venv/lib/python3.11/site-packages/sympy/sets/tests/test_sets.py

from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.containers import TupleKind
from sympy.core.function import Lambda
from sympy.core.kind import NumberKind, UndefinedKind
from sympy.core.numbers import (Float, I, Rational, nan, oo, pi, zoo)
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.core.sympify import sympify
from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.logic.boolalg import (false, true)
from sympy.matrices.kind import MatrixKind
from sympy.matrices.dense import Matrix
from sympy.polys.rootoftools import rootof
from sympy.sets.contains import Contains
from sympy.sets.fancysets import (ImageSet, Range)
from sympy.sets.sets import (Complement, DisjointUnion, FiniteSet, Intersection, Interval, ProductSet, Set, SymmetricDifference, Union, imageset, SetKind)
from mpmath import mpi

from sympy.core.expr import unchanged
from sympy.core.relational import Eq, Ne, Le, Lt, LessThan
from sympy.logic import And, Or, Xor
from sympy.testing.pytest import raises, XFAIL, warns_deprecated_sympy
from sympy.utilities.iterables import cartes

from sympy.abc import x, y, z, m, n

EmptySet = S.EmptySet

def test_imageset():
    ints = S.Integers
    assert imageset(x, x - 1, S.Naturals) is S.Naturals0
    assert imageset(x, x + 1, S.Naturals0) is S.Naturals
    assert imageset(x, abs(x), S.Naturals0) is S.Naturals0
    assert imageset(x, abs(x), S.Naturals) is S.Naturals
    assert imageset(x, abs(x), S.Integers) is S.Naturals0
    # issue 16878a
    r = symbols('r', real=True)
    assert imageset(x, (x, x), S.Reals)._contains((1, r)) == None
    assert imageset(x, (x, x), S.Reals)._contains((1, 2)) == False
    assert (r, r) in imageset(x, (x, x), S.Reals)
    assert 1 + I in imageset(x, x + I, S.Reals)
    assert {1} not in imageset(x, (x,), S.Reals)
    assert (1, 1) not in imageset(x, (x,), S.Reals)
    raises(TypeError, lambda: imageset(x, ints))
    raises(ValueError, lambda: imageset(x, y, z, ints))
    raises(ValueError, lambda: imageset(Lambda(x, cos(x)), y))
    assert (1, 2) in imageset(Lambda((x, y), (x, y)), ints, ints)
    raises(ValueError, lambda: imageset(Lambda(x, x), ints, ints))
    assert imageset(cos, ints) == ImageSet(Lambda(x, cos(x)), ints)
    def f(x):
        return cos(x)
    assert imageset(f, ints) == imageset(x, cos(x), ints)
    f = lambda x: cos(x)
    assert imageset(f, ints) == ImageSet(Lambda(x, cos(x)), ints)
    assert imageset(x, 1, ints) == FiniteSet(1)
    assert imageset(x, y, ints) == {y}
    assert imageset((x, y), (1, z), ints, S.Reals) == {(1, z)}
    clash = Symbol('x', integer=true)
    assert (str(imageset(lambda x: x + clash, Interval(-2, 1)).lamda.expr)
        in ('x0 + x', 'x + x0'))
    x1, x2 = symbols("x1, x2")
    assert imageset(lambda x, y:
        Add(x, y), Interval(1, 2), Interval(2, 3)).dummy_eq(
        ImageSet(Lambda((x1, x2), x1 + x2),
        Interval(1, 2), Interval(2, 3)))


def test_is_empty():
    for s in [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals,
            S.UniversalSet]:
        assert s.is_empty is False

    assert S.EmptySet.is_empty is True


def test_is_finiteset():
    for s in [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals,
            S.UniversalSet]:
        assert s.is_finite_set is False

    assert S.EmptySet.is_finite_set is True

    assert FiniteSet(1, 2).is_finite_set is True
    assert Interval(1, 2).is_finite_set is False
    assert Interval(x, y).is_finite_set is None
    assert ProductSet(FiniteSet(1), FiniteSet(2)).is_finite_set is True
    assert ProductSet(FiniteSet(1), Interval(1, 2)).is_finite_set is False
    assert ProductSet(FiniteSet(1), Interval(x, y)).is_finite_set is None
    assert Union(Interval(0, 1), Interval(2, 3)).is_finite_set is False
    assert Union(FiniteSet(1), Interval(2, 3)).is_finite_set is False
    assert Union(FiniteSet(1), FiniteSet(2)).is_finite_set is True
    assert Union(FiniteSet(1), Interval(x, y)).is_finite_set is None
    assert Intersection(Interval(x, y), FiniteSet(1)).is_finite_set is True
    assert Intersection(Interval(x, y), Interval(1, 2)).is_finite_set is None
    assert Intersection(FiniteSet(x), FiniteSet(y)).is_finite_set is True
    assert Complement(FiniteSet(1), Interval(x, y)).is_finite_set is True
    assert Complement(Interval(x, y), FiniteSet(1)).is_finite_set is None
    assert Complement(Interval(1, 2), FiniteSet(x)).is_finite_set is False
    assert DisjointUnion(Interval(-5, 3), FiniteSet(x, y)).is_finite_set is False
    assert DisjointUnion(S.EmptySet, FiniteSet(x, y), S.EmptySet).is_finite_set is True


def test_deprecated_is_EmptySet():
    with warns_deprecated_sympy():
        S.EmptySet.is_EmptySet

    with warns_deprecated_sympy():
        FiniteSet(1).is_EmptySet


def test_interval_arguments():
    assert Interval(0, oo) == Interval(0, oo, False, True)
    assert Interval(0, oo).right_open is true
    assert Interval(-oo, 0) == Interval(-oo, 0, True, False)
    assert Interval(-oo, 0).left_open is true
    assert Interval(oo, -oo) == S.EmptySet
    assert Interval(oo, oo) == S.EmptySet
    assert Interval(-oo, -oo) == S.EmptySet
    assert Interval(oo, x) == S.EmptySet
    assert Interval(oo, oo) == S.EmptySet
    assert Interval(x, -oo) == S.EmptySet
    assert Interval(x, x) == {x}

    assert isinstance(Interval(1, 1), FiniteSet)
    e = Sum(x, (x, 1, 3))
    assert isinstance(Interval(e, e), FiniteSet)

    assert Interval(1, 0) == S.EmptySet
    assert Interval(1, 1).measure == 0

    assert Interval(1, 1, False, True) == S.EmptySet
    assert Interval(1, 1, True, False) == S.EmptySet
    assert Interval(1, 1, True, True) == S.EmptySet


    assert isinstance(Interval(0, Symbol('a')), Interval)
    assert Interval(Symbol('a', positive=True), 0) == S.EmptySet
    raises(ValueError, lambda: Interval(0, S.ImaginaryUnit))
    raises(ValueError, lambda: Interval(0, Symbol('z', extended_real=False)))
    raises(ValueError, lambda: Interval(x, x + S.ImaginaryUnit))

    raises(NotImplementedError, lambda: Interval(0, 1, And(x, y)))
    raises(NotImplementedError, lambda: Interval(0, 1, False, And(x, y)))
    raises(NotImplementedError, lambda: Interval(0, 1, z, And(x, y)))


def test_interval_symbolic_end_points():
    a = Symbol('a', real=True)

    assert Union(Interval(0, a), Interval(0, 3)).sup == Max(a, 3)
    assert Union(Interval(a, 0), Interval(-3, 0)).inf == Min(-3, a)

    assert Interval(0, a).contains(1) == LessThan(1, a)


def test_interval_is_empty():
    x, y = symbols('x, y')
    r = Symbol('r', real=True)
    p = Symbol('p', positive=True)
    n = Symbol('n', negative=True)
    nn = Symbol('nn', nonnegative=True)
    assert Interval(1, 2).is_empty == False
    assert Interval(3, 3).is_empty == False  # FiniteSet
    assert Interval(r, r).is_empty == False  # FiniteSet
    assert Interval(r, r + nn).is_empty == False
    assert Interval(x, x).is_empty == False
    assert Interval(1, oo).is_empty == False
    assert Interval(-oo, oo).is_empty == False
    assert Interval(-oo, 1).is_empty == False
    assert Interval(x, y).is_empty == None
    assert Interval(r, oo).is_empty == False  # real implies finite
    assert Interval(n, 0).is_empty == False
    assert Interval(n, 0, left_open=True).is_empty == False
    assert Interval(p, 0).is_empty == True  # EmptySet
    assert Interval(nn, 0).is_empty == None
    assert Interval(n, p).is_empty == False
    assert Interval(0, p, left_open=True).is_empty == False
    assert Interval(0, p, right_open=True).is_empty == False
    assert Interval(0, nn, left_open=True).is_empty == None
    assert Interval(0, nn, right_open=True).is_empty == None


def test_union():
    assert Union(Interval(1, 2), Interval(2, 3)) == Interval(1, 3)
    assert Union(Interval(1, 2), Interval(2, 3, True)) == Interval(1, 3)
    assert Union(Interval(1, 3), Interval(2, 4)) == Interval(1, 4)
    assert Union(Interval(1, 2), Interval(1, 3)) == Interval(1, 3)
    assert Union(Interval(1, 3), Interval(1, 2)) == Interval(1, 3)
    assert Union(Interval(1, 3, False, True), Interval(1, 2)) == \
        Interval(1, 3, False, True)
    assert Union(Interval(1, 3), Interval(1, 2, False, True)) == Interval(1, 3)
    assert Union(Interval(1, 2, True), Interval(1, 3)) == Interval(1, 3)
    assert Union(Interval(1, 2, True), Interval(1, 3, True)) == \
        Interval(1, 3, True)
    assert Union(Interval(1, 2, True), Interval(1, 3, True, True)) == \
        Interval(1, 3, True, True)
    assert Union(Interval(1, 2, True, True), Interval(1, 3, True)) == \
        Interval(1, 3, True)
    assert Union(Interval(1, 3), Interval(2, 3)) == Interval(1, 3)
    assert Union(Interval(1, 3, False, True), Interval(2, 3)) == \
        Interval(1, 3)
    assert Union(Interval(1, 2, False, True), Interval(2, 3, True)) != \
        Interval(1, 3)
    assert Union(Interval(1, 2), S.EmptySet) == Interval(1, 2)
    assert Union(S.EmptySet) == S.EmptySet

