Source code for sympy.sets.sets

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)) Union(Interval(0, 1), Interval(2, 3)) >>> Interval(0, 1) + Interval(2, 3) Union(Interval(0, 1), Interval(2, 3)) >>> Interval(1, 2, True, True) + FiniteSet(2, 3) Union(Interval.Lopen(1, 2), {3}) Similarly it is possible to use the '-' operator for set differences: >>> Interval(0, 2) - Interval(0, 1) Interval.Lopen(1, 2) >>> Interval(1, 3) - FiniteSet(2) Union(Interval.Ropen(1, 2), Interval.Lopen(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)) Interval(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): r""" The complement of 'self' w.r.t the given the universe. Examples ======== >>> from sympy import Interval, S >>> Interval(0, 1).complement(S.Reals) Union(Interval.open(-oo, 0), Interval.open(1, oo)) >>> Interval(0, 1).complement(S.UniversalSet) UniversalSet() \ Interval(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])
[docs] def symmetric_difference(self, other): """ Returns symmetric difference of `self` and `other`. Examples ======== >>> from sympy import Interval, S >>> Interval(1, 3).symmetric_difference(S.Reals) Union(Interval.open(-oo, 1), Interval.open(3, oo)) >>> Interval(1, 10).symmetric_difference(S.Reals) Union(Interval.open(-oo, 1), Interval.open(10, oo)) >>> from sympy import S, EmptySet >>> S.Reals.symmetric_difference(EmptySet()) S.Reals References ========== .. [1] https://en.wikipedia.org/wiki/Symmetric_difference """ 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): """ Property method to check whether a set is open. A set is open if and only if it has an empty intersection with its boundary. Examples ======== >>> from sympy import S >>> S.Reals.is_open True """ if not Intersection(self, self.boundary): return True # We can't confidently claim that an intersection exists return None @property def is_closed(self): """ A property method to check whether a set is closed. A set is closed if it's complement is an open set. Examples ======== >>> from sympy import Interval >>> Interval(0, 1).is_closed True """ return self.boundary.is_subset(self) @property def closure(self): """ Property method which returns the closure of a set. The closure is defined as the union of the set itself and its boundary. Examples ======== >>> from sympy import S, Interval >>> S.Reals.closure S.Reals >>> Interval(0, 1).closure Interval(0, 1) """ return self + self.boundary @property def interior(self): """ Property method which returns the interior of a set. The interior of a set S consists all points of S that do not belong to the boundary of S. Examples ======== >>> from sympy import Interval >>> Interval(0, 1).interior Interval.open(0, 1) >>> Interval(0, 1).boundary.interior EmptySet() """ 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) Interval(0, 5) x {1, 2, 3} >>> (2, 2) in ProductSet(I, S) True >>> Interval(0, 1) * Interval(0, 1) # The unit square Interval(0, 1) x Interval(0, 1) >>> coin = FiniteSet('H', 'T') >>> set(coin**2) {(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 other.is_subset(self): return self 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): """ A property method which tests whether a set is iterable or not. Returns True if set is iterable, otherwise returns False. Examples ======== >>> from sympy import FiniteSet, Interval, ProductSet >>> I = Interval(0, 1) >>> A = FiniteSet(1, 2, 3, 4, 5) >>> I.is_iterable False >>> A.is_iterable True """ return all(set.is_iterable for set in self.sets) def __iter__(self): """ A method which implements is_iterable property method. If self.is_iterable returns True (both constituent sets are iterable), then return the Cartesian Product. Otherwise, raise TypeError. """ 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]) def __bool__(self): return all([bool(s) for s in self.args]) __nonzero__ = __bool__
[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) Interval(0, 1) >>> Interval.Ropen(0, 1) Interval.Ropen(0, 1) >>> Interval.Ropen(0, 1) Interval.Ropen(0, 1) >>> Interval.Lopen(0, 1) Interval.Lopen(0, 1) >>> Interval.open(0, 1) Interval.open(0, 1) >>> a = Symbol('a', real=True) >>> Interval(0, a) Interval(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): if start == S.Infinity or start == S.NegativeInfinity: return S.EmptySet 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
[docs] @classmethod def open(cls, a, b): """Return an interval including neither boundary.""" return cls(a, b, True, True)
[docs] @classmethod def Lopen(cls, a, b): """Return an interval not including the left boundary.""" return cls(a, b, True, False)
[docs] @classmethod 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 """ if other.is_EmptySet: return other # 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. # But only in case, when endpoints are finite. Because # interval does not contain oo or -oo. open_left_in_other_and_finite = (self.left_open and sympify(other.contains(self.start)) is S.true and self.start.is_finite) open_right_in_other_and_finite = (self.right_open and sympify(other.contains(self.end)) is S.true and self.end.is_finite) if open_left_in_other_and_finite or open_right_in_other_and_finite: # 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)) Union(Interval(1, 2), Interval(3, 4)) The Union constructor will always try to merge overlapping intervals, if possible. For example: >>> Union(Interval(1, 2), Interval(2, 3)) Interval(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)
[docs] @staticmethod 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)) Interval(2, 3) We often use the .intersect method >>> Interval(1,3).intersect(Interval(2,4)) Interval(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.UniversalSet # 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
[docs] @staticmethod 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: 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): r"""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)
[docs] @staticmethod 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) Interval(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] >>> f = FiniteSet(*members) >>> f {1, 2, 3, 4} >>> f - FiniteSet(2) {1, 3, 4} >>> f + FiniteSet(2, 5) {1, 2, 3, 4, 5} 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)) Interval(0, 4) >>> imageset(lambda x: 2*x, Interval(0, 2)) Interval(0, 4) >>> imageset(Lambda(x, sin(x)), Interval(-2, 1)) ImageSet(Lambda(x, sin(x)), Interval(-2, 1)) >>> imageset(sin, Interval(-2, 1)) ImageSet(Lambda(x, sin(x)), Interval(-2, 1)) >>> imageset(lambda y: x + y, Interval(-2, 1)) ImageSet(Lambda(_x, _x + x), Interval(-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), S.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)