if c:
res.append(x)
elif c is None:
unk.append(x)
else:
pass # drop arg
res = FiniteSet(
*res, evaluate=False) if res else S.EmptySet
if unk:
symbolic_s_list = [x for x in s if x.has(Symbol)]
non_symbolic_s = s - FiniteSet(
*symbolic_s_list, evaluate=False)
while fs_args:
v = fs_args.pop()
if all(i == j for i, j in zip_longest(
symbolic_s_list,
(x for x in v if x.has(Symbol)))):
# all the symbolic elements of `v` are the same
# as in `s` so remove the non-symbol containing
# expressions from `unk`, since they cannot be
# contained
for x in non_symbolic_s:
if x in unk:
unk.remove(x)
else:
# if only a subset of elements in `s` are
# contained in `v` then remove them from `v`
# and add this as a new arg
contained = [x for x in symbolic_s_list
if sympify(v.contains(x)) is S.true]
if contained != symbolic_s_list:
other.append(
v - FiniteSet(
*contained, evaluate=False))
else:
pass # for coverage
other_sets = Intersection(*other)
if not other_sets:
return S.EmptySet # b/c we use evaluate=False below
res += Intersection(
FiniteSet(*unk),
other_sets, evaluate=False)
return res
@staticmethod
[docs] def reduce(args):
"""
Return a simplified intersection by applying rules.
We first start with global rules like
'if any empty sets, return empty set' and 'distribute unions'.
Then we iterate through all pairs and ask the constituent sets if they
can simplify themselves with any other constituent
"""
from sympy.simplify.simplify import clear_coefficients
# ===== Global Rules =====
# If any EmptySets return EmptySet
if any(s.is_EmptySet for s in args):
return S.EmptySet
# Handle Finite sets
rv = Intersection._handle_finite_sets(args)
if rv is not None:
# simplify symbolic intersection between a FiniteSet
# and an interval
if isinstance(rv, Intersection) and len(rv.args) == 2:
ivl, s = rv.args
if isinstance(s, FiniteSet) and len(s) == 1 and isinstance(ivl, Interval):
e = list(s)[0]
if e.free_symbols:
rhs = Dummy()
e, r = clear_coefficients(e, rhs)
if r != rhs:
iargs = list(ivl.args)
iargs[0] = r.subs(rhs, ivl.start)
iargs[1] = r.subs(rhs, ivl.end)
if iargs[0] > iargs[1]:
iargs = iargs[:2][::-1] + iargs[-2:][::-1]
rv = Intersection(FiniteSet(e), Interval(*iargs), evaluate=False)
return rv
# If any of the sets are unions, return a Union of Intersections
for s in args:
if s.is_Union:
other_sets = set(args) - set((s,))
if len(other_sets) > 0:
other = Intersection(other_sets)
return Union(Intersection(arg, other) for arg in s.args)
else:
return Union(arg for arg in s.args)
for s in args:
if s.is_Complement:
args.remove(s)
other_sets = args + [s.args[0]]
return Complement(Intersection(*other_sets), s.args[1])
# At this stage we are guaranteed not to have any
# EmptySets, FiniteSets, or Unions in the intersection
# ===== Pair-wise Rules =====
# Here we depend on rules built into the constituent sets
args = set(args)
new_args = True
while(new_args):
for s in args:
new_args = False
for t in args - set((s,)):
new_set = s._intersect(t)
# This returns None if s does not know how to intersect
# with t. Returns the newly intersected set otherwise
if new_set is not None:
new_args = (args - set((s, t))).union(set((new_set, )))
break
if new_args:
args = new_args
break
if len(args) == 1:
return args.pop()
else:
return Intersection(args, evaluate=False)
[docs] def as_relational(self, symbol):
"""Rewrite an Intersection in terms of equalities and logic operators"""
return And(*[set.as_relational(symbol) for set in self.args])
[docs]class Complement(Set, EvalfMixin):
"""Represents the set difference or relative complement of a set with
another set.
