Source code for sympy.core.sets

from sympy.core.sympify import _sympify, sympify
from sympy.core.basic import Basic
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
from sympy.core.decorators import deprecated

from sympy.mpmath import mpi, mpf
from sympy.assumptions import ask
from sympy.logic.boolalg import And, Or

from sympy.utilities import default_sort_key

[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 FiniteSet for that. Real intervals are represented by the Interval class and unions of sets by the Union class. The empty set is represented by the 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
[docs] def sort_key(self, order=None): """ Give sort_key of infimum (if possible) else sort_key of the set. """ try: infimum = self.inf if infimum.is_comparable: return default_sort_key(infimum, order) except (NotImplementedError, ValueError): pass args = tuple([default_sort_key(a, order) for a in self._sorted_args]) return self.class_key(), (len(args), args), S.One.class_key(), S.One
[docs] def union(self, other): """ Returns the union of 'self' and 'other'. 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] """ return Intersection(self, 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 Union, FiniteSet, or EmptySet Used within the Intersection class """ return None 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 Union class """ return None @property
[docs] def complement(self): """ The complement of 'self'. As a shortcut it is possible to use the '~' or '-' operators: >>> from sympy import Interval >>> Interval(0, 1).complement (-oo, 0) U (1, oo) >>> ~Interval(0, 1) (-oo, 0) U (1, oo) >>> -Interval(0, 1) (-oo, 0) U (1, oo) """ return self._complement
@property def _complement(self): raise NotImplementedError("(%s)._complement" % self) @property
[docs] def inf(self): """ The infimum of 'self' >>> 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
[docs] def sup(self): """ The supremum of 'self' >>> 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: >>> from sympy import Interval >>> Interval(0, 1).contains(0.5) True >>> 0.5 in Interval(0, 1) True """ return self._contains(sympify(other, strict=True))
def _contains(self, other): raise NotImplementedError("(%s)._contains(%s)" % (self, other))
[docs] def subset(self, other): """ Returns True if 'other' is a subset of 'self'. >>> from sympy import Interval >>> Interval(0, 1).contains(0) True >>> Interval(0, 1, left_open=True).contains(0) False """ if isinstance(other, Set): return self.intersect(other) == other else: raise ValueError("Unknown argument '%s'" % other)
@property
[docs] def measure(self): """ The (Lebesgue) measure of 'self' >>> from sympy import Interval, Union >>> Interval(0, 1).measure 1 >>> Union(Interval(0, 1), Interval(2, 3)).measure 2 """ return self._measure
@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 __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 self.intersect(other.complement) def __neg__(self): return self.complement def __invert__(self): return self.complement def __contains__(self, other): symb = self.contains(other) result = ask(symb) if result is None: raise TypeError('contains did not evaluate to a bool: %r' % symb) return result @property def is_real(self): return None
[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 ========== 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(list(map(flatten, arg.args)), []) else: return [arg] elif iterable(arg): return sum(list(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() return Basic.__new__(cls, *sets, **assumptions) def _contains(self, element): """ 'in' operator for ProductSets >>> 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)) @property def sets(self): return self.args @property def _complement(self): # 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 switch_sets = ProductSet(FiniteSet(s, s.complement) for s in self.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 != self) @property def is_real(self): return all(set.is_real for set in self.sets) @property def is_iterable(self): return all(set.is_iterable for set in self.sets) def __iter__(self): if self.is_iterable: from sympy.core.compatibility import product 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
[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, sets >>> Interval(0, 1) [0, 1] >>> Interval(0, 1, False, True) [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 ========== <http://en.wikipedia.org/wiki/Interval_(mathematics)> """ is_Interval = True is_real = True def __new__(cls, start, end, left_open=False, right_open=False): start = _sympify(start) end = _sympify(end) # Only allow real intervals (use symbols with 'is_real=True'). if not start.is_real or not end.is_real: raise ValueError("Only real intervals are supported") # Make sure that the created interval will be valid. if end.is_comparable and start.is_comparable: if end < start: 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
