def test_reduce_rational_inequalities_real_relational():
assert reduce_rational_inequalities([], x) is S.false
assert reduce_rational_inequalities(
[[(x**2 + 3*x + 2)/(x**2 - 16) >= 0]], x, relational=False) == \
Union(Interval.open(-oo, -4), Interval(-2, -1), Interval.open(4, oo))
assert reduce_rational_inequalities(
[[((-2*x - 10)*(3 - x))/((x**2 + 5)*(x - 2)**2) < 0]], x,
relational=False) == \
Union(Interval.open(-5, 2), Interval.open(2, 3))
assert reduce_rational_inequalities([[(x + 1)/(x - 5) <= 0]], x,
relational=False) == \
Interval.Ropen(-1, 5)
assert reduce_rational_inequalities([[(x**2 + 4*x + 3)/(x - 1) > 0]], x,
relational=False) == \
Union(Interval.open(-3, -1), Interval.open(1, oo))
assert reduce_rational_inequalities([[(x**2 - 16)/(x - 1)**2 < 0]], x,
relational=False) == \
Union(Interval.open(-4, 1), Interval.open(1, 4))
assert reduce_rational_inequalities([[(3*x + 1)/(x + 4) >= 1]], x,
relational=False) == \
Union(Interval.open(-oo, -4), Interval.Ropen(Rational(3, 2), oo))
assert reduce_rational_inequalities([[(x - 8)/x <= 3 - x]], x,
relational=False) == \
Union(Interval.Lopen(-oo, -2), Interval.Lopen(0, 4))
def test_intersection():
pytest.raises(TypeError, lambda: Intersection(1))
assert Intersection() == S.UniversalSet
assert Interval(0, 2).intersection(Interval(1, 2)) == Interval(1, 2)
assert Interval(0, 2).intersection(Interval(1, 2, True)) == \
Interval(1, 2, True)
assert Interval(0, 2, True).intersection(Interval(1, 2)) == \
Interval(1, 2, False, False)
assert Interval(0, 2, True, True).intersection(Interval(1, 2)) == \
Interval(1, 2, False, True)
assert Interval(0, 2).intersection(Union(Interval(0, 1), Interval(2, 3))) == \
Union(Interval(0, 1), Interval(2, 2))
x = Symbol('x')
assert Intersection(Interval(0, 1), FiniteSet(x)).is_iterable
assert not Intersection(Interval(0, 1), Interval(0, x)).is_iterable
assert FiniteSet(1, 2, x).intersection(FiniteSet(x)) == FiniteSet(x)
assert FiniteSet('ham', 'eggs').intersection(FiniteSet('ham')) == \
FiniteSet('ham')
assert FiniteSet(1, 2, 3, 4, 5).intersection(S.EmptySet) == S.EmptySet
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))
assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(0, 2)) == \
Union(Interval(0, 1), Interval(2, 2))
assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(1, 2, True, True)) == \
S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).intersection(S.EmptySet) == \
S.EmptySet
assert Union(Interval(0, 5), FiniteSet('ham')).intersection(FiniteSet(2, 3, 4, 5, 6)) == \
Union(FiniteSet(2, 3, 4, 5), Intersection(FiniteSet(6), Union(Interval(0, 5), FiniteSet('ham'))))
# issue sympy/sympy#8217
assert Intersection(FiniteSet(x), FiniteSet(y)) == \
Intersection(FiniteSet(x), FiniteSet(y), evaluate=False)
assert FiniteSet(x).intersection(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))
assert ProductSet(FiniteSet(1),
FiniteSet(2)).intersection(Interval(1,
2)).is_Intersection
# iterable
i = Intersection(FiniteSet(1, 2, 3), Interval(2, 5), evaluate=False)
assert i.is_iterable
assert set(i) == {Integer(2), Integer(3)}
# challenging intervals
x = Symbol('x', real=True)
i = Intersection(Interval(0, 3), Interval(x, 6))
assert (5 in i) is False
pytest.raises(TypeError, lambda: 2 in i)
# Singleton special cases
assert Intersection(Interval(0, 1), S.EmptySet) == S.EmptySet
assert Intersection(S.Reals, Interval(-oo, x,
True)) == Interval(-oo, x, True)
assert Intersection(S.Reals, Interval(-oo, x)) == Interval.Lopen(-oo, x)
assert Intersection(S.Reals, Interval(x, oo)) == Interval.Ropen(x, oo)
assert Intersection(S.Reals, Interval(-oo, 1)) == Interval.Lopen(-oo, 1)
assert Intersection(S.Reals, Interval(1, oo)) == Interval.Ropen(1, oo)
assert Intersection(S.ExtendedReals, Interval(1, 2)) == Interval(1, 2)
# 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
pytest.raises(ValueError, lambda: list(i))
assert (Intersection(Intersection(S.Integers, S.Naturals, evaluate=False),
S.Reals,
evaluate=False) == Intersection(S.Integers,
S.Naturals,
S.Reals,
evaluate=False))
assert (imageset(
Lambda(x, x**2),
Intersection(FiniteSet(1, 2), FiniteSet(2, 3),
evaluate=False)) == FiniteSet(4))
def test_sympyissue_22073():
assert -oo in Interval(-oo, 5)
assert -oo not in Interval.Lopen(-oo, 5)
assert -3 not in Interval.Lopen(-3, 5)
assert -3 in Interval(-3, 5)