from typing import Optional from collections import defaultdict import inspect from sympy.core.basic import Basic from sympy.core.compatibility import iterable, ordered, reduce from sympy.core.containers import Tuple from sympy.core.decorators import (deprecated, sympify_method_args, sympify_return) from sympy.core.evalf import EvalfMixin, prec_to_dps from sympy.core.parameters import global_parameters from sympy.core.expr import Expr from sympy.core.logic import (FuzzyBool, fuzzy_bool, fuzzy_or, fuzzy_and, fuzzy_not) from sympy.core.numbers import Float from sympy.core.operations import LatticeOp from sympy.core.relational import Eq, Ne, is_lt from sympy.core.singleton import Singleton, S from sympy.core.symbol import Symbol, Dummy, uniquely_named_symbol from sympy.core.sympify import _sympify, sympify, converter from sympy.logic.boolalg import And, Or, Not, Xor, true, false from sympy.sets.contains import Contains from sympy.utilities import subsets from sympy.utilities.exceptions import SymPyDeprecationWarning from sympy.utilities.iterables import iproduct, sift, roundrobin from sympy.utilities.misc import func_name, filldedent from mpmath import mpi, mpf tfn = defaultdict(lambda: None, { True: S.true, S.true: S.true, False: S.false, S.false: S.false}) @sympify_method_args class Set(Basic, EvalfMixin): """ The base class for any kind of set. Explanation =========== 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 # type: Optional[bool] is_UniversalSet = None # type: Optional[bool] is_Complement = None # type: Optional[bool] is_ComplexRegion = False is_empty = None # type: FuzzyBool is_finite_set = None # type: FuzzyBool @property # type: ignore @deprecated(useinstead="is S.EmptySet or is_empty", issue=16946, deprecated_since_version="1.5") def is_EmptySet(self): return None @staticmethod def _infimum_key(expr): """ Return infimum (if possible) else S.Infinity. """ try: infimum = expr.inf assert infimum.is_comparable infimum = infimum.evalf() # issue #18505 except (NotImplementedError, AttributeError, AssertionError, ValueError): infimum = S.Infinity return infimum 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({3}, Interval.Lopen(1, 2)) 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) def intersect(self, other): """ Returns the intersection of 'self' and 'other'. Examples ======== >>> 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) def intersection(self, other): """ Alias for :meth:`intersect()` """ return self.intersect(other) 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] https://en.wikipedia.org/wiki/Disjoint_sets """ return self.intersect(other) == S.EmptySet def isdisjoint(self, other): """ Alias for :meth:`is_disjoint()` """ return self.is_disjoint(other) def complement(self, universe): r""" The complement of 'self' w.r.t the given 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) Complement(UniversalSet, Interval(0, 1)) """ return Complement(universe, self) def _complement(self, other): # this behaves as other - self if isinstance(self, ProductSet) and isinstance(other, ProductSet): # If self and other are disjoint then other - self == self if len(self.sets) != len(other.sets): return other # There can be other ways to represent this but this gives: # (A x B) - (C x D) = ((A - C) x B) U (A x (B - D)) overlaps = [] pairs = list(zip(self.sets, other.sets)) for n in range(len(pairs)): sets = (o if i != n else o-s for i, (s, o) in enumerate(pairs)) overlaps.append(ProductSet(*sets)) return Union(*overlaps) 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): from sympy.utilities.iterables import sift sifted = sift(other, lambda x: fuzzy_bool(self.contains(x))) # ignore those that are contained in self return Union(FiniteSet(*(sifted[False])), Complement(FiniteSet(*(sifted[None])), self, evaluate=False) if sifted[None] else S.EmptySet) 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) 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) def contains(self, other): """ Returns a SymPy value indicating whether ``other`` is contained in ``self``: ``true`` if it is, ``false`` if it isn't, else an unevaluated ``Contains`` expression (or, as in the case of ConditionSet and a union of FiniteSet/Intervals, an expression indicating the conditions for containment). Examples ======== >>> from sympy import Interval, S >>> from sympy.abc import x >>> Interval(0, 1).contains(0.5) True As a shortcut it is possible to use the 'in' operator, but that will raise an error unless an affirmative true or false is not obtained. >>> Interval(0, 1).contains(x) (0 <= x) & (x <= 1) >>> x in Interval(0, 1) Traceback (most recent call last): ... TypeError: did not evaluate to a bool: None The result of 'in' is a bool, not a SymPy value >>> 1 in Interval(0, 2) True >>> _ is S.true False """ other = sympify(other, strict=True) c = self._contains(other) if isinstance(c, Contains): return c if c is None: return Contains(other, self, evaluate=False) b = tfn[c] if b is None: return c return b def _contains(self, other): raise NotImplementedError(filldedent(''' (%s)._contains(%s) is not defined. This method, when defined, will receive a sympified object. The method should return True, False, None or something that expresses what must be true for the containment of that object in self to be evaluated. If None is returned then a generic Contains object will be returned by the ``contains`` method.''' % (self, other))) 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 not isinstance(other, Set): raise ValueError("Unknown argument '%s'" % other) # Handle the trivial cases if self == other: return True is_empty = self.is_empty if is_empty is True: return True elif fuzzy_not(is_empty) and other.is_empty: return False if self.is_finite_set is False and other.is_finite_set: return False # Dispatch on subclass rules ret = self._eval_is_subset(other) if ret is not None: return ret ret = other._eval_is_superset(self) if ret is not None: return ret # Use pairwise rules from multiple dispatch from sympy.sets.handlers.issubset import is_subset_sets ret = is_subset_sets(self, other) if ret is not None: return ret # Fall back on computing the intersection # XXX: We shouldn't do this. A query like this should be handled # without evaluating new Set objects. It should be the other way round # so that the intersect method uses is_subset for evaluation. if self.intersect(other) == self: return True def _eval_is_subset(self, other): '''Returns a fuzzy bool for whether self is a subset of other.''' return None def _eval_is_superset(self, other): '''Returns a fuzzy bool for whether self is a subset of other.''' return None # This should be deprecated: def issubset(self, other): """ Alias for :meth:`is_subset()` """ return self.is_subset(other) 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) 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) # This should be deprecated: def issuperset(self, other): """ Alias for :meth:`is_superset()` """ return self.is_superset(other) 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): from .powerset import PowerSet return PowerSet(self) def powerset(self): """ Find the Power set of ``self``. Examples ======== >>> from sympy import EmptySet, FiniteSet, Interval A power set of an empty set: >>> A = EmptySet >>> A.powerset() {EmptySet} A power set of a finite set: >>> A = FiniteSet(1, 2) >>> a, b, c = FiniteSet(1), FiniteSet(2), FiniteSet(1, 2) >>> A.powerset() == FiniteSet(a, b, c, EmptySet) True A power set of an interval: >>> Interval(1, 2).powerset() PowerSet(Interval(1, 2)) References ========== .. [1] https://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. Explanation =========== 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. Explanation =========== A set is open if and only if it has an empty intersection with its boundary. In particular, a subset A of the reals is open if and only if each one of its points is contained in an open interval that is a subset of A. Examples ======== >>> from sympy import S >>> S.Reals.is_open True >>> S.Rationals.is_open False """ return Intersection(self, self.boundary).is_empty @property def is_closed(self): """ A property method to check whether a set is closed. Explanation =========== A set is closed if its complement is an open set. The closedness of a subset of the reals is determined with respect to R and its standard topology. 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 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() @property def _measure(self): raise NotImplementedError("(%s)._measure" % self) def _eval_evalf(self, prec): return self.func(*[arg.evalf(n=prec_to_dps(prec)) for arg in self.args]) @sympify_return([('other', 'Set')], NotImplemented) def __add__(self, other): return self.union(other) @sympify_return([('other', 'Set')], NotImplemented) def __or__(self, other): return self.union(other) @sympify_return([('other', 'Set')], NotImplemented) def __and__(self, other): return self.intersect(other) @sympify_return([('other', 'Set')], NotImplemented) def __mul__(self, other): return ProductSet(self, other) @sympify_return([('other', 'Set')], NotImplemented) def __xor__(self, other): return SymmetricDifference(self, other) @sympify_return([('exp', Expr)], NotImplemented) def __pow__(self, exp): if not (exp.is_Integer and exp >= 0): raise ValueError("%s: Exponent must be a positive Integer" % exp) return ProductSet(*[self]*exp) @sympify_return([('other', 'Set')], NotImplemented) def __sub__(self, other): return Complement(self, other) def __contains__(self, other): other = _sympify(other) c = self._contains(other) b = tfn[c] if b is None: # x in y must evaluate to T or F; to entertain a None # result with Set use y.contains(x) raise TypeError('did not evaluate to a bool: %r' % c) return b class ProductSet(Set): """ Represents a Cartesian Product of Sets. Explanation =========== 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) ProductSet(Interval(0, 5), {1, 2, 3}) >>> (2, 2) in ProductSet(I, S) True >>> Interval(0, 1) * Interval(0, 1) # The unit square ProductSet(Interval(0, 1), Interval(0, 1)) >>> coin = FiniteSet('H', 'T') >>> set(coin**2) {(H, H), (H, T), (T, H), (T, T)} The Cartesian product is not commutative or associative e.g.: >>> I*S == S*I False >>> (I*I)*I == I*(I*I) False Notes ===== - Passes most operations down to the argument sets References ========== .. [1] https://en.wikipedia.org/wiki/Cartesian_product """ is_ProductSet = True def __new__(cls, *sets, **assumptions): if len(sets) == 1 and iterable(sets[0]) and not isinstance(sets[0], (Set, set)): SymPyDeprecationWarning( feature="ProductSet(iterable)", useinstead="ProductSet(*iterable)", issue=17557, deprecated_since_version="1.5" ).warn() sets = tuple(sets[0]) sets = [sympify(s) for s in sets] if not all(isinstance(s, Set) for s in sets): raise TypeError("Arguments to ProductSet should be of type Set") # Nullary product of sets is *not* the empty set if len(sets) == 0: return FiniteSet(()) if S.EmptySet in sets: return S.EmptySet return Basic.__new__(cls, *sets, **assumptions) @property def sets(self): return self.args def flatten(self): def _flatten(sets): for s in sets: if s.is_ProductSet: yield from _flatten(s.sets) else: yield s return ProductSet(*_flatten(self.sets)) 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 """ if element.is_Symbol: return None if not isinstance(element, Tuple) or len(element) != len(self.sets): return False return fuzzy_and(s._contains(e) for s, e in zip(self.sets, element)) def as_relational(self, *symbols): symbols = [_sympify(s) for s in symbols] if len(symbols) != len(self.sets) or not all( i.is_Symbol for i in symbols): raise ValueError( 'number of symbols must match the number of sets') return And(*[s.as_relational(i) for s, i in zip(self.sets, symbols)]) @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 >>> 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. """ return iproduct(*self.sets) @property def is_empty(self): return fuzzy_or(s.is_empty for s in self.sets) @property def is_finite_set(self): all_finite = fuzzy_and(s.is_finite_set for s in self.sets) return fuzzy_or([self.is_empty, all_finite]) @property def _measure(self): measure = 1 for s in self.sets: measure *= s.measure return measure def __len__(self): return reduce(lambda a, b: a*b, (len(s) for s in self.args)) def __bool__(self): return all([bool(s) for s in self.sets]) class Interval(Set): """ 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] 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) 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)) # Only allow real intervals if fuzzy_not(fuzzy_and(i.is_extended_real for i in (start, end, end-start))): raise ValueError("Non-real intervals are not supported") # evaluate if possible if is_lt(end, start): 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 is S.Infinity or start is S.NegativeInfinity: return S.EmptySet return FiniteSet(end) # Make sure infinite interval end points are open. if start is S.NegativeInfinity: left_open = true if end is S.Infinity: right_open = true if start == S.Infinity or end == S.NegativeInfinity: return S.EmptySet 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] @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] @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] @classmethod def open(cls, a, b): """Return an interval including neither boundary.""" return cls(a, b, True, True) @classmethod def Lopen(cls, a, b): """Return an interval not including the left boundary.""" return cls(a, b, True, False) @classmethod def Ropen(cls, a, b): """Return an interval not including the right boundary.""" return cls(a, b, False, True) @property def _inf(self): return self.start @property def _sup(self): return self.end @property def left(self): return self.start @property def right(self): return self.end @property