Source code for sympy.sets.sets

from __future__ import print_function, division

from itertools import product

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, with_metaclass, ordered
from sympy.core.evaluate import global_evaluate
from sympy.core.decorators import deprecated
from sympy.core.mul import Mul
from sympy.sets.contains import Contains

from sympy.mpmath import mpi, mpf
from sympy.logic.boolalg import And, Or, Not, true, false
from sympy.utilities import default_sort_key, 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 @staticmethod def _infimum_key(expr): """ Return infimum (if possible) else S.Infinity. """ try: infimum = expr.inf assert infimum.is_comparable except (NotImplementedError, AttributeError, AssertionError, ValueError): infimum = S.Infinity return infimum
[docs] def union(self, other): """ Returns the union of 'self' and 'other'. Examples ======== As a shortcut it is possible to use the '+' operator: >>> from sympy import Interval, FiniteSet >>> Interval(0, 1).union(Interval(2, 3)) [0, 1] U [2, 3] >>> Interval(0, 1) + Interval(2, 3) [0, 1] U [2, 3] >>> Interval(1, 2, True, True) + FiniteSet(2, 3) (1, 2] U {3} Similarly it is possible to use the '-' operator for set differences: >>> Interval(0, 2) - Interval(0, 1) (1, 2] >>> Interval(1, 3) - FiniteSet(2) [1, 2) U (2, 3] """ return Union(self, other)
[docs] def intersect(self, other): """ Returns the intersection of 'self' and 'other'. >>> from sympy import Interval >>> Interval(1, 3).intersect(Interval(1, 2)) [1, 2] """ return Intersection(self, other)
[docs] def intersection(self, other): """ Alias for :meth:`intersect()` """ return self.intersect(other)
def _intersect(self, other): """ This function should only be used internally self._intersect(other) returns a new, intersected set if self knows how to intersect itself with other, otherwise it returns ``None`` When making a new set class you can be assured that other will not be a :class:`Union`, :class:`FiniteSet`, or :class:`EmptySet` Used within the :class:`Intersection` class """ return None
[docs] def is_disjoint(self, other): """ Returns True if 'self' and 'other' are disjoint Examples ======== >>> from sympy import Interval >>> Interval(0, 2).is_disjoint(Interval(1, 2)) False >>> Interval(0, 2).is_disjoint(Interval(3, 4)) True References ========== .. [1] http://en.wikipedia.org/wiki/Disjoint_sets """ return self.intersect(other) == S.EmptySet
[docs] def isdisjoint(self, other): """ Alias for :meth:`is_disjoint()` """ return self.is_disjoint(other)
def _union(self, other): """ This function should only be used internally self._union(other) returns a new, joined set if self knows how to join itself with other, otherwise it returns ``None``. It may also return a python set of SymPy Sets if they are somehow simpler. If it does this it must be idempotent i.e. the sets returned must return ``None`` with _union'ed with each other Used within the :class:`Union` class """ return None
[docs] def complement(self, universe): """ The complement of 'self' w.r.t the given the universe. Examples ======== >>> from sympy import Interval, S >>> Interval(0, 1).complement(S.Reals) (-oo, 0) U (1, oo) >>> Interval(0, 1).complement(S.UniversalSet) UniversalSet() \ [0, 1] """ return Complement(universe, self)
def _complement(self, other): # this behaves as other - self if isinstance(other, ProductSet): # For each set consider it or it's complement # We need at least one of the sets to be complemented # Consider all 2^n combinations. # We can conveniently represent these options easily using a ProductSet # XXX: this doesn't work if the dimentions of the sets isn't same. # A - B is essentially same as A if B has a different # dimentionality than A switch_sets = ProductSet(FiniteSet(o, o - s) for s, o in zip(self.sets, other.sets)) product_sets = (ProductSet(*set) for set in switch_sets) # Union of all combinations but this one return Union(p for p in product_sets if p != other) elif isinstance(other, Interval): if isinstance(self, Interval) or isinstance(self, FiniteSet): return Intersection(other, self.complement(S.Reals)) elif isinstance(other, Union): return Union(o - self for o in other.args) elif isinstance(other, Complement): return Complement(other.args[0], Union(other.args[1], self)) elif isinstance(other, EmptySet): return S.EmptySet elif isinstance(other, FiniteSet): return FiniteSet(*[el for el in other if el not in self]) return None @property
[docs] 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
[docs] 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 """ ret = self._contains(sympify(other, strict=True)) if ret is None: ret = Contains(other, self, evaluate=False) return ret
def _contains(self, other): raise NotImplementedError("(%s)._contains(%s)" % (self, other)) @deprecated(useinstead="is_subset", issue=7460, deprecated_since_version="0.7.6")
[docs] def subset(self, other): """ Returns True if 'other' is a subset of 'self'. """ return other.is_subset(self)