    assert Union(Interval(0, 1), *[FiniteSet(1.0/n) for n in range(1, 10)]) == \
        Interval(0, 1)
    # issue #18241:
    x = Symbol('x')
    assert Union(Interval(0, 1), FiniteSet(1, x)) == Union(
        Interval(0, 1), FiniteSet(x))
    assert unchanged(Union, Interval(0, 1), FiniteSet(2, x))

    assert Interval(1, 2).union(Interval(2, 3)) == \
        Interval(1, 2) + Interval(2, 3)

    assert Interval(1, 2).union(Interval(2, 3)) == Interval(1, 3)

    assert Union(Set()) == Set()

    assert FiniteSet(1) + FiniteSet(2) + FiniteSet(3) == FiniteSet(1, 2, 3)
    assert FiniteSet('ham') + FiniteSet('eggs') == FiniteSet('ham', 'eggs')
    assert FiniteSet(1, 2, 3) + S.EmptySet == FiniteSet(1, 2, 3)

    assert FiniteSet(1, 2, 3) & FiniteSet(2, 3, 4) == FiniteSet(2, 3)
    assert FiniteSet(1, 2, 3) | FiniteSet(2, 3, 4) == FiniteSet(1, 2, 3, 4)

    assert FiniteSet(1, 2, 3) & S.EmptySet == S.EmptySet
    assert FiniteSet(1, 2, 3) | S.EmptySet == FiniteSet(1, 2, 3)

    x = Symbol("x")
    y = Symbol("y")
    z = Symbol("z")
    assert S.EmptySet | FiniteSet(x, FiniteSet(y, z)) == \
        FiniteSet(x, FiniteSet(y, z))

    # Test that Intervals and FiniteSets play nicely
    assert Interval(1, 3) + FiniteSet(2) == Interval(1, 3)
    assert Interval(1, 3, True, True) + FiniteSet(3) == \
        Interval(1, 3, True, False)
    X = Interval(1, 3) + FiniteSet(5)
    Y = Interval(1, 2) + FiniteSet(3)
    XandY = X.intersect(Y)
    assert 2 in X and 3 in X and 3 in XandY
    assert XandY.is_subset(X) and XandY.is_subset(Y)

    raises(TypeError, lambda: Union(1, 2, 3))

    assert X.is_iterable is False

    # issue 7843
    assert Union(S.EmptySet, FiniteSet(-sqrt(-I), sqrt(-I))) == \
        FiniteSet(-sqrt(-I), sqrt(-I))

    assert Union(S.Reals, S.Integers) == S.Reals


def test_union_iter():
    # Use Range because it is ordered
    u = Union(Range(3), Range(5), Range(4), evaluate=False)

    # Round robin
    assert list(u) == [0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4]


def test_union_is_empty():
    assert (Interval(x, y) + FiniteSet(1)).is_empty == False
    assert (Interval(x, y) + Interval(-x, y)).is_empty == None


def test_difference():
    assert Interval(1, 3) - Interval(1, 2) == Interval(2, 3, True)
    assert Interval(1, 3) - Interval(2, 3) == Interval(1, 2, False, True)
    assert Interval(1, 3, True) - Interval(2, 3) == Interval(1, 2, True, True)
    assert Interval(1, 3, True) - Interval(2, 3, True) == \
        Interval(1, 2, True, False)
    assert Interval(0, 2) - FiniteSet(1) == \
        Union(Interval(0, 1, False, True), Interval(1, 2, True, False))

    # issue #18119
    assert S.Reals - FiniteSet(I) == S.Reals
    assert S.Reals - FiniteSet(-I, I) == S.Reals
    assert Interval(0, 10) - FiniteSet(-I, I) == Interval(0, 10)
    assert Interval(0, 10) - FiniteSet(1, I) == Union(
        Interval.Ropen(0, 1), Interval.Lopen(1, 10))
    assert S.Reals - FiniteSet(1, 2 + I, x, y**2) == Complement(
        Union(Interval.open(-oo, 1), Interval.open(1, oo)), FiniteSet(x, y**2),
        evaluate=False)

    assert FiniteSet(1, 2, 3) - FiniteSet(2) == FiniteSet(1, 3)
    assert FiniteSet('ham', 'eggs') - FiniteSet('eggs') == FiniteSet('ham')
    assert FiniteSet(1, 2, 3, 4) - Interval(2, 10, True, False) == \
        FiniteSet(1, 2)
    assert FiniteSet(1, 2, 3, 4) - S.EmptySet == FiniteSet(1, 2, 3, 4)
    assert Union(Interval(0, 2), FiniteSet(2, 3, 4)) - Interval(1, 3) == \
        Union(Interval(0, 1, False, True), FiniteSet(4))

    assert -1 in S.Reals - S.Naturals


def test_Complement():
    A = FiniteSet(1, 3, 4)
    B = FiniteSet(3, 4)
    C = Interval(1, 3)
    D = Interval(1, 2)

    assert Complement(A, B, evaluate=False).is_iterable is True
    assert Complement(A, C, evaluate=False).is_iterable is True
    assert Complement(C, D, evaluate=False).is_iterable is None

    assert FiniteSet(*Complement(A, B, evaluate=False)) == FiniteSet(1)
    assert FiniteSet(*Complement(A, C, evaluate=False)) == FiniteSet(4)
    raises(TypeError, lambda: FiniteSet(*Complement(C, A, evaluate=False)))

    assert Complement(Interval(1, 3), Interval(1, 2)) == Interval(2, 3, True)
    assert Complement(FiniteSet(1, 3, 4), FiniteSet(3, 4)) == FiniteSet(1)
    assert Complement(Union(Interval(0, 2), FiniteSet(2, 3, 4)),
                      Interval(1, 3)) == \
        Union(Interval(0, 1, False, True), FiniteSet(4))

    assert 3 not in Complement(Interval(0, 5), Interval(1, 4), evaluate=False)
    assert -1 in Complement(S.Reals, S.Naturals, evaluate=False)
    assert 1 not in Complement(S.Reals, S.Naturals, evaluate=False)

    assert Complement(S.Integers, S.UniversalSet) == EmptySet
    assert S.UniversalSet.complement(S.Integers) == EmptySet

    assert (0 not in S.Reals.intersect(S.Integers - FiniteSet(0)))

    assert S.EmptySet - S.Integers == S.EmptySet

    assert (S.Integers - FiniteSet(0)) - FiniteSet(1) == S.Integers - FiniteSet(0, 1)

    assert S.Reals - Union(S.Naturals, FiniteSet(pi)) == \
            Intersection(S.Reals - S.Naturals, S.Reals - FiniteSet(pi))
    # issue 12712
    assert Complement(FiniteSet(x, y, 2), Interval(-10, 10)) == \
            Complement(FiniteSet(x, y), Interval(-10, 10))

    A = FiniteSet(*symbols('a:c'))
    B = FiniteSet(*symbols('d:f'))
    assert unchanged(Complement, ProductSet(A, A), B)

    A2 = ProductSet(A, A)
    B3 = ProductSet(B, B, B)
    assert A2 - B3 == A2
    assert B3 - A2 == B3


def test_set_operations_nonsets():
    '''Tests that e.g. FiniteSet(1) * 2 raises TypeError'''
    ops = [
        lambda a, b: a + b,
        lambda a, b: a - b,
        lambda a, b: a * b,
        lambda a, b: a / b,
        lambda a, b: a // b,
        lambda a, b: a | b,
        lambda a, b: a & b,
        lambda a, b: a ^ b,
        # FiniteSet(1) ** 2 gives a ProductSet
        #lambda a, b: a ** b,
    ]
    Sx = FiniteSet(x)
    Sy = FiniteSet(y)
    sets = [
        {1},
        FiniteSet(1),
        Interval(1, 2),
        Union(Sx, Interval(1, 2)),
        Intersection(Sx, Sy),
        Complement(Sx, Sy),
        ProductSet(Sx, Sy),
        S.EmptySet,
    ]
    nums = [0, 1, 2, S(0), S(1), S(2)]

    for si in sets:
        for ni in nums:
            for op in ops:
                raises(TypeError, lambda : op(si, ni))
                raises(TypeError, lambda : op(ni, si))
        raises(TypeError, lambda: si ** object())
        raises(TypeError, lambda: si ** {1})


def test_complement():
    assert Complement({1, 2}, {1}) == {2}
    assert Interval(0, 1).complement(S.Reals) == \
        Union(Interval(-oo, 0, True, True), Interval(1, oo, True, True))
    assert Interval(0, 1, True, False).complement(S.Reals) == \
        Union(Interval(-oo, 0, True, False), Interval(1, oo, True, True))
    assert Interval(0, 1, False, True).complement(S.Reals) == \
        Union(Interval(-oo, 0, True, True), Interval(1, oo, False, True))
    assert Interval(0, 1, True, True).complement(S.Reals) == \
        Union(Interval(-oo, 0, True, False), Interval(1, oo, False, True))

    assert S.UniversalSet.complement(S.EmptySet) == S.EmptySet
    assert S.UniversalSet.complement(S.Reals) == S.EmptySet
    assert S.UniversalSet.complement(S.UniversalSet) == S.EmptySet

    assert S.EmptySet.complement(S.Reals) == S.Reals

    assert Union(Interval(0, 1), Interval(2, 3)).complement(S.Reals) == \
        Union(Interval(-oo, 0, True, True), Interval(1, 2, True, True),
              Interval(3, oo, True, True))

    assert FiniteSet(0).complement(S.Reals) ==  \
        Union(Interval(-oo, 0, True, True), Interval(0, oo, True, True))

    assert (FiniteSet(5) + Interval(S.NegativeInfinity,
                                    0)).complement(S.Reals) == \
        Interval(0, 5, True, True) + Interval(5, S.Infinity, True, True)

    assert FiniteSet(1, 2, 3).complement(S.Reals) == \
        Interval(S.NegativeInfinity, 1, True, True) + \
        Interval(1, 2, True, True) + Interval(2, 3, True, True) +\
        Interval(3, S.Infinity, True, True)

    assert FiniteSet(x).complement(S.Reals) == Complement(S.Reals, FiniteSet(x))

    assert FiniteSet(0, x).complement(S.Reals) == Complement(Interval(-oo, 0, True, True) +
                                                             Interval(0, oo, True, True)
                                                             , FiniteSet(x), evaluate=False)

