Source code for diofant.sets.sets

import itertools

from mpmath import mpf, mpi

from ..core import Basic, Eq, Expr, Mul, S, nan, oo, sympify, zoo
from ..core.compatibility import iterable, ordered
from ..core.evalf import EvalfMixin
from ..core.evaluate import global_evaluate
from ..core.singleton import Singleton
from ..logic import And, Not, Or, false, true
from ..utilities import subsets
from .contains import Contains


[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_SymmetricDifference = None @staticmethod def _infimum_key(expr): """ Return infimum (if possible) else oo. """ try: infimum = expr.inf assert infimum.is_comparable except (NotImplementedError, AttributeError, AssertionError, ValueError): infimum = oo 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: >>> 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 intersection(self, other): """ Returns the intersection of 'self' and 'other'. >>> Interval(1, 3).intersection(Interval(1, 2)) [1, 2] """ return Intersection(self, other)
def _intersection(self, other): """ This function should only be used internally self._intersection(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
[docs] def is_disjoint(self, other): """ Returns True if 'self' and 'other' are disjoint Examples ======== >>> Interval(0, 2).is_disjoint(Interval(1, 2)) False >>> Interval(0, 2).is_disjoint(Interval(3, 4)) True References ========== * https//en.wikipedia.org/wiki/Disjoint_sets """ return self.intersection(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 Diofant 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
[docs] def complement(self, universe): r""" The complement of 'self' w.r.t the given the universe. Examples ======== >>> 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 other.is_subset(self): return S.EmptySet elif 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 # dimensionality 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, FiniteSet): unks = FiniteSet(*[el for el in other if self.contains(el) not in [S.true, S.false]]) other = FiniteSet(*[el for el in other if self.contains(el) is not S.true]) ret = FiniteSet(*[el for el in other if self.contains(el) is S.false]) if unks: ret |= Complement(FiniteSet(*unks), self, evaluate=False) return ret 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 ======== >>> 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 ======== >>> 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 ======== >>> Interval(0, 1).contains(0.5) true >>> 0.5 in Interval(0, 1) True """ other = sympify(other, strict=True) ret = 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 ======== >>> 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.intersection(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 ======== >>> 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 ======== >>> 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 ======== >>> 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 ======== >>> 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 ========== * https//en.wikipedia.org/wiki/Power_set """ return self._eval_powerset()
@property def measure(self): """ The (Lebesgue) measure of 'self' Examples ======== >>> 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 ======== >>> Interval(0, 1).boundary {0, 1} >>> Interval(0, 1, True, False).boundary {0, 1} """ return self._boundary @property def is_open(self): """ Test if a set is open. A set is open if it has an empty intersection with its boundary. Examples ======== >>> S.Reals.is_open True See Also ======== boundary """ if not Intersection(self, self.boundary): return True @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 .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.intersection(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 or 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 = self.contains(other) if symb not in (true, 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 ======== >>> 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] >>> H, T = Symbol('H'), Symbol('T') >>> 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 ========== * https//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 ======== >>> (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(*[s.contains(i) for s, i in zip(self.sets, element)]) def _intersection(self, other): """ This function should only be used internally See Set._intersection for docstring """ if not other.is_ProductSet: return if len(other.args) != len(self.args): return S.EmptySet return ProductSet(a.intersection(b) for a, b in zip(self.sets, other.sets)) def _union(self, other): if not other.is_ProductSet: return if len(other.args) != len(self.args): return 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] @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): from ..utilities.iterables import cantor_product if self.is_iterable: return cantor_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 int(Mul(*[len(s) for s in self.args]))