[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.powerset() == FiniteSet(FiniteSet(1), FiniteSet(2), FiniteSet(1, 2), EmptySet()) True References ========== .. [1] http://en.wikipedia.org/wiki/Power_set """ return self._eval_powerset()
@property
[docs] 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
[docs] def boundary(self): """ The boundary or frontier of a set A point x is on the boundary of a set S if 1. x is in the closure of S. I.e. Every neighborhood of x contains a point in S. 2. x is not in the interior of S. I.e. There does not exist an open set centered on x contained entirely within S. There are the points on the outer rim of S. If S is open then these points need not actually be contained within S. For example, the boundary of an interval is its start and end points. This is true regardless of whether or not the interval is open. Examples ======== >>> from sympy import Interval >>> Interval(0, 1).boundary {0, 1} >>> Interval(0, 1, True, False).boundary {0, 1} """ return self._boundary
@property def is_open(self): if not Intersection(self, self.boundary): return True # We can't confidently claim that an intersection exists return None @property def is_closed(self): return self.boundary.is_subset(self) @property def closure(self): return self + self.boundary @property def interior(self): return self - self.boundary @property def _boundary(self): raise NotImplementedError() def _eval_imageset(self, f): from sympy.sets.fancysets import ImageSet return ImageSet(f, self) @property def _measure(self): raise NotImplementedError("(%s)._measure" % self) def __add__(self, other): return self.union(other) def __or__(self, other): return self.union(other) def __and__(self, other): return self.intersect(other) def __mul__(self, other): return ProductSet(self, other) def __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 = self.contains(other) if symb not in (true, false): raise TypeError('contains did not evaluate to a bool: %r' % symb) return bool(symb) @property @deprecated(useinstead="is_subset(Reals)", issue=6212, deprecated_since_version="0.7.6") 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 ========== .. [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 _contains(self, element): """ 'in' operator for ProductSets Examples ======== >>> from sympy import Interval >>> (2, 3) in Interval(0, 5) * Interval(0, 5) True >>> (10, 10) in Interval(0, 5) * Interval(0, 5) False Passes operation on to constituent sets """ try: if len(element) != len(self.args): return false except TypeError: # maybe element isn't an iterable return false return And(*[set.contains(item) for set, item in zip(self.sets, element)]) def _intersect(self, other): """ This function should only be used internally See Set._intersect for docstring """ if not other.is_ProductSet: return None if len(other.args) != len(self.args): return S.EmptySet return ProductSet(a.intersect(b) for a, b in zip(self.sets, other.sets)) def _union(self, other): if not other.is_ProductSet: return None if len(other.args) != len(self.args): return None if self.args[0] == other.args[0]: return self.args[0] * Union(ProductSet(self.args[1:]), ProductSet(other.args[1:])) if self.args[-1] == other.args[-1]: return Union(ProductSet(self.args[:-1]), ProductSet(other.args[:-1])) * self.args[-1] return None @property def sets(self): return self.args @property def _boundary(self): return Union(ProductSet(b + b.boundary if i != j else b.boundary for j, b in enumerate(self.sets)) for i, a in enumerate(self.sets)) @property @deprecated(useinstead="is_subset(Reals)", issue=6212, deprecated_since_version="0.7.6") 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: return product(*self.sets) else: raise TypeError("Not all constituent sets are iterable") @property def _measure(self): measure = 1 for set in self.sets: measure *= set.measure return measure def __len__(self): return Mul(*[len(s) for s in self.args])
[docs]class Interval(Set, EvalfMixin): """ Represents a real interval as a Set. Usage: Returns an interval with end points "start" and "end". For left_open=True (default left_open is False) the interval will be open on the left. Similarly, for right_open=True the interval will be open on the right. Examples ======== >>> from sympy import Symbol, Interval >>> Interval(0, 1) [0, 1] >>> Interval(0, 1, False, True) [0, 1) >>> a = Symbol('a', real=True) >>> Interval(0, a) [0, a] Notes ===== - Only real end points are supported - Interval(a, b) with a > b will return the empty set - Use the evalf() method to turn an Interval into an mpmath 'mpi' interval instance References ========== .. [1] http://en.wikipedia.org/wiki/Interval_%28mathematics%29 """ is_Interval = True @property @deprecated(useinstead="is_subset(Reals)", issue=6212, deprecated_since_version="0.7.6") def is_real(self): return 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 (start.is_real or start in inftys) or not (end.is_real or end in inftys): 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. Examples ======== >>> 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. Examples ======== >>> 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. 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