    square = Interval(0, 1) * Interval(0, 1)
    notsquare = square.complement(S.Reals*S.Reals)

    assert all(pt in square for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)])
    assert not any(
        pt in notsquare for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)])
    assert not any(pt in square for pt in [(-1, 0), (1.5, .5), (10, 10)])
    assert all(pt in notsquare for pt in [(-1, 0), (1.5, .5), (10, 10)])


def test_intersect1():
    assert all(S.Integers.intersection(i) is i for i in
        (S.Naturals, S.Naturals0))
    assert all(i.intersection(S.Integers) is i for i in
        (S.Naturals, S.Naturals0))
    s =  S.Naturals0
    assert S.Naturals.intersection(s) is S.Naturals
    assert s.intersection(S.Naturals) is S.Naturals
    x = Symbol('x')
    assert Interval(0, 2).intersect(Interval(1, 2)) == Interval(1, 2)
    assert Interval(0, 2).intersect(Interval(1, 2, True)) == \
        Interval(1, 2, True)
    assert Interval(0, 2, True).intersect(Interval(1, 2)) == \
        Interval(1, 2, False, False)
    assert Interval(0, 2, True, True).intersect(Interval(1, 2)) == \
        Interval(1, 2, False, True)
    assert Interval(0, 2).intersect(Union(Interval(0, 1), Interval(2, 3))) == \
        Union(Interval(0, 1), Interval(2, 2))

    assert FiniteSet(1, 2).intersect(FiniteSet(1, 2, 3)) == FiniteSet(1, 2)
    assert FiniteSet(1, 2, x).intersect(FiniteSet(x)) == FiniteSet(x)
    assert FiniteSet('ham', 'eggs').intersect(FiniteSet('ham')) == \
        FiniteSet('ham')
    assert FiniteSet(1, 2, 3, 4, 5).intersect(S.EmptySet) == S.EmptySet

    assert Interval(0, 5).intersect(FiniteSet(1, 3)) == FiniteSet(1, 3)
    assert Interval(0, 1, True, True).intersect(FiniteSet(1)) == S.EmptySet

    assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(1, 2)) == \
        Union(Interval(1, 1), Interval(2, 2))
    assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(0, 2)) == \
        Union(Interval(0, 1), Interval(2, 2))
    assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(1, 2, True, True)) == \
        S.EmptySet
    assert Union(Interval(0, 1), Interval(2, 3)).intersect(S.EmptySet) == \
        S.EmptySet
    assert Union(Interval(0, 5), FiniteSet('ham')).intersect(FiniteSet(2, 3, 4, 5, 6)) == \
        Intersection(FiniteSet(2, 3, 4, 5, 6), Union(FiniteSet('ham'), Interval(0, 5)))
    assert Intersection(FiniteSet(1, 2, 3), Interval(2, x), Interval(3, y)) == \
        Intersection(FiniteSet(3), Interval(2, x), Interval(3, y), evaluate=False)
    assert Intersection(FiniteSet(1, 2), Interval(0, 3), Interval(x, y)) == \
        Intersection({1, 2}, Interval(x, y), evaluate=False)
    assert Intersection(FiniteSet(1, 2, 4), Interval(0, 3), Interval(x, y)) == \
        Intersection({1, 2}, Interval(x, y), evaluate=False)
    # XXX: Is the real=True necessary here?
    # https://github.com/sympy/sympy/issues/17532
    m, n = symbols('m, n', real=True)
    assert Intersection(FiniteSet(m), FiniteSet(m, n), Interval(m, m+1)) == \
        FiniteSet(m)

    # issue 8217
    assert Intersection(FiniteSet(x), FiniteSet(y)) == \
        Intersection(FiniteSet(x), FiniteSet(y), evaluate=False)
    assert FiniteSet(x).intersect(S.Reals) == \
        Intersection(S.Reals, FiniteSet(x), evaluate=False)

    # tests for the intersection alias
    assert Interval(0, 5).intersection(FiniteSet(1, 3)) == FiniteSet(1, 3)
    assert Interval(0, 1, True, True).intersection(FiniteSet(1)) == S.EmptySet

    assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(1, 2)) == \
        Union(Interval(1, 1), Interval(2, 2))

    # canonical boundary selected
    a = sqrt(2*sqrt(6) + 5)
    b = sqrt(2) + sqrt(3)
    assert Interval(a, 4).intersection(Interval(b, 5)) == Interval(b, 4)
    assert Interval(1, a).intersection(Interval(0, b)) == Interval(1, b)


def test_intersection_interval_float():
    # intersection of Intervals with mixed Rational/Float boundaries should
    # lead to Float boundaries in all cases regardless of which Interval is
    # open or closed.
    typs = [
        (Interval, Interval, Interval),
        (Interval, Interval.open, Interval.open),
        (Interval, Interval.Lopen, Interval.Lopen),
        (Interval, Interval.Ropen, Interval.Ropen),
        (Interval.open, Interval.open, Interval.open),
        (Interval.open, Interval.Lopen, Interval.open),
        (Interval.open, Interval.Ropen, Interval.open),
        (Interval.Lopen, Interval.Lopen, Interval.Lopen),
        (Interval.Lopen, Interval.Ropen, Interval.open),
        (Interval.Ropen, Interval.Ropen, Interval.Ropen),
    ]

    as_float = lambda a1, a2: a2 if isinstance(a2, float) else a1

    for t1, t2, t3 in typs:
        for t1i, t2i in [(t1, t2), (t2, t1)]:
            for a1, a2, b1, b2 in cartes([2, 2.0], [2, 2.0], [3, 3.0], [3, 3.0]):
                I1 = t1(a1, b1)
                I2 = t2(a2, b2)
                I3 = t3(as_float(a1, a2), as_float(b1, b2))
                assert I1.intersect(I2) == I3


def test_intersection():
    # iterable
    i = Intersection(FiniteSet(1, 2, 3), Interval(2, 5), evaluate=False)
    assert i.is_iterable
    assert set(i) == {S(2), S(3)}

    # challenging intervals
    x = Symbol('x', real=True)
    i = Intersection(Interval(0, 3), Interval(x, 6))
    assert (5 in i) is False
    raises(TypeError, lambda: 2 in i)

    # Singleton special cases
    assert Intersection(Interval(0, 1), S.EmptySet) == S.EmptySet
    assert Intersection(Interval(-oo, oo), Interval(-oo, x)) == Interval(-oo, x)

    # Products
    line = Interval(0, 5)
    i = Intersection(line**2, line**3, evaluate=False)
    assert (2, 2) not in i
    assert (2, 2, 2) not in i
    raises(TypeError, lambda: list(i))

    a = Intersection(Intersection(S.Integers, S.Naturals, evaluate=False), S.Reals, evaluate=False)
    assert a._argset == frozenset([Intersection(S.Naturals, S.Integers, evaluate=False), S.Reals])

    assert Intersection(S.Complexes, FiniteSet(S.ComplexInfinity)) == S.EmptySet

    # issue 12178
    assert Intersection() == S.UniversalSet

    # issue 16987
    assert Intersection({1}, {1}, {x}) == Intersection({1}, {x})


def test_issue_9623():
    n = Symbol('n')

    a = S.Reals
    b = Interval(0, oo)
    c = FiniteSet(n)

    assert Intersection(a, b, c) == Intersection(b, c)
    assert Intersection(Interval(1, 2), Interval(3, 4), FiniteSet(n)) == EmptySet


def test_is_disjoint():
    assert Interval(0, 2).is_disjoint(Interval(1, 2)) == False
    assert Interval(0, 2).is_disjoint(Interval(3, 4)) == True


def test_ProductSet__len__():
    A = FiniteSet(1, 2)
    B = FiniteSet(1, 2, 3)
    assert ProductSet(A).__len__() == 2
    assert ProductSet(A).__len__() is not S(2)
    assert ProductSet(A, B).__len__() == 6
    assert ProductSet(A, B).__len__() is not S(6)


def test_ProductSet():
    # ProductSet is always a set of Tuples
    assert ProductSet(S.Reals) == S.Reals ** 1
    assert ProductSet(S.Reals, S.Reals) == S.Reals ** 2
    assert ProductSet(S.Reals, S.Reals, S.Reals) == S.Reals ** 3

    assert ProductSet(S.Reals) != S.Reals
    assert ProductSet(S.Reals, S.Reals) == S.Reals * S.Reals
    assert ProductSet(S.Reals, S.Reals, S.Reals) != S.Reals * S.Reals * S.Reals
    assert ProductSet(S.Reals, S.Reals, S.Reals) == (S.Reals * S.Reals * S.Reals).flatten()

    assert 1 not in ProductSet(S.Reals)
    assert (1,) in ProductSet(S.Reals)

    assert 1 not in ProductSet(S.Reals, S.Reals)
    assert (1, 2) in ProductSet(S.Reals, S.Reals)
    assert (1, I) not in ProductSet(S.Reals, S.Reals)

    assert (1, 2, 3) in ProductSet(S.Reals, S.Reals, S.Reals)
    assert (1, 2, 3) in S.Reals ** 3
    assert (1, 2, 3) not in S.Reals * S.Reals * S.Reals
    assert ((1, 2), 3) in S.Reals * S.Reals * S.Reals
    assert (1, (2, 3)) not in S.Reals * S.Reals * S.Reals
    assert (1, (2, 3)) in S.Reals * (S.Reals * S.Reals)

    assert ProductSet() == FiniteSet(())
    assert ProductSet(S.Reals, S.EmptySet) == S.EmptySet

    # See GH-17458

    for ni in range(5):
        Rn = ProductSet(*(S.Reals,) * ni)
        assert (1,) * ni in Rn
        assert 1 not in Rn

    assert (S.Reals * S.Reals) * S.Reals != S.Reals * (S.Reals * S.Reals)

    S1 = S.Reals
    S2 = S.Integers
    x1 = pi
    x2 = 3
    assert x1 in S1
    assert x2 in S2
    assert (x1, x2) in S1 * S2
    S3 = S1 * S2
    x3 = (x1, x2)
    assert x3 in S3
    assert (x3, x3) in S3 * S3
    assert x3 + x3 not in S3 * S3

    raises(ValueError, lambda: S.Reals**-1)
    with warns_deprecated_sympy():
        ProductSet(FiniteSet(s) for s in range(2))
    raises(TypeError, lambda: ProductSet(None))