[docs]class Interval(Set, EvalfMixin): """ Represents a real interval as a Set. 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 ======== >>> 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 ========== * https//en.wikipedia.org/wiki/Interval_%28mathematics%29 """ is_Interval = True def __new__(cls, start, end, left_open=False, right_open=False): start = sympify(start, strict=True) end = sympify(end, strict=True) left_open = sympify(left_open, strict=True) right_open = sympify(right_open, strict=True) 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)) if not all(i.is_extended_real is not False for i in (start, end)): raise ValueError("Non-real intervals are not supported") if (end - start).is_negative: return S.EmptySet is_open = left_open or right_open if end == start and is_open: return S.EmptySet if end == start and not is_open: return FiniteSet(end) 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 ======== >>> 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 ======== >>> Interval(0, 1).end 1 """ return self.args[1] _sup = right = end @property def left_open(self): """ True if 'self' is left-open. Examples ======== >>> 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 ======== >>> Interval(0, 1, right_open=True).right_open true >>> Interval(0, 1, right_open=False).right_open false """ return self.args[3] def _intersection(self, other): """ This function should only be used internally See Set._intersection for docstring """ # We only know how to intersect with other intervals if not other.is_Interval: return # handle (-oo, oo) if Eq(self, S.Reals) is S.true: 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 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 is S.Reals: a = Interval(-oo, self.start, True, not self.left_open) b = Interval(self.end, oo, not self.right_open, True) return Union(a, b) 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 ..functions 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 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 {new_self, other} @property def _boundary(self): return FiniteSet(self.start, self.end) def _contains(self, other): if not isinstance(other, Expr) or other in (nan, zoo): return false if other.is_extended_real is False: return false 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, strict=True) def _eval_imageset(self, f): from ..functions import Min, Max from ..solvers import solve from ..core import diff, Lambda from ..series import limit from ..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 S.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 (p_expr, p_cond) != expr.args[-1] and domain_set.is_EmptySet: break return result try: sing = [x for x in singularities(expr, var) if x.is_extended_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 = solve(diff(expr, var), var) extr = [_start, _end] + [f(x[var]) for x in solns if (x[var].is_extended_real and x[var] 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(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 == -oo @property def is_right_unbounded(self): """Return ``True`` if the right endpoint is positive infinity. """ return self.right == oo
[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 ======== >>> 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 ========== * https//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 - {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 = {new_set} new_args = (args - {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 ..functions 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 ..functions import Max return Max(*[set.sup for set in self.args]) 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.intersection(B).intersection(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.intersection(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): return Union(set._eval_evalf(prec) for set in self.args) def __iter__(self): # roundrobin recipe taken from itertools documentation: # https://docs.python.org/3/library/itertools.html#itertools-recipes def roundrobin(*iterables): """roundrobin('ABC', 'D', 'EF') --> A D E B F C""" sentinel = object() it = itertools.chain.from_iterable(itertools.zip_longest(fillvalue=sentinel, *iterables)) return (i for i in it if i is not sentinel) 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 ======== >>> Intersection(Interval(1, 3), Interval(2, 4)) [2, 3] We often use the .intersect method >>> Interval(1, 3).intersection(Interval(2, 4)) [2, 3] See Also ======== Union References ========== * https//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: raise TypeError("Intersection expected at least one argument") 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) 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): for s in self.args: if s.is_iterable: other_sets = set(self.args) - {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")