`A - B = \{x \in A| x \\notin B\}`
Examples
========
>>> from sympy import Complement, FiniteSet
>>> Complement(FiniteSet(0, 1, 2), FiniteSet(1))
{0, 2}
See Also
=========
Intersection, Union
References
==========
.. [1] http://mathworld.wolfram.com/ComplementSet.html
"""
is_Complement = True
def __new__(cls, a, b, evaluate=True):
if evaluate:
return Complement.reduce(a, b)
return Basic.__new__(cls, a, b)
@staticmethod
[docs] def reduce(A, B):
"""
Simplify a :class:`Complement`.
"""
if B == S.UniversalSet or A.is_subset(B):
return EmptySet()
if isinstance(B, Union):
return Intersection(s.complement(A) for s in B.args)
result = B._complement(A)
if result != None:
return result
else:
return Complement(A, B, evaluate=False)
def _contains(self, other):
A = self.args[0]
B = self.args[1]
return And(A.contains(other), Not(B.contains(other)))
[docs]class EmptySet(with_metaclass(Singleton, Set)):
"""
Represents the empty set. The empty set is available as a singleton
as S.EmptySet.
Examples
========
>>> from sympy import S, Interval
>>> S.EmptySet
EmptySet()
>>> Interval(1, 2).intersect(S.EmptySet)
EmptySet()
See Also
========
UniversalSet
References
==========
.. [1] http://en.wikipedia.org/wiki/Empty_set
"""
is_EmptySet = True
is_FiniteSet = True
def _intersect(self, other):
return S.EmptySet
@property
def _measure(self):
return 0
def _contains(self, other):
return false
def as_relational(self, symbol):
return false
def __len__(self):
return 0
def _union(self, other):
return other
def __iter__(self):
return iter([])
def _eval_imageset(self, f):
return self
def _eval_powerset(self):
return FiniteSet(self)
@property
def _boundary(self):
return self
def _complement(self, other):
return other
def _symmetric_difference(self, other):
return other
[docs]class UniversalSet(with_metaclass(Singleton, Set)):
"""
Represents the set of all things.
The universal set is available as a singleton as S.UniversalSet
Examples
========
>>> from sympy import S, Interval
>>> S.UniversalSet
UniversalSet()
>>> Interval(1, 2).intersect(S.UniversalSet)
[1, 2]
See Also
========
EmptySet
References
==========
.. [1] http://en.wikipedia.org/wiki/Universal_set
"""
is_UniversalSet = True
def _intersect(self, other):
return other
def _complement(self, other):
return S.EmptySet
def _symmetric_difference(self, other):
return other
@property
def _measure(self):
return S.Infinity
def _contains(self, other):
return true
def as_relational(self, symbol):
return true
def _union(self, other):
return self
@property
def _boundary(self):
return EmptySet()
[docs]class FiniteSet(Set, EvalfMixin):
"""
Represents a finite set of discrete numbers
Examples
========
>>> from sympy import FiniteSet
>>> FiniteSet(1, 2, 3, 4)
{1, 2, 3, 4}
>>> 3 in FiniteSet(1, 2, 3, 4)
True
>>> members = [1, 2, 3, 4]
>>> FiniteSet(*members)
{1, 2, 3, 4}
References
==========
.. [1] http://en.wikipedia.org/wiki/Finite_set
"""
is_FiniteSet = True
is_iterable = True
def __new__(cls, *args, **kwargs):
evaluate = kwargs.get('evaluate', global_evaluate[0])
if evaluate:
args = list(map(sympify, args))
if len(args) == 0:
return EmptySet()
else:
args = list(map(sympify, args))
args = list(ordered(frozenset(tuple(args)), Set._infimum_key))
obj = Basic.__new__(cls, *args)
obj._elements = frozenset(args)
return obj
def _eval_Eq(self, other):
if not other.is_FiniteSet:
if (other.is_Union or other.is_Complement or
other.is_Intersection or other.is_ProductSet):
return
return false
if len(self) != len(other):
return false
return And(*(Eq(x, y) for x, y in zip(self.args, other.args)))
def __iter__(self):
return iter(self.args)
def _intersect(self, other):
"""
This function should only be used internally
See Set._intersect for docstring
"""
if isinstance(other, self.__class__):
return self.__class__(*(self._elements & other._elements))
return self.__class__(*[el for el in self if el in other])
def _complement(self, other):
if isinstance(other, Interval):
nums = sorted(m for m in self.args if m.is_number)
if other == S.Reals and nums != []:
syms = [m for m in self.args if m.is_Symbol]