def is_empty(self): if self.left_open or self.right_open: cond = self.start >= self.end # One/both bounds open else: cond = self.start > self.end # Both bounds closed return fuzzy_bool(cond) @property def is_finite_set(self): return self.measure.is_zero 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) @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.NaN or other.is_real is False): return false if self.start is S.NegativeInfinity and self.end is S.Infinity: if other.is_real is not None: return other.is_real d = Dummy() return self.as_relational(d).subs(d, other) 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) @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._evalf(prec), self.right._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") def _eval_Eq(self, other): if not isinstance(other, Interval): if isinstance(other, FiniteSet): return false elif isinstance(other, Set): return None return false class Union(Set, LatticeOp): """ 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] https://en.wikipedia.org/wiki/Union_%28set_theory%29 """ is_Union = True @property def identity(self): return S.EmptySet @property def zero(self): return S.UniversalSet def __new__(cls, *args, **kwargs): evaluate = kwargs.get('evaluate', global_parameters.evaluate) # flatten inputs to merge intersections and iterables args = _sympify(args) # Reduce sets using known rules if evaluate: args = list(cls._new_args_filter(args)) return simplify_union(args) args = list(ordered(args, Set._infimum_key)) obj = Basic.__new__(cls, *args) obj._argset = frozenset(args) return obj @property def args(self): return self._args 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]) @property def is_empty(self): return fuzzy_and(set.is_empty for set in self.args) @property def is_finite_set(self): return fuzzy_and(set.is_finite_set 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 _contains(self, other): return Or(*[s.contains(other) for s in self.args]) def is_subset(self, other): return fuzzy_and(s.is_subset(other) for s in self.args) def as_relational(self, symbol): """Rewrite a Union in terms of equalities and logic operators. """ if (len(self.args) == 2 and all(isinstance(i, Interval) for i in self.args)): # optimization to give 3 args as (x > 1) & (x < 5) & Ne(x, 3) # instead of as 4, ((1 <= x) & (x < 3)) | ((x <= 5) & (3 < x)) a, b = self.args if (a.sup == b.inf and not any(a.sup in i for i in self.args)): return And(Ne(symbol, a.sup), symbol < b.sup, symbol > a.inf) return Or(*[i.as_relational(symbol) for i in self.args]) @property def is_iterable(self): return all(arg.is_iterable for arg in self.args) def __iter__(self): return roundrobin(*(iter(arg) for arg in self.args)) class Intersection(Set, LatticeOp): """ 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] https://en.wikipedia.org/wiki/Intersection_%28set_theory%29 """ is_Intersection = True @property def identity(self): return S.UniversalSet @property def zero(self): return S.EmptySet def __new__(cls, *args, **kwargs): evaluate = kwargs.get('evaluate', global_parameters.evaluate) # flatten inputs to merge intersections and iterables args = list(ordered(set(_sympify(args)))) # Reduce sets using known rules if evaluate: args = list(cls._new_args_filter(args)) return simplify_intersection(args) args = list(ordered(args, Set._infimum_key)) obj = Basic.__new__(cls, *args) obj._argset = frozenset(args) return obj @property def args(self): return self._args @property def is_iterable(self): return any(arg.is_iterable for arg in self.args) @property def is_finite_set(self): if fuzzy_or(arg.is_finite_set for arg in self.args): return True @property def _inf(self): raise NotImplementedError() @property def _sup(self): raise NotImplementedError() def _contains(self, other): return And(*[set.contains(other) for set in self.args]) def __iter__(self): sets_sift = sift(self.args, lambda x: x.is_iterable) completed = False candidates = sets_sift[True] + sets_sift[None] finite_candidates, others = [], [] for candidate in candidates: length = None try: length = len(candidate) except TypeError: others.append(candidate) if length is not None: finite_candidates.append(candidate) finite_candidates.sort(key=len) for s in finite_candidates + others: other_sets = set(self.args) - {s} other = Intersection(*other_sets, evaluate=False) completed = True for x in s: try: if x in other: yield x except TypeError: completed = False if completed: return if not completed: if not candidates: raise TypeError("None of the constituent sets are iterable") raise TypeError( "The computation had not completed because of the " "undecidable set membership is found in every candidates.") @staticmethod def _handle_finite_sets(args): '''Simplify intersection of one or more FiniteSets and other sets''' # First separate the FiniteSets from the others fs_args, others = sift(args, lambda x: x.is_FiniteSet, binary=True) # Let the caller handle intersection of non-FiniteSets if not fs_args: return # Convert to Python sets and build the set of all elements fs_sets = [set(fs) for fs in fs_args] all_elements = reduce(lambda a, b: a | b, fs_sets, set()) # Extract elements that are definitely in or definitely not in the # intersection. Here we check contains for all of args. definite = set() for e in all_elements: inall = fuzzy_and(s.contains(e) for s in args) if inall is True: definite.add(e) if inall is not None: for s in fs_sets: s.discard(e) # At this point all elements in all of fs_sets are possibly in the # intersection. In some cases this is because they are definitely in # the intersection of the finite sets but it's not clear if they are # members of others. We might have {m, n}, {m}, and Reals where we # don't know if m or n is real. We want to remove n here but it is # possibly in because it might be equal to m. So what we do now is # extract the elements that are definitely in the remaining finite # sets iteratively until we end up with {n}, {}. At that point if we # get any empty set all remaining elements are discarded. fs_elements = reduce(lambda a, b: a | b, fs_sets, set()) # Need fuzzy containment testing fs_symsets = [FiniteSet(*s) for s in fs_sets] while fs_elements: for e in fs_elements: infs = fuzzy_and(s.contains(e) for s in fs_symsets) if infs is True: definite.add(e) if infs is not None: for n, s in enumerate(fs_sets): # Update Python set and FiniteSet if e in s: s.remove(e) fs_symsets[n] = FiniteSet(*s) fs_elements.remove(e) break # If we completed the for loop without removing anything we are # done so quit the outer while loop else: break # If any of the sets of remainder elements is empty then we discard # all of them for the intersection. if not all(fs_sets): fs_sets = [set()] # Here we fold back the definitely included elements into each fs. # Since they are definitely included they must have been members of # each FiniteSet to begin with. We could instead fold these in with a # Union at the end to get e.g. {3}|({x}&{y}) rather than {3,x}&{3,y}. if definite: fs_sets = [fs | definite for fs in fs_sets] if fs_sets == [set()]: return S.EmptySet sets = [FiniteSet(*s) for s in fs_sets] # Any set in others is redundant if it contains all the elements that # are in the finite sets so we don't need it in the Intersection all_elements = reduce(lambda a, b: a | b, fs_sets, set()) is_redundant = lambda o: all(fuzzy_bool(o.contains(e)) for e in all_elements) others = [o for o in others if not is_redundant(o)] if others: rest = Intersection(*others) # XXX: Maybe this shortcut should be at the beginning. For large # FiniteSets it could much more efficient to process the other # sets first... if rest is S.EmptySet: return S.EmptySet # Flatten the Intersection if rest.is_Intersection: sets.extend(rest.args) else: sets.append(rest) if len(sets) == 1: return sets[0] else: return Intersection(*sets, evaluate=False) 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]) class Complement(Set): r"""Represents the set difference or relative complement of a set with another set. `A - B = \{x \in A \mid x \notin B\}` Examples ======== >>> from sympy import Complement, FiniteSet >>> Complement(FiniteSet(0, 1, 2), FiniteSet(1)) {0, 2} See Also ========= Intersection, Union References ========== .. [1] http://mathworld.wolfram.com/ComplementSet.html """ is_Complement = True def __new__(cls, a, b, evaluate=True): if evaluate: return Complement.reduce(a, b) return Basic.__new__(cls, a, b) @staticmethod def reduce(A, B): """ Simplify a :class:`Complement`. """ if B == S.UniversalSet or A.is_subset(B): return S.EmptySet if isinstance(B, Union): return Intersection(*(s.complement(A) for s in B.args)) 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))) def as_relational(self, symbol): """Rewrite a complement in terms of equalities and logic operators""" A, B = self.args A_rel = A.as_relational(symbol) B_rel = Not(B.as_relational(symbol)) return And(A_rel, B_rel) @property def is_iterable(self): if self.args[0].is_iterable: return True @property def is_finite_set(self): A, B = self.args a_finite = A.is_finite_set if a_finite is True: return True elif a_finite is False and B.is_finite_set: return False def __iter__(self): A, B = self.args for a in A: if a not in B: yield a else: continue class EmptySet(Set, metaclass=Singleton): """ Represents the empty set. The empty set is available as a singleton as S.EmptySet. Examples ======== >>> from sympy import S, Interval >>> S.EmptySet EmptySet >>> Interval(1, 2).intersect(S.EmptySet) EmptySet See Also ======== UniversalSet References ========== .. [1] https://en.wikipedia.org/wiki/Empty_set """ is_empty = True is_finite_set = True is_FiniteSet = True @property # type: ignore @deprecated(useinstead="is S.EmptySet or is_empty", issue=16946, deprecated_since_version="1.5") def is_EmptySet(self): return 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 __iter__(self): return iter([]) 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 class UniversalSet(Set, metaclass=Singleton): """ Represents the set of all things. The universal set is available as a singleton as S.UniversalSet. Examples ======== >>> from sympy import S, Interval >>> S.UniversalSet UniversalSet >>> Interval(1, 2).intersect(S.UniversalSet) Interval(1, 2) See Also ======== EmptySet References ========== .. [1] https://en.wikipedia.org/wiki/Universal_set """ is_UniversalSet = True is_empty = False is_finite_set = False 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 @property def _boundary(self): return S.EmptySet class FiniteSet(Set): """ 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] https://en.wikipedia.org/wiki/Finite_set """ is_FiniteSet = True is_iterable = True is_empty = False is_finite_set = True def __new__(cls, *args, **kwargs): evaluate = kwargs.get('evaluate', global_parameters.evaluate) if evaluate: args = list(map(sympify, args)) if len(args) == 0: return S.EmptySet else: args = list(map(sympify, args)) # keep the form of the first canonical arg dargs = {} for i in reversed(list(ordered(args))): if i.is_Symbol: dargs[i] = i else: try: dargs[i.as_dummy()] = i except TypeError: # e.g. i = class without args like `Interval` dargs[i] = i _args_set = set(dargs.values()) args = list(ordered(_args_set, Set._infimum_key)) obj = Basic.__new__(cls, *args) obj._args_set = _args_set return obj def __iter__(self): return iter(self.args) def _complement(self, other): if isinstance(other, Interval): # Splitting in sub-intervals is only done for S.Reals; # other cases that need splitting will first pass through # Set._complement(). nums, syms = [], [] for m in self.args: if m.is_number and m.is_real: nums.append(m) elif m.is_real == False: pass # drop non-reals else: syms.append(m) # various symbolic expressions if other == S.Reals and nums != []: nums.sort() 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 == []: # no splitting necessary or possible: if syms: return Complement(other, FiniteSet(*syms), evaluate=False) else: return other 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 _contains(self, other): """ Tests whether an element, other, is in the set. Explanation =========== The actual test is for mathematical equality (as opposed to syntactical equality). In the worst case all elements of the set must be checked. Examples ======== >>> from sympy import FiniteSet >>> 1 in FiniteSet(1, 2) True >>> 5 in FiniteSet(1, 2) False """ if other in self._args_set: return True else: # evaluate=True is needed to override evaluate=False context; # we need Eq to do the evaluation return fuzzy_or(fuzzy_bool(Eq(e, other, evaluate=True)) for e in self.args) def _eval_is_subset(self, other): return fuzzy_and(other._contains(e) for e in self.args) @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) 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.evalf(n=prec_to_dps(prec)) for elem in self]) def _eval_simplify(self, **kwargs): from sympy.simplify import simplify return FiniteSet(*[simplify(elem, **kwargs) for elem in self]) @property def _sorted_args(self): return self.args def _eval_powerset(self): return self.func(*[self.func(*s) for s in subsets(self.args)]) def _eval_rewrite_as_PowerSet(self, *args, **kwargs): """Rewriting method for a finite set to a power set.""" from .powerset import PowerSet is2pow = lambda n: bool(n and not n & (n - 1)) if not is2pow(len(self)): return None fs_test = lambda arg: isinstance(arg, Set) and arg.is_FiniteSet if not all(fs_test(arg) for arg in args): return None biggest = max(args, key=len) for arg in subsets(biggest.args): arg_set = FiniteSet(*arg) if arg_set not in args: return None return PowerSet(biggest) 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] 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) def as_relational(self, symbol): """Rewrite a symmetric_difference in terms of equalities and logic operators""" A, B = self.args