    S1 = FiniteSet(1, 2)
    S2 = FiniteSet(3, 4)
    S3 = ProductSet(S1, S2)
    assert (S3.as_relational(x, y)
            == And(S1.as_relational(x), S2.as_relational(y))
            == And(Or(Eq(x, 1), Eq(x, 2)), Or(Eq(y, 3), Eq(y, 4))))
    raises(ValueError, lambda: S3.as_relational(x))
    raises(ValueError, lambda: S3.as_relational(x, 1))
    raises(ValueError, lambda: ProductSet(Interval(0, 1)).as_relational(x, y))

    Z2 = ProductSet(S.Integers, S.Integers)
    assert Z2.contains((1, 2)) is S.true
    assert Z2.contains((1,)) is S.false
    assert Z2.contains(x) == Contains(x, Z2, evaluate=False)
    assert Z2.contains(x).subs(x, 1) is S.false
    assert Z2.contains((x, 1)).subs(x, 2) is S.true
    assert Z2.contains((x, y)) == Contains(x, S.Integers) & Contains(y, S.Integers)
    assert unchanged(Contains, (x, y), Z2)
    assert Contains((1, 2), Z2) is S.true


def test_ProductSet_of_single_arg_is_not_arg():
    assert unchanged(ProductSet, Interval(0, 1))
    assert unchanged(ProductSet, ProductSet(Interval(0, 1)))


def test_ProductSet_is_empty():
    assert ProductSet(S.Integers, S.Reals).is_empty == False
    assert ProductSet(Interval(x, 1), S.Reals).is_empty == None


def test_interval_subs():
    a = Symbol('a', real=True)

    assert Interval(0, a).subs(a, 2) == Interval(0, 2)
    assert Interval(a, 0).subs(a, 2) == S.EmptySet


def test_interval_to_mpi():
    assert Interval(0, 1).to_mpi() == mpi(0, 1)
    assert Interval(0, 1, True, False).to_mpi() == mpi(0, 1)
    assert type(Interval(0, 1).to_mpi()) == type(mpi(0, 1))


def test_set_evalf():
    assert Interval(S(11)/64, S.Half).evalf() == Interval(
        Float('0.171875'), Float('0.5'))
    assert Interval(x, S.Half, right_open=True).evalf() == Interval(
        x, Float('0.5'), right_open=True)
    assert Interval(-oo, S.Half).evalf() == Interval(-oo, Float('0.5'))
    assert FiniteSet(2, x).evalf() == FiniteSet(Float('2.0'), x)


def test_measure():
    a = Symbol('a', real=True)

    assert Interval(1, 3).measure == 2
    assert Interval(0, a).measure == a
    assert Interval(1, a).measure == a - 1

    assert Union(Interval(1, 2), Interval(3, 4)).measure == 2
    assert Union(Interval(1, 2), Interval(3, 4), FiniteSet(5, 6, 7)).measure \
        == 2

    assert FiniteSet(1, 2, oo, a, -oo, -5).measure == 0

    assert S.EmptySet.measure == 0

    square = Interval(0, 10) * Interval(0, 10)
    offsetsquare = Interval(5, 15) * Interval(5, 15)
    band = Interval(-oo, oo) * Interval(2, 4)

    assert square.measure == offsetsquare.measure == 100
    assert (square + offsetsquare).measure == 175  # there is some overlap
    assert (square - offsetsquare).measure == 75
    assert (square * FiniteSet(1, 2, 3)).measure == 0
    assert (square.intersect(band)).measure == 20
    assert (square + band).measure is oo
    assert (band * FiniteSet(1, 2, 3)).measure is nan


def test_is_subset():
    assert Interval(0, 1).is_subset(Interval(0, 2)) is True
    assert Interval(0, 3).is_subset(Interval(0, 2)) is False
    assert Interval(0, 1).is_subset(FiniteSet(0, 1)) is False

    assert FiniteSet(1, 2).is_subset(FiniteSet(1, 2, 3, 4))
    assert FiniteSet(4, 5).is_subset(FiniteSet(1, 2, 3, 4)) is False
    assert FiniteSet(1).is_subset(Interval(0, 2))
    assert FiniteSet(1, 2).is_subset(Interval(0, 2, True, True)) is False
    assert (Interval(1, 2) + FiniteSet(3)).is_subset(
        Interval(0, 2, False, True) + FiniteSet(2, 3))

    assert Interval(3, 4).is_subset(Union(Interval(0, 1), Interval(2, 5))) is True
    assert Interval(3, 6).is_subset(Union(Interval(0, 1), Interval(2, 5))) is False

    assert FiniteSet(1, 2, 3, 4).is_subset(Interval(0, 5)) is True
    assert S.EmptySet.is_subset(FiniteSet(1, 2, 3)) is True

    assert Interval(0, 1).is_subset(S.EmptySet) is False
    assert S.EmptySet.is_subset(S.EmptySet) is True

    raises(ValueError, lambda: S.EmptySet.is_subset(1))

    # tests for the issubset alias
    assert FiniteSet(1, 2, 3, 4).issubset(Interval(0, 5)) is True
    assert S.EmptySet.issubset(FiniteSet(1, 2, 3)) is True

    assert S.Naturals.is_subset(S.Integers)
    assert S.Naturals0.is_subset(S.Integers)

    assert FiniteSet(x).is_subset(FiniteSet(y)) is None
    assert FiniteSet(x).is_subset(FiniteSet(y).subs(y, x)) is True
    assert FiniteSet(x).is_subset(FiniteSet(y).subs(y, x+1)) is False

    assert Interval(0, 1).is_subset(Interval(0, 1, left_open=True)) is False
    assert Interval(-2, 3).is_subset(Union(Interval(-oo, -2), Interval(3, oo))) is False

    n = Symbol('n', integer=True)
    assert Range(-3, 4, 1).is_subset(FiniteSet(-10, 10)) is False
    assert Range(S(10)**100).is_subset(FiniteSet(0, 1, 2)) is False
    assert Range(6, 0, -2).is_subset(FiniteSet(2, 4, 6)) is True
    assert Range(1, oo).is_subset(FiniteSet(1, 2)) is False
    assert Range(-oo, 1).is_subset(FiniteSet(1)) is False
    assert Range(3).is_subset(FiniteSet(0, 1, n)) is None
    assert Range(n, n + 2).is_subset(FiniteSet(n, n + 1)) is True
    assert Range(5).is_subset(Interval(0, 4, right_open=True)) is False
    #issue 19513
    assert imageset(Lambda(n, 1/n), S.Integers).is_subset(S.Reals) is None

def test_is_proper_subset():
    assert Interval(0, 1).is_proper_subset(Interval(0, 2)) is True
    assert Interval(0, 3).is_proper_subset(Interval(0, 2)) is False
    assert S.EmptySet.is_proper_subset(FiniteSet(1, 2, 3)) is True

    raises(ValueError, lambda: Interval(0, 1).is_proper_subset(0))


def test_is_superset():
    assert Interval(0, 1).is_superset(Interval(0, 2)) == False
    assert Interval(0, 3).is_superset(Interval(0, 2))

    assert FiniteSet(1, 2).is_superset(FiniteSet(1, 2, 3, 4)) == False
    assert FiniteSet(4, 5).is_superset(FiniteSet(1, 2, 3, 4)) == False
    assert FiniteSet(1).is_superset(Interval(0, 2)) == False
    assert FiniteSet(1, 2).is_superset(Interval(0, 2, True, True)) == False
    assert (Interval(1, 2) + FiniteSet(3)).is_superset(
        Interval(0, 2, False, True) + FiniteSet(2, 3)) == False

    assert Interval(3, 4).is_superset(Union(Interval(0, 1), Interval(2, 5))) == False

    assert FiniteSet(1, 2, 3, 4).is_superset(Interval(0, 5)) == False
    assert S.EmptySet.is_superset(FiniteSet(1, 2, 3)) == False

    assert Interval(0, 1).is_superset(S.EmptySet) == True
    assert S.EmptySet.is_superset(S.EmptySet) == True

    raises(ValueError, lambda: S.EmptySet.is_superset(1))

    # tests for the issuperset alias
    assert Interval(0, 1).issuperset(S.EmptySet) == True
    assert S.EmptySet.issuperset(S.EmptySet) == True


def test_is_proper_superset():
    assert Interval(0, 1).is_proper_superset(Interval(0, 2)) is False
    assert Interval(0, 3).is_proper_superset(Interval(0, 2)) is True
    assert FiniteSet(1, 2, 3).is_proper_superset(S.EmptySet) is True

    raises(ValueError, lambda: Interval(0, 1).is_proper_superset(0))


def test_contains():
    assert Interval(0, 2).contains(1) is S.true
    assert Interval(0, 2).contains(3) is S.false
    assert Interval(0, 2, True, False).contains(0) is S.false
    assert Interval(0, 2, True, False).contains(2) is S.true
    assert Interval(0, 2, False, True).contains(0) is S.true
    assert Interval(0, 2, False, True).contains(2) is S.false
    assert Interval(0, 2, True, True).contains(0) is S.false
    assert Interval(0, 2, True, True).contains(2) is S.false

    assert (Interval(0, 2) in Interval(0, 2)) is False

    assert FiniteSet(1, 2, 3).contains(2) is S.true
    assert FiniteSet(1, 2, Symbol('x')).contains(Symbol('x')) is S.true

    assert FiniteSet(y)._contains(x) == Eq(y, x, evaluate=False)
    raises(TypeError, lambda: x in FiniteSet(y))
    assert FiniteSet({x, y})._contains({x}) == Eq({x, y}, {x}, evaluate=False)
    assert FiniteSet({x, y}).subs(y, x)._contains({x}) is S.true
    assert FiniteSet({x, y}).subs(y, x+1)._contains({x}) is S.false

    # issue 8197
    from sympy.abc import a, b
    assert FiniteSet(b).contains(-a) == Eq(b, -a)
    assert FiniteSet(b).contains(a) == Eq(b, a)
    assert FiniteSet(a).contains(1) == Eq(a, 1)
    raises(TypeError, lambda: 1 in FiniteSet(a))