# Reals cannot contain elements other than numbers and symbols.
intervals = [] # Build up a list of intervals between the elements
intervals += [Interval(S.NegativeInfinity, nums[0], True, True)]
for a, b in zip(nums[:-1], nums[1:]):
intervals.append(Interval(a, b, True, True)) # both open
intervals.append(Interval(nums[-1], S.Infinity, True, True))
if syms != []:
return Complement(Union(intervals, evaluate=False),
FiniteSet(*syms), evaluate=False)
else:
return Union(intervals, evaluate=False)
elif nums == []:
return None
elif isinstance(other, FiniteSet):
unk = []
for i in self:
c = sympify(other.contains(i))
if c is not S.true and c is not S.false:
unk.append(i)
unk = FiniteSet(*unk)
if unk == self:
return
not_true = []
for i in other:
c = sympify(self.contains(i))
if c is not S.true:
not_true.append(i)
return Complement(FiniteSet(*not_true), unk)
return Set._complement(self, other)
def _union(self, other):
"""
This function should only be used internally
See Set._union for docstring
"""
if other.is_FiniteSet:
return FiniteSet(*(self._elements | other._elements))
# If other set contains one of my elements, remove it from myself
if any(sympify(other.contains(x)) is S.true for x in self):
return set((
FiniteSet(*[x for x in self
if other.contains(x) != True]), other))
return None
def _contains(self, other):
"""
Tests whether an element, other, is in the set.
Relies on Python's set class. This tests for object equality
All inputs are sympified
Examples
========
>>> from sympy import FiniteSet
>>> 1 in FiniteSet(1, 2)
True
>>> 5 in FiniteSet(1, 2)
False
"""
r = false
for e in self._elements:
t = Eq(e, other, evaluate=True)
if isinstance(t, Eq):
t = t.simplify()
if t == true:
return t
elif t != false:
r = None
return r
def _eval_imageset(self, f):
return FiniteSet(*map(f, self))
@property
def _boundary(self):
return self
@property
def _inf(self):
from sympy.functions.elementary.miscellaneous import Min
return Min(*self)
@property
def _sup(self):
from sympy.functions.elementary.miscellaneous import Max
return Max(*self)
@property
def measure(self):
return 0
def __len__(self):
return len(self.args)
[docs] def as_relational(self, symbol):
"""Rewrite a FiniteSet in terms of equalities and logic operators. """
from sympy.core.relational import Eq
return Or(*[Eq(symbol, elem) for elem in self])
def compare(self, other):
return (hash(self) - hash(other))
def _eval_evalf(self, prec):
return FiniteSet(*[elem._eval_evalf(prec) for elem in self])
def _hashable_content(self):
return (self._elements,)
@property
def _sorted_args(self):
return tuple(ordered(self.args, Set._infimum_key))
def _eval_powerset(self):
return self.func(*[self.func(*s) for s in subsets(self.args)])
def __ge__(self, other):
if not isinstance(other, Set):
raise TypeError("Invalid comparison of set with %s" % func_name(other))
return other.is_subset(self)
def __gt__(self, other):
if not isinstance(other, Set):
raise TypeError("Invalid comparison of set with %s" % func_name(other))
return self.is_proper_superset(other)
def __le__(self, other):
if not isinstance(other, Set):
raise TypeError("Invalid comparison of set with %s" % func_name(other))
return self.is_subset(other)
def __lt__(self, other):
if not isinstance(other, Set):
raise TypeError("Invalid comparison of set with %s" % func_name(other))
return self.is_proper_subset(other)
converter[set] = lambda x: FiniteSet(*x)
converter[frozenset] = lambda x: FiniteSet(*x)
class SymmetricDifference(Set):
"""Represents the set of elements which are in either of the
sets and not in their intersection.