A_rel = A.as_relational(symbol) B_rel = B.as_relational(symbol) return Xor(A_rel, B_rel) @property def is_iterable(self): if all(arg.is_iterable for arg in self.args): return True def __iter__(self): args = self.args union = roundrobin(*(iter(arg) for arg in args)) for item in union: count = 0 for s in args: if item in s: count += 1 if count % 2 == 1: yield item class DisjointUnion(Set): """ Represents the disjoint union (also known as the external disjoint union) of a finite number of sets. Examples ======== >>> from sympy import DisjointUnion, FiniteSet, Interval, Union, Symbol >>> A = FiniteSet(1, 2, 3) >>> B = Interval(0, 5) >>> DisjointUnion(A, B) DisjointUnion({1, 2, 3}, Interval(0, 5)) >>> DisjointUnion(A, B).rewrite(Union) Union(ProductSet({1, 2, 3}, {0}), ProductSet(Interval(0, 5), {1})) >>> C = FiniteSet(Symbol('x'), Symbol('y'), Symbol('z')) >>> DisjointUnion(C, C) DisjointUnion({x, y, z}, {x, y, z}) >>> DisjointUnion(C, C).rewrite(Union) ProductSet({x, y, z}, {0, 1}) References ========== https://en.wikipedia.org/wiki/Disjoint_union """ def __new__(cls, *sets): dj_collection = [] for set_i in sets: if isinstance(set_i, Set): dj_collection.append(set_i) else: raise TypeError("Invalid input: '%s', input args \ to DisjointUnion must be Sets" % set_i) obj = Basic.__new__(cls, *dj_collection) return obj @property def sets(self): return self.args @property def is_empty(self): return fuzzy_and(s.is_empty for s in self.sets) @property def is_finite_set(self): all_finite = fuzzy_and(s.is_finite_set for s in self.sets) return fuzzy_or([self.is_empty, all_finite]) @property def is_iterable(self): if self.is_empty: return False iter_flag = True for set_i in self.sets: if not set_i.is_empty: iter_flag = iter_flag and set_i.is_iterable return iter_flag def _eval_rewrite_as_Union(self, *sets): """ Rewrites the disjoint union as the union of (``set`` x {``i``}) where ``set`` is the element in ``sets`` at index = ``i`` """ dj_union = EmptySet() index = 0 for set_i in sets: if isinstance(set_i, Set): cross = ProductSet(set_i, FiniteSet(index)) dj_union = Union(dj_union, cross) index = index + 1 return dj_union def _contains(self, element): """ 'in' operator for DisjointUnion Examples ======== >>> from sympy import Interval, DisjointUnion >>> D = DisjointUnion(Interval(0, 1), Interval(0, 2)) >>> (0.5, 0) in D True >>> (0.5, 1) in D True >>> (1.5, 0) in D False >>> (1.5, 1) in D True Passes operation on to constituent sets """ if not isinstance(element, Tuple) or len(element) != 2: return False if not element[1].is_Integer: return False if element[1] >= len(self.sets) or element[1] < 0: return False return element[0] in self.sets[element[1]] def __iter__(self): if self.is_iterable: from sympy.core.numbers import Integer iters = [] for i, s in enumerate(self.sets): iters.append(iproduct(s, {Integer(i)})) return iter(roundrobin(*iters)) else: raise ValueError("'%s' is not iterable." % self) def __len__(self): """ Returns the length of the disjoint union, i.e., the number of elements in the set. Examples ======== >>> from sympy import FiniteSet, DisjointUnion, EmptySet >>> D1 = DisjointUnion(FiniteSet(1, 2, 3, 4), EmptySet, FiniteSet(3, 4, 5)) >>> len(D1) 7 >>> D2 = DisjointUnion(FiniteSet(3, 5, 7), EmptySet, FiniteSet(3, 5, 7)) >>> len(D2) 6 >>> D3 = DisjointUnion(EmptySet, EmptySet) >>> len(D3) 0 Adds up the lengths of the constituent sets. """ if self.is_finite_set: size = 0 for set in self.sets: size += len(set) return size else: raise ValueError("'%s' is not a finite set." % self) def imageset(*args): r""" Return an image of the set under transformation ``f``. Explanation =========== If this function can't compute the image, it returns an unevaluated ImageSet object. .. math:: \{ f(x) \mid x \in \mathrm{self} \} Examples ======== >>> from sympy import S, Interval, imageset, sin, Lambda >>> from sympy.abc import x >>> 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(y, x + y), 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), Integers) See Also ======== sympy.sets.fancysets.ImageSet """ from sympy.core import Lambda from sympy.sets.fancysets import ImageSet from sympy.sets.setexpr import set_function if len(args) < 2: raise ValueError('imageset expects at least 2 args, got: %s' % len(args)) if isinstance(args[0], (Symbol, tuple)) and len(args) > 2: f = Lambda(args[0], args[1]) set_list = args[2:] else: f = args[0] set_list = args[1:] if isinstance(f, Lambda): pass elif callable(f): nargs = getattr(f, 'nargs', {}) if nargs: if len(nargs) != 1: raise NotImplementedError(filldedent(''' This function can take more than 1 arg but the