    # issue 8209
    rad1 = Pow(Pow(2, Rational(1, 3)) - 1, Rational(1, 3))
    rad2 = Pow(Rational(1, 9), Rational(1, 3)) - Pow(Rational(2, 9), Rational(1, 3)) + Pow(Rational(4, 9), Rational(1, 3))
    s1 = FiniteSet(rad1)
    s2 = FiniteSet(rad2)
    assert s1 - s2 == S.EmptySet

    items = [1, 2, S.Infinity, S('ham'), -1.1]
    fset = FiniteSet(*items)
    assert all(item in fset for item in items)
    assert all(fset.contains(item) is S.true for item in items)

    assert Union(Interval(0, 1), Interval(2, 5)).contains(3) is S.true
    assert Union(Interval(0, 1), Interval(2, 5)).contains(6) is S.false
    assert Union(Interval(0, 1), FiniteSet(2, 5)).contains(3) is S.false

    assert S.EmptySet.contains(1) is S.false
    assert FiniteSet(rootof(x**3 + x - 1, 0)).contains(S.Infinity) is S.false

    assert rootof(x**5 + x**3 + 1, 0) in S.Reals
    assert not rootof(x**5 + x**3 + 1, 1) in S.Reals

    # non-bool results
    assert Union(Interval(1, 2), Interval(3, 4)).contains(x) == \
        Or(And(S.One <= x, x <= 2), And(S(3) <= x, x <= 4))
    assert Intersection(Interval(1, x), Interval(2, 3)).contains(y) == \
        And(y <= 3, y <= x, S.One <= y, S(2) <= y)

    assert (S.Complexes).contains(S.ComplexInfinity) == S.false


def test_interval_symbolic():
    x = Symbol('x')
    e = Interval(0, 1)
    assert e.contains(x) == And(S.Zero <= x, x <= 1)
    raises(TypeError, lambda: x in e)
    e = Interval(0, 1, True, True)
    assert e.contains(x) == And(S.Zero < x, x < 1)
    c = Symbol('c', real=False)
    assert Interval(x, x + 1).contains(c) == False
    e = Symbol('e', extended_real=True)
    assert Interval(-oo, oo).contains(e) == And(
        S.NegativeInfinity < e, e < S.Infinity)


def test_union_contains():
    x = Symbol('x')
    i1 = Interval(0, 1)
    i2 = Interval(2, 3)
    i3 = Union(i1, i2)
    assert i3.as_relational(x) == Or(And(S.Zero <= x, x <= 1), And(S(2) <= x, x <= 3))
    raises(TypeError, lambda: x in i3)
    e = i3.contains(x)
    assert e == i3.as_relational(x)
    assert e.subs(x, -0.5) is false
    assert e.subs(x, 0.5) is true
    assert e.subs(x, 1.5) is false
    assert e.subs(x, 2.5) is true
    assert e.subs(x, 3.5) is false

    U = Interval(0, 2, True, True) + Interval(10, oo) + FiniteSet(-1, 2, 5, 6)
    assert all(el not in U for el in [0, 4, -oo])
    assert all(el in U for el in [2, 5, 10])


def test_is_number():
    assert Interval(0, 1).is_number is False
    assert Set().is_number is False


def test_Interval_is_left_unbounded():
    assert Interval(3, 4).is_left_unbounded is False
    assert Interval(-oo, 3).is_left_unbounded is True
    assert Interval(Float("-inf"), 3).is_left_unbounded is True


def test_Interval_is_right_unbounded():
    assert Interval(3, 4).is_right_unbounded is False
    assert Interval(3, oo).is_right_unbounded is True
    assert Interval(3, Float("+inf")).is_right_unbounded is True


def test_Interval_as_relational():
    x = Symbol('x')

    assert Interval(-1, 2, False, False).as_relational(x) == \
        And(Le(-1, x), Le(x, 2))
    assert Interval(-1, 2, True, False).as_relational(x) == \
        And(Lt(-1, x), Le(x, 2))
    assert Interval(-1, 2, False, True).as_relational(x) == \
        And(Le(-1, x), Lt(x, 2))
    assert Interval(-1, 2, True, True).as_relational(x) == \
        And(Lt(-1, x), Lt(x, 2))

    assert Interval(-oo, 2, right_open=False).as_relational(x) == And(Lt(-oo, x), Le(x, 2))
    assert Interval(-oo, 2, right_open=True).as_relational(x) == And(Lt(-oo, x), Lt(x, 2))

    assert Interval(-2, oo, left_open=False).as_relational(x) == And(Le(-2, x), Lt(x, oo))
    assert Interval(-2, oo, left_open=True).as_relational(x) == And(Lt(-2, x), Lt(x, oo))

    assert Interval(-oo, oo).as_relational(x) == And(Lt(-oo, x), Lt(x, oo))
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    assert Interval(x, y).as_relational(x) == (x <= y)
    assert Interval(y, x).as_relational(x) == (y <= x)


def test_Finite_as_relational():
    x = Symbol('x')
    y = Symbol('y')

    assert FiniteSet(1, 2).as_relational(x) == Or(Eq(x, 1), Eq(x, 2))
    assert FiniteSet(y, -5).as_relational(x) == Or(Eq(x, y), Eq(x, -5))


def test_Union_as_relational():
    x = Symbol('x')
    assert (Interval(0, 1) + FiniteSet(2)).as_relational(x) == \
        Or(And(Le(0, x), Le(x, 1)), Eq(x, 2))
    assert (Interval(0, 1, True, True) + FiniteSet(1)).as_relational(x) == \
        And(Lt(0, x), Le(x, 1))
    assert Or(x < 0, x > 0).as_set().as_relational(x) == \
        And((x > -oo), (x < oo), Ne(x, 0))
    assert (Interval.Ropen(1, 3) + Interval.Lopen(3, 5)
        ).as_relational(x) == And(Ne(x,3),(x>=1),(x<=5))


def test_Intersection_as_relational():
    x = Symbol('x')
    assert (Intersection(Interval(0, 1), FiniteSet(2),
            evaluate=False).as_relational(x)
            == And(And(Le(0, x), Le(x, 1)), Eq(x, 2)))


def test_Complement_as_relational():
    x = Symbol('x')
    expr = Complement(Interval(0, 1), FiniteSet(2), evaluate=False)
    assert expr.as_relational(x) == \
        And(Le(0, x), Le(x, 1), Ne(x, 2))


@XFAIL
def test_Complement_as_relational_fail():
    x = Symbol('x')
    expr = Complement(Interval(0, 1), FiniteSet(2), evaluate=False)
    # XXX This example fails because 0 <= x changes to x >= 0
    # during the evaluation.
    assert expr.as_relational(x) == \
            (0 <= x) & (x <= 1) & Ne(x, 2)


def test_SymmetricDifference_as_relational():
    x = Symbol('x')
    expr = SymmetricDifference(Interval(0, 1), FiniteSet(2), evaluate=False)
    assert expr.as_relational(x) == Xor(Eq(x, 2), Le(0, x) & Le(x, 1))


def test_EmptySet():
    assert S.EmptySet.as_relational(Symbol('x')) is S.false
    assert S.EmptySet.intersect(S.UniversalSet) == S.EmptySet
    assert S.EmptySet.boundary == S.EmptySet


def test_finite_basic():
    x = Symbol('x')
    A = FiniteSet(1, 2, 3)
    B = FiniteSet(3, 4, 5)
    AorB = Union(A, B)
    AandB = A.intersect(B)
    assert A.is_subset(AorB) and B.is_subset(AorB)
    assert AandB.is_subset(A)
    assert AandB == FiniteSet(3)

    assert A.inf == 1 and A.sup == 3
    assert AorB.inf == 1 and AorB.sup == 5
    assert FiniteSet(x, 1, 5).sup == Max(x, 5)
    assert FiniteSet(x, 1, 5).inf == Min(x, 1)

    # issue 7335
    assert FiniteSet(S.EmptySet) != S.EmptySet
    assert FiniteSet(FiniteSet(1, 2, 3)) != FiniteSet(1, 2, 3)
    assert FiniteSet((1, 2, 3)) != FiniteSet(1, 2, 3)

    # Ensure a variety of types can exist in a FiniteSet
    assert FiniteSet((1, 2), A, -5, x, 'eggs', x**2)

    assert (A > B) is False
    assert (A >= B) is False
    assert (A < B) is False
    assert (A <= B) is False
    assert AorB > A and AorB > B
    assert AorB >= A and AorB >= B
    assert A >= A and A <= A
    assert A >= AandB and B >= AandB
    assert A > AandB and B > AandB


def test_product_basic():
    H, T = 'H', 'T'
    unit_line = Interval(0, 1)
    d6 = FiniteSet(1, 2, 3, 4, 5, 6)
    d4 = FiniteSet(1, 2, 3, 4)
    coin = FiniteSet(H, T)

    square = unit_line * unit_line

    assert (0, 0) in square
    assert 0 not in square
    assert (H, T) in coin ** 2
    assert (.5, .5, .5) in (square * unit_line).flatten()
    assert ((.5, .5), .5) in square * unit_line
    assert (H, 3, 3) in (coin * d6 * d6).flatten()
    assert ((H, 3), 3) in coin * d6 * d6
    HH, TT = sympify(H), sympify(T)
    assert set(coin**2) == {(HH, HH), (HH, TT), (TT, HH), (TT, TT)}

    assert (d4*d4).is_subset(d6*d6)

    assert square.complement(Interval(-oo, oo)*Interval(-oo, oo)) == Union(
        (Interval(-oo, 0, True, True) +
         Interval(1, oo, True, True))*Interval(-oo, oo),
         Interval(-oo, oo)*(Interval(-oo, 0, True, True) +
                  Interval(1, oo, True, True)))

    assert (Interval(-5, 5)**3).is_subset(Interval(-10, 10)**3)
    assert not (Interval(-10, 10)**3).is_subset(Interval(-5, 5)**3)
    assert not (Interval(-5, 5)**2).is_subset(Interval(-10, 10)**3)

    assert (Interval(.2, .5)*FiniteSet(.5)).is_subset(square)  # segment in square

    assert len(coin*coin*coin) == 8
    assert len(S.EmptySet*S.EmptySet) == 0
    assert len(S.EmptySet*coin) == 0
    raises(TypeError, lambda: len(coin*Interval(0, 2)))


def test_real():
    x = Symbol('x', real=True)