Examples
========
>>> from sympy import SymmetricDifference, FiniteSet
>>> SymmetricDifference(FiniteSet(1, 2, 3), FiniteSet(3, 4, 5))
{1, 2, 4, 5}
See Also
========
Complement, Union
References
==========
.. [1] http://en.wikipedia.org/wiki/Symmetric_difference
"""
is_SymmetricDifference = True
def __new__(cls, a, b, evaluate=True):
if evaluate:
return SymmetricDifference.reduce(a, b)
return Basic.__new__(cls, a, b)
@staticmethod
def reduce(A, B):
result = B._symmetric_difference(A)
if result is not None:
return result
else:
return SymmetricDifference(A, B, evaluate=False)
[docs]def imageset(*args):
r"""
Return an image of the set under transformation ``f``.
If this function can't compute the image, it returns an
unevaluated ImageSet object.
.. math::
{ f(x) | x \in self }
Examples
========
>>> from sympy import S, Interval, Symbol, imageset, sin, Lambda
>>> from sympy.abc import x, y
>>> imageset(x, 2*x, Interval(0, 2))
[0, 4]
>>> imageset(lambda x: 2*x, Interval(0, 2))
[0, 4]
>>> imageset(Lambda(x, sin(x)), Interval(-2, 1))
ImageSet(Lambda(x, sin(x)), [-2, 1])
>>> imageset(sin, Interval(-2, 1))
ImageSet(Lambda(x, sin(x)), [-2, 1])
>>> imageset(lambda y: x + y, Interval(-2, 1))
ImageSet(Lambda(_x, _x + x), [-2, 1])
Expressions applied to the set of Integers are simplified
to show as few negatives as possible and linear expressions
are converted to a canonical form. If this is not desirable
then the unevaluated ImageSet should be used.
>>> imageset(x, -2*x + 5, S.Integers)
ImageSet(Lambda(x, 2*x + 1), Integers())
See Also
========
sympy.sets.fancysets.ImageSet
"""
from sympy.core import Lambda
from sympy.sets.fancysets import ImageSet
from sympy.geometry.util import _uniquely_named_symbol
if len(args) not in (2, 3):
raise ValueError('imageset expects 2 or 3 args, got: %s' % len(args))
set = args[-1]
if not isinstance(set, Set):
name = func_name(set)
raise ValueError(
'last argument should be a set, not %s' % name)
if len(args) == 3:
f = Lambda(*args[:2])
elif len(args) == 2:
f = args[0]
if isinstance(f, Lambda):
pass
elif (
isinstance(f, FunctionClass) # like cos
or func_name(f) == '<lambda>'
):
var = _uniquely_named_symbol(Symbol('x'), f(Dummy()))
expr = f(var)
f = Lambda(var, expr)
else:
raise TypeError(filldedent('''
expecting lambda, Lambda, or FunctionClass, not \'%s\'''' %
func_name(f)))
r = set._eval_imageset(f)
if isinstance(r, ImageSet):
f, set = r.args
if f.variables[0] == f.expr:
return set
if isinstance(set, ImageSet):
if len(set.lamda.variables) == 1 and len(f.variables) == 1:
return imageset(Lambda(set.lamda.variables[0],
f.expr.subs(f.variables[0], set.lamda.expr)),
set.base_set)
if r is not None:
return r
return ImageSet(f, set)