[docs] @staticmethod 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 FiniteSets see which elements of that finite set occur within # all other sets in the intersection for s in args: if s.is_FiniteSet: args = [a for a in args if a != s] res = s.func(*[x for x in s if all(other.contains(x) is S.true for other in args)]) unk = [x for x in s if any(other.contains(x) not in (S.true, S.false) for other in args)] if unk: res += Intersection(*([s.func(*unk)] + args), evaluate=False) return res # If any of the sets are unions, return a Union of Intersections for s in args: if s.is_Union: other_sets = set(args) - {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: other_sets = args + [s.args[0]] other_sets.remove(s) 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 - {s}: new_set = s._intersection(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 - {s, t}).union({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 relative complement of a set with another set. `A - B = \{x \in A| x \notin B\}` Examples ======== >>> Complement(FiniteSet(0, 1, 2), FiniteSet(1)) {0, 2} See Also ========= Intersection, Union References ========== * 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`. """ result = B._complement(A) if result is not 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(Set, metaclass=Singleton): """ Represents the empty set. The empty set is available as a singleton as S.EmptySet. Examples ======== >>> S.EmptySet EmptySet() >>> Interval(1, 2).intersection(S.EmptySet) EmptySet() See Also ======== UniversalSet References ========== * https//en.wikipedia.org/wiki/Empty_set """ is_EmptySet = True is_FiniteSet = True @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(Set, metaclass=Singleton): """ Represents the set of all things. The universal set is available as a singleton as S.UniversalSet Examples ======== >>> S.UniversalSet UniversalSet() >>> Interval(1, 2).intersection(S.UniversalSet) [1, 2] See Also ======== EmptySet References ========== * https//en.wikipedia.org/wiki/Universal_set """ is_UniversalSet = True def _intersection(self, other): return other def _complement(self, other): return S.EmptySet def _symmetric_difference(self, other): return other @property def _measure(self): return oo def _contains(self, other): return true def as_relational(self, symbol): return true @property def _boundary(self): return EmptySet()
[docs]class FiniteSet(Set, EvalfMixin): """ Represents a finite set of discrete numbers Examples ======== >>> FiniteSet(1, 2, 3, 4) {1, 2, 3, 4} >>> 3 in FiniteSet(1, 2, 3, 4) True References ========== * https//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 _complement(self, other): if other.is_Interval: nums = sorted(m for m in self.args if m.is_number and m in other) syms = [m for m in self.args if m.is_Symbol] # Intervals cannot contain elements other than numbers and symbols. intervals = S.EmptySet # Build up a list of intervals between the elements if nums: intervals |= Interval(other.left, nums[0], other.left_open, True) for a, b in zip(nums[:-1], nums[1:]): intervals |= Interval(a, b, True, True) # both open intervals |= Interval(nums[-1], other.right, True, other.right_open) else: intervals |= other if syms: return Complement(intervals, FiniteSet(*syms), evaluate=False) else: return intervals elif other.is_FiniteSet: common = FiniteSet(*[el for el in other if self.contains(el) is S.true]) self2 = FiniteSet(*[el for el in self if common.contains(el) is not S.true]) if self2.is_EmptySet: self2 = common other = FiniteSet(*[el for el in other if common.contains(el) is not S.true]) return Set._complement(FiniteSet(*[el for el in self2 if other.contains(el) is not S.false]), other) return Set._complement(self, other) def _union(self, other): """ This function should only be used internally See Set._union for docstring """ # If other set contains one of my elements, remove it from myself if any(other.contains(x) is true for x in self): return {FiniteSet(*[x for x in self if other.contains(x) is not true]), other} 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 ======== >>> 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 ..functions import Min return Min(*self) @property def _sup(self): from ..functions 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. """ return Or(*[Eq(symbol, elem) for elem in self])
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" % repr(other)) return other.is_subset(self) def __gt__(self, other): if not isinstance(other, Set): raise TypeError("Invalid comparison of set with %s" % repr(other)) return self.is_proper_superset(other) def __le__(self, other): if not isinstance(other, Set): raise TypeError("Invalid comparison of set with %s" % repr(other)) return self.is_subset(other) def __lt__(self, other): if not isinstance(other, Set): raise TypeError("Invalid comparison of set with %s" % repr(other)) return self.is_proper_subset(other)
class SymmetricDifference(Set): """ Represents the symmetric difference of two sets. The set of elements which are in either of the sets and not in their intersection. Examples ======== >>> SymmetricDifference(FiniteSet(1, 2, 3), FiniteSet(3, 4, 5)) {1, 2, 4, 5} See Also ======== Complement, Union References ========== * https//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""" Image of 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 ======== >>> 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]) See Also ======== diofant.sets.fancysets.ImageSet """ from ..core import Dummy, Lambda from .fancysets import ImageSet if len(args) == 3: f = Lambda(*args[:2]) else: # var and expr are being defined this way to # support Python lambda and not just diofant Lambda f = args[0] if not isinstance(f, Lambda): var = Dummy() expr = args[0](var) f = Lambda(var, expr) set = args[-1] 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)