    I = Interval(0, 5)
    J = Interval(10, 20)
    A = FiniteSet(1, 2, 30, x, S.Pi)
    B = FiniteSet(-4, 0)
    C = FiniteSet(100)
    D = FiniteSet('Ham', 'Eggs')

    assert all(s.is_subset(S.Reals) for s in [I, J, A, B, C])
    assert not D.is_subset(S.Reals)
    assert all((a + b).is_subset(S.Reals) for a in [I, J, A, B, C] for b in [I, J, A, B, C])
    assert not any((a + D).is_subset(S.Reals) for a in [I, J, A, B, C, D])

    assert not (I + A + D).is_subset(S.Reals)


def test_supinf():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)

    assert (Interval(0, 1) + FiniteSet(2)).sup == 2
    assert (Interval(0, 1) + FiniteSet(2)).inf == 0
    assert (Interval(0, 1) + FiniteSet(x)).sup == Max(1, x)
    assert (Interval(0, 1) + FiniteSet(x)).inf == Min(0, x)
    assert FiniteSet(5, 1, x).sup == Max(5, x)
    assert FiniteSet(5, 1, x).inf == Min(1, x)
    assert FiniteSet(5, 1, x, y).sup == Max(5, x, y)
    assert FiniteSet(5, 1, x, y).inf == Min(1, x, y)
    assert FiniteSet(5, 1, x, y, S.Infinity, S.NegativeInfinity).sup == \
        S.Infinity
    assert FiniteSet(5, 1, x, y, S.Infinity, S.NegativeInfinity).inf == \
        S.NegativeInfinity
    assert FiniteSet('Ham', 'Eggs').sup == Max('Ham', 'Eggs')


def test_universalset():
    U = S.UniversalSet
    x = Symbol('x')
    assert U.as_relational(x) is S.true
    assert U.union(Interval(2, 4)) == U

    assert U.intersect(Interval(2, 4)) == Interval(2, 4)
    assert U.measure is S.Infinity
    assert U.boundary == S.EmptySet
    assert U.contains(0) is S.true


def test_Union_of_ProductSets_shares():
    line = Interval(0, 2)
    points = FiniteSet(0, 1, 2)
    assert Union(line * line, line * points) == line * line


def test_Interval_free_symbols():
    # issue 6211
    assert Interval(0, 1).free_symbols == set()
    x = Symbol('x', real=True)
    assert Interval(0, x).free_symbols == {x}


def test_image_interval():
    x = Symbol('x', real=True)
    a = Symbol('a', real=True)
    assert imageset(x, 2*x, Interval(-2, 1)) == Interval(-4, 2)
    assert imageset(x, 2*x, Interval(-2, 1, True, False)) == \
        Interval(-4, 2, True, False)
    assert imageset(x, x**2, Interval(-2, 1, True, False)) == \
        Interval(0, 4, False, True)
    assert imageset(x, x**2, Interval(-2, 1)) == Interval(0, 4)
    assert imageset(x, x**2, Interval(-2, 1, True, False)) == \
        Interval(0, 4, False, True)
    assert imageset(x, x**2, Interval(-2, 1, True, True)) == \
        Interval(0, 4, False, True)
    assert imageset(x, (x - 2)**2, Interval(1, 3)) == Interval(0, 1)
    assert imageset(x, 3*x**4 - 26*x**3 + 78*x**2 - 90*x, Interval(0, 4)) == \
        Interval(-35, 0)  # Multiple Maxima
    assert imageset(x, x + 1/x, Interval(-oo, oo)) == Interval(-oo, -2) \
        + Interval(2, oo)  # Single Infinite discontinuity
    assert imageset(x, 1/x + 1/(x-1)**2, Interval(0, 2, True, False)) == \
        Interval(Rational(3, 2), oo, False)  # Multiple Infinite discontinuities

    # Test for Python lambda
    assert imageset(lambda x: 2*x, Interval(-2, 1)) == Interval(-4, 2)

    assert imageset(Lambda(x, a*x), Interval(0, 1)) == \
            ImageSet(Lambda(x, a*x), Interval(0, 1))

    assert imageset(Lambda(x, sin(cos(x))), Interval(0, 1)) == \
            ImageSet(Lambda(x, sin(cos(x))), Interval(0, 1))


def test_image_piecewise():
    f = Piecewise((x, x <= -1), (1/x**2, x <= 5), (x**3, True))
    f1 = Piecewise((0, x <= 1), (1, x <= 2), (2, True))
    assert imageset(x, f, Interval(-5, 5)) == Union(Interval(-5, -1), Interval(Rational(1, 25), oo))
    assert imageset(x, f1, Interval(1, 2)) == FiniteSet(0, 1)


@XFAIL  # See: https://github.com/sympy/sympy/pull/2723#discussion_r8659826
def test_image_Intersection():
    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    assert imageset(x, x**2, Interval(-2, 0).intersect(Interval(x, y))) == \
           Interval(0, 4).intersect(Interval(Min(x**2, y**2), Max(x**2, y**2)))


def test_image_FiniteSet():
    x = Symbol('x', real=True)
    assert imageset(x, 2*x, FiniteSet(1, 2, 3)) == FiniteSet(2, 4, 6)


def test_image_Union():
    x = Symbol('x', real=True)
    assert imageset(x, x**2, Interval(-2, 0) + FiniteSet(1, 2, 3)) == \
            (Interval(0, 4) + FiniteSet(9))


def test_image_EmptySet():
    x = Symbol('x', real=True)
    assert imageset(x, 2*x, S.EmptySet) == S.EmptySet


def test_issue_5724_7680():
    assert I not in S.Reals  # issue 7680
    assert Interval(-oo, oo).contains(I) is S.false


def test_boundary():
    assert FiniteSet(1).boundary == FiniteSet(1)
    assert all(Interval(0, 1, left_open, right_open).boundary == FiniteSet(0, 1)
            for left_open in (true, false) for right_open in (true, false))


def test_boundary_Union():
    assert (Interval(0, 1) + Interval(2, 3)).boundary == FiniteSet(0, 1, 2, 3)
    assert ((Interval(0, 1, False, True)
           + Interval(1, 2, True, False)).boundary == FiniteSet(0, 1, 2))

    assert (Interval(0, 1) + FiniteSet(2)).boundary == FiniteSet(0, 1, 2)
    assert Union(Interval(0, 10), Interval(5, 15), evaluate=False).boundary \
            == FiniteSet(0, 15)

    assert Union(Interval(0, 10), Interval(0, 1), evaluate=False).boundary \
            == FiniteSet(0, 10)
    assert Union(Interval(0, 10, True, True),
                 Interval(10, 15, True, True), evaluate=False).boundary \
            == FiniteSet(0, 10, 15)


@XFAIL
def test_union_boundary_of_joining_sets():
    """ Testing the boundary of unions is a hard problem """
    assert Union(Interval(0, 10), Interval(10, 15), evaluate=False).boundary \
            == FiniteSet(0, 15)


def test_boundary_ProductSet():
    open_square = Interval(0, 1, True, True) ** 2
    assert open_square.boundary == (FiniteSet(0, 1) * Interval(0, 1)
                                  + Interval(0, 1) * FiniteSet(0, 1))

    second_square = Interval(1, 2, True, True) * Interval(0, 1, True, True)
    assert (open_square + second_square).boundary == (
                FiniteSet(0, 1) * Interval(0, 1)
              + FiniteSet(1, 2) * Interval(0, 1)
              + Interval(0, 1) * FiniteSet(0, 1)
              + Interval(1, 2) * FiniteSet(0, 1))


def test_boundary_ProductSet_line():
    line_in_r2 = Interval(0, 1) * FiniteSet(0)
    assert line_in_r2.boundary == line_in_r2


def test_is_open():
    assert Interval(0, 1, False, False).is_open is False
    assert Interval(0, 1, True, False).is_open is False
    assert Interval(0, 1, True, True).is_open is True
    assert FiniteSet(1, 2, 3).is_open is False


def test_is_closed():
    assert Interval(0, 1, False, False).is_closed is True
    assert Interval(0, 1, True, False).is_closed is False
    assert FiniteSet(1, 2, 3).is_closed is True


def test_closure():
    assert Interval(0, 1, False, True).closure == Interval(0, 1, False, False)


def test_interior():
    assert Interval(0, 1, False, True).interior == Interval(0, 1, True, True)


def test_issue_7841():
    raises(TypeError, lambda: x in S.Reals)


def test_Eq():
    assert Eq(Interval(0, 1), Interval(0, 1))
    assert Eq(Interval(0, 1), Interval(0, 2)) == False

    s1 = FiniteSet(0, 1)
    s2 = FiniteSet(1, 2)

    assert Eq(s1, s1)
    assert Eq(s1, s2) == False

    assert Eq(s1*s2, s1*s2)
    assert Eq(s1*s2, s2*s1) == False

    assert unchanged(Eq, FiniteSet({x, y}), FiniteSet({x}))
    assert Eq(FiniteSet({x, y}).subs(y, x), FiniteSet({x})) is S.true
    assert Eq(FiniteSet({x, y}), FiniteSet({x})).subs(y, x) is S.true
    assert Eq(FiniteSet({x, y}).subs(y, x+1), FiniteSet({x})) is S.false
    assert Eq(FiniteSet({x, y}), FiniteSet({x})).subs(y, x+1) is S.false

    assert Eq(ProductSet({1}, {2}), Interval(1, 2)) is S.false
    assert Eq(ProductSet({1}), ProductSet({1}, {2})) is S.false

    assert Eq(FiniteSet(()), FiniteSet(1)) is S.false
    assert Eq(ProductSet(), FiniteSet(1)) is S.false

    i1 = Interval(0, 1)
    i2 = Interval(x, y)
    assert unchanged(Eq, ProductSet(i1, i1), ProductSet(i2, i2))


def test_SymmetricDifference():
    A = FiniteSet(0, 1, 2, 3, 4, 5)
    B = FiniteSet(2, 4, 6, 8, 10)
    C = Interval(8, 10)

    assert SymmetricDifference(A, B, evaluate=False).is_iterable is True
    assert SymmetricDifference(A, C, evaluate=False).is_iterable is None
    assert FiniteSet(*SymmetricDifference(A, B, evaluate=False)) == \
        FiniteSet(0, 1, 3, 5, 6, 8, 10)
    raises(TypeError,
        lambda: FiniteSet(*SymmetricDifference(A, C, evaluate=False)))