potentially complicated set input has not been analyzed at this point to know its dimensions. TODO ''')) N = nargs.args[0] if N == 1: s = 'x' else: s = [Symbol('x%i' % i) for i in range(1, N + 1)] else: s = inspect.signature(f).parameters dexpr = _sympify(f(*[Dummy() for i in s])) var = tuple(uniquely_named_symbol( Symbol(i), dexpr) for i in s) f = Lambda(var, f(*var)) else: raise TypeError(filldedent(''' expecting lambda, Lambda, or FunctionClass, not \'%s\'.''' % func_name(f))) if any(not isinstance(s, Set) for s in set_list): name = [func_name(s) for s in set_list] raise ValueError( 'arguments after mapping should be sets, not %s' % name) if len(set_list) == 1: set = set_list[0] try: # TypeError if arg count != set dimensions r = set_function(f, set) if r is None: raise TypeError if not r: return r except TypeError: r = ImageSet(f, set) if isinstance(r, ImageSet): f, set = r.args if f.variables[0] == f.expr: return set if isinstance(set, ImageSet): # XXX: Maybe this should just be: # f2 = set.lambda # fun = Lambda(f2.signature, f(*f2.expr)) # return imageset(fun, *set.base_sets) if len(set.lamda.variables) == 1 and len(f.variables) == 1: x = set.lamda.variables[0] y = f.variables[0] return imageset( Lambda(x, f.expr.subs(y, set.lamda.expr)), *set.base_sets) if r is not None: return r return ImageSet(f, *set_list) def is_function_invertible_in_set(func, setv): """ Checks whether function ``func`` is invertible when the domain is restricted to set ``setv``. """ from sympy import exp, log # Functions known to always be invertible: if func in (exp, log): return True u = Dummy("u") fdiff = func(u).diff(u) # monotonous functions: # TODO: check subsets (`func` in `setv`) if (fdiff > 0) == True or (fdiff < 0) == True: return True # TODO: support more return None def simplify_union(args): """ Simplify a :class:`Union` using known rules. Explanation =========== 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. This process depends on ``union_sets(a, b)`` functions. """ from sympy.sets.handlers.union import union_sets # ===== Global Rules ===== if not args: return S.EmptySet for arg in args: if not isinstance(arg, Set): raise TypeError("Input args to Union must be Sets") # 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 = union_sets(s, 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 simplify_intersection(args): """ Simplify an intersection using known rules. Explanation =========== 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 not args: return S.UniversalSet for arg in args: if not isinstance(arg, Set): raise TypeError("Input args to Union must be Sets") # If any EmptySets return EmptySet if S.EmptySet 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) - {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]) from sympy.sets.handlers.intersection import intersection_sets # 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 = intersection_sets(s, 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) def _handle_finite_sets(op, x, y, commutative): # Handle finite sets: fs_args, other = sift([x, y], lambda x: isinstance(x, FiniteSet), binary=True) if len(fs_args) == 2: return FiniteSet(*[op(i, j) for i in fs_args[0] for j in fs_args[1]]) elif len(fs_args) == 1: sets = [_apply_operation(op, other[0], i, commutative) for i in fs_args[0]] return Union(*sets) else: return None def _apply_operation(op, x, y, commutative): from sympy.sets import ImageSet from sympy import symbols,Lambda d = Dummy('d') out = _handle_finite_sets(op, x, y, commutative) if out is None: out = op(x, y) if out is None and commutative: out = op(y, x) if out is None: _x, _y = symbols("x y") if isinstance(x, Set) and not isinstance(y, Set): out = ImageSet(Lambda(d, op(d, y)), x).doit() elif not isinstance(x, Set) and isinstance(y, Set): out = ImageSet(Lambda(d, op(x, d)), y).doit() else: out = ImageSet(Lambda((_x, _y), op(_x, _y)), x, y) return out def set_add(x, y): from sympy.sets.handlers.add import _set_add return _apply_operation(_set_add, x, y, commutative=True) def set_sub(x, y): from sympy.sets.handlers.add import _set_sub return _apply_operation(_set_sub, x, y, commutative=False) def set_mul(x, y): from sympy.sets.handlers.mul import _set_mul return _apply_operation(_set_mul, x, y, commutative=True) def set_div(x, y): from sympy.sets.handlers.mul import _set_div return _apply_operation(_set_div, x, y, commutative=False) def set_pow(x, y): from sympy.sets.handlers.power import _set_pow return _apply_operation(_set_pow, x, y, commutative=False) def set_function(f, x): from sympy.sets.handlers.functions import _set_function return _set_function(f, x)