[docs] def right_open(self): """ True if 'self' is right-open. Examples ======== >>> from sympy import Interval >>> Interval(0, 1, right_open=True).right_open True >>> Interval(0, 1, right_open=False).right_open False """ return self._args[3]
def _intersect(self, other): """ This function should only be used internally See Set._intersect for docstring """ # We only know how to intersect with other intervals if not other.is_Interval: return None # 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 is 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) return Set._complement(self, other) 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 _boundary(self): return FiniteSet(self.start, self.end) def _contains(self, other): if other.is_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) def _eval_imageset(self, f): from sympy.functions.elementary.miscellaneous import Min, Max from sympy.solvers import solve from sympy.core.function import diff from sympy.series import limit from sympy.calculus.singularities import singularities # TODO: handle piecewise defined functions # 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 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 = solve(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(1, 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.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. """ other = sympify(symbol) if self.right_open: right = other < self.end else: right = other <= self.end if right == True: if self.left_open: return other > self.start else: return other >= self.start if self.left_open: left = self.start < other else: left = self.start <= other return And(left, right)
[docs]class Union(Set, EvalfMixin): """ Represents a union of sets as a :class:`Set`. Examples ======== >>> from sympy import Union, Interval >>> Union(Interval(1, 2), Interval(3, 4)) [1, 2] U [3, 4] The Union constructor will always try to merge overlapping intervals, if possible. For example: >>> Union(Interval(1, 2), Interval(2, 3)) [1, 3] See Also ======== Intersection References ========== .. [1] http://en.wikipedia.org/wiki/Union_%28set_theory%29 """ is_Union = True def __new__(cls, *args, **kwargs): evaluate = kwargs.get('evaluate', global_evaluate[0]) # flatten inputs to merge intersections and iterables args = list(args) def flatten(arg): if isinstance(arg, Set): if arg.is_Union: return sum(map(flatten, arg.args), []) else: return [arg] if iterable(arg): # and not isinstance(arg, Set) (implicit) return sum(map(flatten, arg), []) raise TypeError("Input must be Sets or iterables of Sets") args = flatten(args) # Union of no sets is EmptySet if len(args) == 0: return S.EmptySet # Reduce sets using known rules if evaluate: return Union.reduce(args) args = list(ordered(args, Set._infimum_key)) return Basic.__new__(cls, *args) @staticmethod
[docs] def reduce(args): """ Simplify a :class:`Union` using known rules We first start with global rules like 'Merge all FiniteSets' Then we iterate through all pairs and ask the constituent sets if they can simplify themselves with any other constituent """ # ===== Global Rules ===== # Merge all finite sets finite_sets = [x for x in args if x.is_FiniteSet] if len(finite_sets) > 1: a = (x for set in finite_sets for x in set) finite_set = FiniteSet(*a) args = [finite_set] + [x for x in args if not x.is_FiniteSet] # ===== Pair-wise Rules ===== # Here we depend on rules built into the constituent sets args = set(args) new_args = True while(new_args): for s in args: new_args = False for t in args - set((s,)): new_set = s._union(t) # This returns None if s does not know how to intersect # with t. Returns the newly intersected set otherwise if new_set is not None: if not isinstance(new_set, set): new_set = set((new_set, )) new_args = (args - set((s, t))).union(new_set) break if new_args: args = new_args break if len(args) == 1: return args.pop() else: return Union(args, evaluate=False)
def complement(self, universe): # DeMorgan's Law return Intersection(s.complement(universe) for s in self.args) @property def _inf(self): # We use Min so that sup is meaningful in combination with symbolic # interval end points. from sympy.functions.elementary.miscellaneous import Min return Min(*[set.inf for set in self.args]) @property def _sup(self): # We use Max so that sup is meaningful in combination with symbolic # end points. from sympy.functions.elementary.miscellaneous import Max return Max(*[set.sup for set in self.args]) def _contains(self, other): 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 @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.evalf() for set in self.args) except Exception: 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 @deprecated(useinstead="is_subset(Reals)", issue=6212, deprecated_since_version="0.7.6") 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 :class:`Set`. Examples ======== >>> from sympy import Intersection, Interval >>> Intersection(Interval(1, 3), Interval(2, 4)) [2, 3] We often use the .intersect method >>> Interval(1,3).intersect(Interval(2,4)) [2, 3] See Also ======== Union References ========== .. [1] http://en.wikipedia.org/wiki/Intersection_%28set_theory%29 """ is_Intersection = True def __new__(cls, *args, **kwargs): evaluate = kwargs.get('evaluate', global_evaluate[0]) # flatten inputs to merge intersections and iterables args = list(args) def flatten(arg): if isinstance(arg, Set): if arg.is_Intersection: return sum(map(flatten, arg.args), []) else: return [arg] if iterable(arg): # and not isinstance(arg, Set) (implicit) return sum(map(flatten, arg), []) raise TypeError("Input must be Sets or iterables of Sets") args = flatten(args) if len(args) == 0: raise TypeError("Intersection expected at least one argument") # Reduce sets using known rules if evaluate: return Intersection.reduce(args) args = list(ordered(args, Set._infimum_key)) 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): 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.func(*[x for x in s if all(other.contains(x) == True 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,)) 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 - set((s,)): new_set = s._intersect(t) # This returns None if s does not know how to intersect # with t. Returns the newly intersected set otherwise if new_set is not None: new_args = (args - set((s, t))).union(set((new_set, ))) break if new_args: args = new_args break if len(args) == 1: return args.pop() else: return Intersection(args, evaluate=False)
[docs] def as_relational(self, symbol): """Rewrite an Intersection in terms of equalities and logic operators""" return And(*[set.as_relational(symbol) for set in self.args])
[docs]class Complement(Set, EvalfMixin): """ Represents the set difference or relative complement of a set with another set. `A - B = \{x \in A| x \\notin B\}` Examples ======== >>> from sympy import Complement, FiniteSet >>> Complement(FiniteSet(0, 1, 2), FiniteSet(1)) {0, 2} See Also ========= Intersection, Union References ========== http://mathworld.wolfram.com/SetComplement.html """ is_Complement = True def __new__(cls, a, b, evaluate=True): if evaluate: return Complement.reduce(a, b) return Basic.__new__(cls, a, b) @staticmethod
[docs] def reduce(A, B): """ Simplify a :class:`Complement`. """ if B == S.UniversalSet: 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
[docs]class UniversalSet(with_metaclass(Singleton, Set)): """ Represents the set of all things. The universal set is available as a singleton as S.UniversalSet Examples ======== >>> from sympy import S, Interval >>> S.UniversalSet UniversalSet() >>> Interval(1, 2).intersect(S.UniversalSet) [1, 2] See Also ======== EmptySet References ========== .. [1] http://en.wikipedia.org/wiki/Universal_set """ is_UniversalSet = True def _intersect(self, other): return other def complement(self, universal_set): 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 @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 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 __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 other is S.Reals: nums = sorted(m for m in self.args if m.is_number) 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 if nums != []: intervals += [Interval(S.NegativeInfinity, nums[0], True, True)] for a, b in zip(nums[:-1], nums[1:]): intervals.append(Interval(a, b, True, True)) # open intervals 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) 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(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 Examples ======== >>> from sympy import FiniteSet >>> 1 in FiniteSet(1, 2) True >>> 5 in FiniteSet(1, 2) False """ if other in self._elements: return true else: if not other.free_symbols: return false elif all(e.is_Symbol for e in self._elements): return false 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])
@property @deprecated(useinstead="is_subset(Reals)", issue=6212, deprecated_since_version="0.7.6") 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 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): return other.is_subset(self) def __gt__(self, other): return self.is_proper_superset(other) def __le__(self, other): return self.is_subset(other) def __lt__(self, other): return self.is_proper_subset(other)
[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 ======== >>> from sympy import Interval, Symbol, imageset, sin, Lambda >>> x = Symbol('x') >>> 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 ======== sympy.sets.fancysets.ImageSet """ from sympy.core import Dummy, Lambda from sympy.sets.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 sympy 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)