    assert SymmetricDifference(FiniteSet(0, 1, 2, 3, 4, 5), \
            FiniteSet(2, 4, 6, 8, 10)) == FiniteSet(0, 1, 3, 5, 6, 8, 10)
    assert SymmetricDifference(FiniteSet(2, 3, 4), FiniteSet(2, 3, 4 ,5)) \
            == FiniteSet(5)
    assert FiniteSet(1, 2, 3, 4, 5) ^ FiniteSet(1, 2, 5, 6) == \
            FiniteSet(3, 4, 6)
    assert Set(S(1), S(2), S(3)) ^ Set(S(2), S(3), S(4)) == Union(Set(S(1), S(2), S(3)) - Set(S(2), S(3), S(4)), \
            Set(S(2), S(3), S(4)) - Set(S(1), S(2), S(3)))
    assert Interval(0, 4) ^ Interval(2, 5) == Union(Interval(0, 4) - \
            Interval(2, 5), Interval(2, 5) - Interval(0, 4))


def test_issue_9536():
    from sympy.functions.elementary.exponential import log
    a = Symbol('a', real=True)
    assert FiniteSet(log(a)).intersect(S.Reals) == Intersection(S.Reals, FiniteSet(log(a)))


def test_issue_9637():
    n = Symbol('n')
    a = FiniteSet(n)
    b = FiniteSet(2, n)
    assert Complement(S.Reals, a) == Complement(S.Reals, a, evaluate=False)
    assert Complement(Interval(1, 3), a) == Complement(Interval(1, 3), a, evaluate=False)
    assert Complement(Interval(1, 3), b) == \
        Complement(Union(Interval(1, 2, False, True), Interval(2, 3, True, False)), a)
    assert Complement(a, S.Reals) == Complement(a, S.Reals, evaluate=False)
    assert Complement(a, Interval(1, 3)) == Complement(a, Interval(1, 3), evaluate=False)


def test_issue_9808():
    # See https://github.com/sympy/sympy/issues/16342
    assert Complement(FiniteSet(y), FiniteSet(1)) == Complement(FiniteSet(y), FiniteSet(1), evaluate=False)
    assert Complement(FiniteSet(1, 2, x), FiniteSet(x, y, 2, 3)) == \
        Complement(FiniteSet(1), FiniteSet(y), evaluate=False)


def test_issue_9956():
    assert Union(Interval(-oo, oo), FiniteSet(1)) == Interval(-oo, oo)
    assert Interval(-oo, oo).contains(1) is S.true


def test_issue_Symbol_inter():
    i = Interval(0, oo)
    r = S.Reals
    mat = Matrix([0, 0, 0])
    assert Intersection(r, i, FiniteSet(m), FiniteSet(m, n)) == \
        Intersection(i, FiniteSet(m))
    assert Intersection(FiniteSet(1, m, n), FiniteSet(m, n, 2), i) == \
        Intersection(i, FiniteSet(m, n))
    assert Intersection(FiniteSet(m, n, x), FiniteSet(m, z), r) == \
        Intersection(Intersection({m, z}, {m, n, x}), r)
    assert Intersection(FiniteSet(m, n, 3), FiniteSet(m, n, x), r) == \
        Intersection(FiniteSet(3, m, n), FiniteSet(m, n, x), r, evaluate=False)
    assert Intersection(FiniteSet(m, n, 3), FiniteSet(m, n, 2, 3), r) == \
        Intersection(FiniteSet(3, m, n), r)
    assert Intersection(r, FiniteSet(mat, 2, n), FiniteSet(0, mat, n)) == \
        Intersection(r, FiniteSet(n))
    assert Intersection(FiniteSet(sin(x), cos(x)), FiniteSet(sin(x), cos(x), 1), r) == \
        Intersection(r, FiniteSet(sin(x), cos(x)))
    assert Intersection(FiniteSet(x**2, 1, sin(x)), FiniteSet(x**2, 2, sin(x)), r) == \
        Intersection(r, FiniteSet(x**2, sin(x)))


def test_issue_11827():
    assert S.Naturals0**4


def test_issue_10113():
    f = x**2/(x**2 - 4)
    assert imageset(x, f, S.Reals) == Union(Interval(-oo, 0), Interval(1, oo, True, True))
    assert imageset(x, f, Interval(-2, 2)) == Interval(-oo, 0)
    assert imageset(x, f, Interval(-2, 3)) == Union(Interval(-oo, 0), Interval(Rational(9, 5), oo))


def test_issue_10248():
    raises(
        TypeError, lambda: list(Intersection(S.Reals, FiniteSet(x)))
    )
    A = Symbol('A', real=True)
    assert list(Intersection(S.Reals, FiniteSet(A))) == [A]


def test_issue_9447():
    a = Interval(0, 1) + Interval(2, 3)
    assert Complement(S.UniversalSet, a) == Complement(
            S.UniversalSet, Union(Interval(0, 1), Interval(2, 3)), evaluate=False)
    assert Complement(S.Naturals, a) == Complement(
            S.Naturals, Union(Interval(0, 1), Interval(2, 3)), evaluate=False)


def test_issue_10337():
    assert (FiniteSet(2) == 3) is False
    assert (FiniteSet(2) != 3) is True
    raises(TypeError, lambda: FiniteSet(2) < 3)
    raises(TypeError, lambda: FiniteSet(2) <= 3)
    raises(TypeError, lambda: FiniteSet(2) > 3)
    raises(TypeError, lambda: FiniteSet(2) >= 3)


def test_issue_10326():
    bad = [
        EmptySet,
        FiniteSet(1),
        Interval(1, 2),
        S.ComplexInfinity,
        S.ImaginaryUnit,
        S.Infinity,
        S.NaN,
        S.NegativeInfinity,
        ]
    interval = Interval(0, 5)
    for i in bad:
        assert i not in interval

    x = Symbol('x', real=True)
    nr = Symbol('nr', extended_real=False)
    assert x + 1 in Interval(x, x + 4)
    assert nr not in Interval(x, x + 4)
    assert Interval(1, 2) in FiniteSet(Interval(0, 5), Interval(1, 2))
    assert Interval(-oo, oo).contains(oo) is S.false
    assert Interval(-oo, oo).contains(-oo) is S.false


def test_issue_2799():
    U = S.UniversalSet
    a = Symbol('a', real=True)
    inf_interval = Interval(a, oo)
    R = S.Reals

    assert U + inf_interval == inf_interval + U
    assert U + R == R + U
    assert R + inf_interval == inf_interval + R


def test_issue_9706():
    assert Interval(-oo, 0).closure == Interval(-oo, 0, True, False)
    assert Interval(0, oo).closure == Interval(0, oo, False, True)
    assert Interval(-oo, oo).closure == Interval(-oo, oo)


def test_issue_8257():
    reals_plus_infinity = Union(Interval(-oo, oo), FiniteSet(oo))
    reals_plus_negativeinfinity = Union(Interval(-oo, oo), FiniteSet(-oo))
    assert Interval(-oo, oo) + FiniteSet(oo) == reals_plus_infinity
    assert FiniteSet(oo) + Interval(-oo, oo) == reals_plus_infinity
    assert Interval(-oo, oo) + FiniteSet(-oo) == reals_plus_negativeinfinity
    assert FiniteSet(-oo) + Interval(-oo, oo) == reals_plus_negativeinfinity


def test_issue_10931():
    assert S.Integers - S.Integers == EmptySet
    assert S.Integers - S.Reals == EmptySet


def test_issue_11174():
    soln = Intersection(Interval(-oo, oo), FiniteSet(-x), evaluate=False)
    assert Intersection(FiniteSet(-x), S.Reals) == soln

    soln = Intersection(S.Reals, FiniteSet(x), evaluate=False)
    assert Intersection(FiniteSet(x), S.Reals) == soln


def test_issue_18505():
    assert ImageSet(Lambda(n, sqrt(pi*n/2 - 1 + pi/2)), S.Integers).contains(0) == \
            Contains(0, ImageSet(Lambda(n, sqrt(pi*n/2 - 1 + pi/2)), S.Integers))


def test_finite_set_intersection():
    # The following should not produce recursion errors
    # Note: some of these are not completely correct. See
    # https://github.com/sympy/sympy/issues/16342.
    assert Intersection(FiniteSet(-oo, x), FiniteSet(x)) == FiniteSet(x)
    assert Intersection._handle_finite_sets([FiniteSet(-oo, x), FiniteSet(0, x)]) == FiniteSet(x)

    assert Intersection._handle_finite_sets([FiniteSet(-oo, x), FiniteSet(x)]) == FiniteSet(x)
    assert Intersection._handle_finite_sets([FiniteSet(2, 3, x, y), FiniteSet(1, 2, x)]) == \
        Intersection._handle_finite_sets([FiniteSet(1, 2, x), FiniteSet(2, 3, x, y)]) == \
        Intersection(FiniteSet(1, 2, x), FiniteSet(2, 3, x, y)) == \
        Intersection(FiniteSet(1, 2, x), FiniteSet(2, x, y))

    assert FiniteSet(1+x-y) & FiniteSet(1) == \
        FiniteSet(1) & FiniteSet(1+x-y) == \
        Intersection(FiniteSet(1+x-y), FiniteSet(1), evaluate=False)

    assert FiniteSet(1) & FiniteSet(x) == FiniteSet(x) & FiniteSet(1) == \
        Intersection(FiniteSet(1), FiniteSet(x), evaluate=False)

    assert FiniteSet({x}) & FiniteSet({x, y}) == \
        Intersection(FiniteSet({x}), FiniteSet({x, y}), evaluate=False)


def test_union_intersection_constructor():
    # The actual exception does not matter here, so long as these fail
    sets = [FiniteSet(1), FiniteSet(2)]
    raises(Exception, lambda: Union(sets))
    raises(Exception, lambda: Intersection(sets))
    raises(Exception, lambda: Union(tuple(sets)))
    raises(Exception, lambda: Intersection(tuple(sets)))
    raises(Exception, lambda: Union(i for i in sets))
    raises(Exception, lambda: Intersection(i for i in sets))

    # Python sets are treated the same as FiniteSet
    # The union of a single set (of sets) is the set (of sets) itself
    assert Union(set(sets)) == FiniteSet(*sets)
    assert Intersection(set(sets)) == FiniteSet(*sets)

    assert Union({1}, {2}) == FiniteSet(1, 2)
    assert Intersection({1, 2}, {2, 3}) == FiniteSet(2)


def test_Union_contains():
    assert zoo not in Union(
        Interval.open(-oo, 0), Interval.open(0, oo))