[docs] def start(self): """ The left end point of 'self'. This property takes the same value as the 'inf' property. >>> from sympy import Interval >>> Interval(0, 1).start 0 """ return self._args[0]
_inf = left = start @property
[docs] def end(self): """ The right end point of 'self'. This property takes the same value as the 'sup' property. >>> from sympy import Interval >>> Interval(0, 1).end 1 """ return self._args[1]
_sup = right = end @property
[docs] def left_open(self): """ True if 'self' is left-open. >>> 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
[docs] def right_open(self): """ True if 'self' is right-open. >>> 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 # 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 _union(self, other): """ This function should only be used internally See Set._union for docstring """ 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 other.contains(self.start) is True) or (self.right_open and other.contains(self.end) is 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 _complement(self): 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) def _contains(self, other): 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 expr @property def _measure(self): return self.end - self.start def to_mpi(self, prec=53): return mpi(mpf(self.start.evalf(prec)), mpf(self.end.evalf(prec))) def _eval_evalf(self, prec): return Interval(self.left.evalf(), self.right.evalf(), 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
[docs] 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
[docs] 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, symbol): """Rewrite an interval in terms of inequalities and logic operators. """ from sympy.core.relational import Lt, Le if not self.is_left_unbounded: if self.left_open: left = Lt(self.start, symbol) else: left = Le(self.start, symbol) if not self.is_right_unbounded: if self.right_open: right = Lt(symbol, self.right) else: right = Le(symbol, self.right) if self.is_left_unbounded and self.is_right_unbounded: return True # XXX: Contained(symbol, Floats) elif self.is_left_unbounded: return right elif self.is_right_unbounded: return left else: return And(left, right)
[docs]class Union(Set, EvalfMixin): """ Represents a union of sets as a 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 ========== <http://en.wikipedia.org/wiki/Union_(set_theory)> """ is_Union = True def __new__(cls, *args, **kwargs): evaluate = kwargs.get('evaluate', True) # flatten inputs to merge intersections and iterables args = list(args) def flatten(arg): if isinstance(arg, Set): if arg.is_Union: return sum(list(map(flatten, arg.args)), []) else: return [arg] if iterable(arg): # and not isinstance(arg, Set) (implicit) return sum(list(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 args = sorted(args, key=default_sort_key) # Reduce sets using known rules if evaluate: return Union.reduce(args) return Basic.__new__(cls, *args) @staticmethod
[docs] def reduce(args): """ Simplify a 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: finite_set = FiniteSet(x for set in finite_sets for x in set) 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)
@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]) @property def _complement(self): # De Morgan's formula. complement = self.args[0].complement for set in self.args[1:]: complement = complement.intersect(set.complement) return complement def _contains(self, other): or_args = [the_set.contains(other) for the_set in self.args] return Or(*or_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
[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.evalf() for set in self.args) except: raise TypeError("Not all sets are evalf-able") def __iter__(self): import itertools if all(set.is_iterable for set in self.args): return itertools.chain(*(iter(arg) for arg in self.args)) else: raise TypeError("Not all constituent sets are iterable") @property def is_real(self): return all(set.is_real for set in self.args)
