text: from typing import Optional
  from functools import reduce
  from collections import defaultdict
  import inspect
  
  from sympy.core.basic import Basic
  from sympy.core.compatibility import iterable, ordered
  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, 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):
              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):
          dps = prec_to_dps(prec)
          return self.func(*[arg.evalf(n=dps) 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(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 the interval.
  
          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 the interval.
  
          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 interval 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 interval 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 or other.has(S.ComplexInfinity)):
                  # if an expression has zoo it will be zoo or nan
                  # and neither of those is real
                  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. """
          return Or(*[Eq(symbol, elem) for elem in self])
  
      def compare(self, other):
          return (hash(self) - hash(other))
  
      def _eval_evalf(self, prec):
          dps = prec_to_dps(prec)
          return FiniteSet(*[elem.evalf(n=dps) 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 cannot 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)
  