@XFAIL
def test_issue_16878b():
    # in intersection_sets for (ImageSet, Set) there is no code
    # that handles the base_set of S.Reals like there is
    # for Integers
    assert imageset(x, (x, x), S.Reals).is_subset(S.Reals**2) is True

def test_DisjointUnion():
    assert DisjointUnion(FiniteSet(1, 2, 3), FiniteSet(1, 2, 3), FiniteSet(1, 2, 3)).rewrite(Union) == (FiniteSet(1, 2, 3) * FiniteSet(0, 1, 2))
    assert DisjointUnion(Interval(1, 3), Interval(2, 4)).rewrite(Union) == Union(Interval(1, 3) * FiniteSet(0), Interval(2, 4) * FiniteSet(1))
    assert DisjointUnion(Interval(0, 5), Interval(0, 5)).rewrite(Union) == Union(Interval(0, 5) * FiniteSet(0), Interval(0, 5) * FiniteSet(1))
    assert DisjointUnion(Interval(-1, 2), S.EmptySet, S.EmptySet).rewrite(Union) == Interval(-1, 2) * FiniteSet(0)
    assert DisjointUnion(Interval(-1, 2)).rewrite(Union) == Interval(-1, 2) * FiniteSet(0)
    assert DisjointUnion(S.EmptySet, Interval(-1, 2), S.EmptySet).rewrite(Union) == Interval(-1, 2) * FiniteSet(1)
    assert DisjointUnion(Interval(-oo, oo)).rewrite(Union) == Interval(-oo, oo) * FiniteSet(0)
    assert DisjointUnion(S.EmptySet).rewrite(Union) == S.EmptySet
    assert DisjointUnion().rewrite(Union) == S.EmptySet
    raises(TypeError, lambda: DisjointUnion(Symbol('n')))

    x = Symbol("x")
    y = Symbol("y")
    z = Symbol("z")
    assert DisjointUnion(FiniteSet(x), FiniteSet(y, z)).rewrite(Union) == (FiniteSet(x) * FiniteSet(0)) + (FiniteSet(y, z) * FiniteSet(1))

def test_DisjointUnion_is_empty():
    assert DisjointUnion(S.EmptySet).is_empty is True
    assert DisjointUnion(S.EmptySet, S.EmptySet).is_empty is True
    assert DisjointUnion(S.EmptySet, FiniteSet(1, 2, 3)).is_empty is False

def test_DisjointUnion_is_iterable():
    assert DisjointUnion(S.Integers, S.Naturals, S.Rationals).is_iterable is True
    assert DisjointUnion(S.EmptySet, S.Reals).is_iterable is False
    assert DisjointUnion(FiniteSet(1, 2, 3), S.EmptySet, FiniteSet(x, y)).is_iterable is True
    assert DisjointUnion(S.EmptySet, S.EmptySet).is_iterable is False

def test_DisjointUnion_contains():
    assert (0, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (0, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (0, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (1, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (1, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (1, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (2, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (2, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (2, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (0, 1, 2) not in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (0, 0.5) not in DisjointUnion(FiniteSet(0.5))
    assert (0, 5) not in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
    assert (x, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
    assert (y, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
    assert (z, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
    assert (y, 2) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
    assert (0.5, 0) in DisjointUnion(Interval(0, 1), Interval(0, 2))
    assert (0.5, 1) in DisjointUnion(Interval(0, 1), Interval(0, 2))
    assert (1.5, 0) not in DisjointUnion(Interval(0, 1), Interval(0, 2))
    assert (1.5, 1) in DisjointUnion(Interval(0, 1), Interval(0, 2))

def test_DisjointUnion_iter():
    D = DisjointUnion(FiniteSet(3, 5, 7, 9), FiniteSet(x, y, z))
    it = iter(D)
    L1 = [(x, 1), (y, 1), (z, 1)]
    L2 = [(3, 0), (5, 0), (7, 0), (9, 0)]
    nxt = next(it)
    assert nxt in L2
    L2.remove(nxt)
    nxt = next(it)
    assert nxt in L1
    L1.remove(nxt)
    nxt = next(it)
    assert nxt in L2
    L2.remove(nxt)
    nxt = next(it)
    assert nxt in L1
    L1.remove(nxt)
    nxt = next(it)
    assert nxt in L2
    L2.remove(nxt)
    nxt = next(it)
    assert nxt in L1
    L1.remove(nxt)
    nxt = next(it)
    assert nxt in L2
    L2.remove(nxt)
    raises(StopIteration, lambda: next(it))

    raises(ValueError, lambda: iter(DisjointUnion(Interval(0, 1), S.EmptySet)))

def test_DisjointUnion_len():
    assert len(DisjointUnion(FiniteSet(3, 5, 7, 9), FiniteSet(x, y, z))) == 7
    assert len(DisjointUnion(S.EmptySet, S.EmptySet, FiniteSet(x, y, z), S.EmptySet)) == 3
    raises(ValueError, lambda: len(DisjointUnion(Interval(0, 1), S.EmptySet)))

def test_SetKind_ProductSet():
    p = ProductSet(FiniteSet(Matrix([1, 2])), FiniteSet(Matrix([1, 2])))
    mk = MatrixKind(NumberKind)
    k = SetKind(TupleKind(mk, mk))
    assert p.kind is k
    assert ProductSet(Interval(1, 2), FiniteSet(Matrix([1, 2]))).kind is SetKind(TupleKind(NumberKind, mk))

def test_SetKind_Interval():
    assert Interval(1, 2).kind is SetKind(NumberKind)

def test_SetKind_EmptySet_UniversalSet():
    assert S.UniversalSet.kind is SetKind(UndefinedKind)
    assert EmptySet.kind is SetKind()

def test_SetKind_FiniteSet():
    assert FiniteSet(1, Matrix([1, 2])).kind is SetKind(UndefinedKind)
    assert FiniteSet(1, 2).kind is SetKind(NumberKind)

def test_SetKind_Unions():
    assert Union(FiniteSet(Matrix([1, 2])), Interval(1, 2)).kind is SetKind(UndefinedKind)
    assert Union(Interval(1, 2), Interval(1, 7)).kind is SetKind(NumberKind)

def test_SetKind_DisjointUnion():
    A = FiniteSet(1, 2, 3)
    B = Interval(0, 5)
    assert DisjointUnion(A, B).kind is SetKind(NumberKind)

def test_SetKind_evaluate_False():
    U = lambda *args: Union(*args, evaluate=False)
    assert U({1}, EmptySet).kind is SetKind(NumberKind)
    assert U(Interval(1, 2), EmptySet).kind is SetKind(NumberKind)
    assert U({1}, S.UniversalSet).kind is SetKind(UndefinedKind)
    assert U(Interval(1, 2), Interval(4, 5),
            FiniteSet(1)).kind is SetKind(NumberKind)
    I = lambda *args: Intersection(*args, evaluate=False)
    assert I({1}, S.UniversalSet).kind is SetKind(NumberKind)
    assert I({1}, EmptySet).kind is SetKind()
    C = lambda *args: Complement(*args, evaluate=False)
    assert C(S.UniversalSet, {1, 2, 4, 5}).kind is SetKind(UndefinedKind)
    assert C({1, 2, 3, 4, 5}, EmptySet).kind is SetKind(NumberKind)
    assert C(EmptySet, {1, 2, 3, 4, 5}).kind is SetKind()

def test_SetKind_ImageSet_Special():
    f = ImageSet(Lambda(n, n ** 2), Interval(1, 4))
    assert (f - FiniteSet(3)).kind is SetKind(NumberKind)
    assert (f + Interval(16, 17)).kind is SetKind(NumberKind)
    assert (f + FiniteSet(17)).kind is SetKind(NumberKind)

def test_issue_20089():
    B = FiniteSet(FiniteSet(1, 2), FiniteSet(1))
    assert 1 not in B
    assert 1.0 not in B
    assert not Eq(1, FiniteSet(1, 2))
    assert FiniteSet(1) in B
    A = FiniteSet(1, 2)
    assert A in B
    assert B.issubset(B)
    assert not A.issubset(B)
    assert 1 in A
    C = FiniteSet(FiniteSet(1, 2), FiniteSet(1), 1, 2)
    assert A.issubset(C)
    assert B.issubset(C)

def test_issue_19378():
    a = FiniteSet(1, 2)
    b = ProductSet(a, a)
    c = FiniteSet((1, 1), (1, 2), (2, 1), (2, 2))
    assert b.is_subset(c) is True
    d = FiniteSet(1)
    assert b.is_subset(d) is False
    assert Eq(c, b).simplify() is S.true
    assert Eq(a, c).simplify() is S.false
    assert Eq({1}, {x}).simplify() == Eq({1}, {x})

def test_intersection_symbolic():
    n = Symbol('n')
    # These should not throw an error
    assert isinstance(Intersection(Range(n), Range(100)), Intersection)
    assert isinstance(Intersection(Range(n), Interval(1, 100)), Intersection)
    assert isinstance(Intersection(Range(100), Interval(1, n)), Intersection)


@XFAIL
def test_intersection_symbolic_failing():
    n = Symbol('n', integer=True, positive=True)
    assert Intersection(Range(10, n), Range(4, 500, 5)) == Intersection(
        Range(14, n), Range(14, 500, 5))
    assert Intersection(Interval(10, n), Range(4, 500, 5)) == Intersection(
        Interval(14, n), Range(14, 500, 5))


def test_issue_20379():
    #https://github.com/sympy/sympy/issues/20379
    x = pi - 3.14159265358979
    assert FiniteSet(x).evalf(2) == FiniteSet(Float('3.23108914886517e-15', 2))

def test_finiteset_simplify():
    S = FiniteSet(1, cos(1)**2 + sin(1)**2)
    assert S.simplify() == {1}

def test_issue_14336():
    #https://github.com/sympy/sympy/issues/14336
    U = S.Complexes
    x = Symbol("x")
    U -= U.intersect(Ne(x, 1).as_set())
    U -= U.intersect(S.true.as_set())

def test_issue_9855():
    #https://github.com/sympy/sympy/issues/9855
    x, y, z = symbols('x, y, z', real=True)
    s1 = Interval(1, x) & Interval(y, 2)
    s2 = Interval(1, 2)
    assert s1.is_subset(s2) == None