[docs]class Intersection(Set): """ Represents an intersection of sets as a 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 ========== <http://en.wikipedia.org/wiki/Intersection_(set_theory)> """ is_Intersection = True def __new__(cls, *args, **kwargs): evaluate = kwargs.get('evaluate', True) # flatten inputs to merge intersections and iterables args = list(args) def flatten(arg): if isinstance(arg, Set): if arg.is_Intersection: return sum(list(map(flatten, arg.args)), []) else: return [arg] if iterable(arg): # and not isinstance(arg, Set) (implicit) return sum(list(map(flatten, arg)), []) raise TypeError("Input must be Sets or iterables of Sets") args = flatten(args) # Intersection of no sets is everything if len(args)==0: return S.UniversalSet args = sorted(args, key=default_sort_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() @property def _complement(self): raise NotImplementedError() def _contains(self, other): from sympy.logic.boolalg import And return And(*[set.contains(other) for set in self.args]) def __iter__(self): for s in self.args: if s.is_iterable: other_sets = set(self.args) - set((s,)) other = Intersection(other_sets, evaluate=False) return (x for x in s if x in other) raise ValueError("None of the constituent sets are iterable") @staticmethod
[docs] def reduce(args): """ Simplify an intersection using known rules We first start with global rules like 'if any empty sets return empty set' and 'distribute any unions' Then we iterate through all pairs and ask the constituent sets if they can simplify themselves with any other constituent """ # ===== Global Rules ===== # If any EmptySets return EmptySet if any(s.is_EmptySet for s in args): return S.EmptySet # If any FiniteSets see which elements of that finite set occur within # all other sets in the intersection for s in args: if s.is_FiniteSet: return s.__class__(x for x in s if all(x in other for other in args)) # 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,)) other = Intersection(other_sets) return Union(Intersection(arg, other) for arg in s.args) # 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 EmptySet(Set, metaclass=Singleton): """ 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 ========== http://en.wikipedia.org/wiki/Empty_set """ is_EmptySet = True def _intersect(self, other): return S.EmptySet @property def _complement(self): return S.UniversalSet @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([])
[docs]class UniversalSet(Set, metaclass=Singleton): """ 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 ========== http://en.wikipedia.org/wiki/Universal_set """ is_UniversalSet = True def _intersect(self, other): return other @property def _complement(self): return S.EmptySet @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
[docs]class FiniteSet(Set, EvalfMixin): """ Represents a finite set of discrete numbers Examples ======== >>> from sympy import Symbol, FiniteSet, sets >>> FiniteSet(1, 2, 3, 4) {1, 2, 3, 4} >>> 3 in FiniteSet(1, 2, 3, 4) True References ========== http://en.wikipedia.org/wiki/Finite_set """ is_FiniteSet = True is_iterable = True def __new__(cls, *args): if len(args)==1 and iterable(args[0]): args = args[0] args = list(map(sympify, args)) if len(args) == 0: return EmptySet() args = frozenset(args) # remove duplicates obj = Basic.__new__(cls, *args) obj._elements = args return obj 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 _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(other.contains(x) is True for x in self): return set(( FiniteSet(x for x in self if other.contains(x) is not 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 >>> from sympy import FiniteSet >>> 1 in FiniteSet(1, 2) True >>> 5 in FiniteSet(1, 2) False """ return other in self._elements @property def _complement(self): """ The complement of a real finite set is the Union of open Intervals between the elements of the set. >>> from sympy import FiniteSet >>> FiniteSet(1, 2, 3).complement (-oo, 1) U (1, 2) U (2, 3) U (3, oo) """ if not all(elem.is_number for elem in self): raise ValueError("%s: Complement not defined for symbolic inputs" %self) # as there are only numbers involved, a straight sort is sufficient; # default_sort_key is not needed args = sorted(self.args) intervals = [] # Build up a list of intervals between the elements intervals += [Interval(S.NegativeInfinity, args[0], True, True)] for a, b in zip(args[:-1], args[1:]): intervals.append(Interval(a, b, True, True)) # open intervals intervals.append(Interval(args[-1], S.Infinity, True, True)) return Union(intervals, evaluate=False) @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) def __sub__(self, other): return FiniteSet(el for el in self if el not in other)
[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])
@property def is_real(self): return all(el.is_real for el in self) def compare(self, other): return (hash(self) - hash(other)) def _eval_evalf(self, prec): return FiniteSet(elem.evalf(prec) for elem in self) def _hashable_content(self): return (self._elements,) @property def _sorted_args(self): from sympy.utilities import default_sort_key return sorted(self.args, key=default_sort_key)