def find_curve_range_intersection(curve_1, curve_2, cut_at_inflection=False):
"""
Return intersections of x- and y-ranges of two real curves,
which are parametric curves on the xy-plane given as
(x_array, y_array), a tuple of NumPy arrays.
"""
x1, y1 = curve_1
x2, y2 = curve_2
if cut_at_inflection is True:
x1_min, x1_max = sorted([x1[0], x1[-1]])
x2_min, x2_max = sorted([x2[0], x2[-1]])
y1_min, y1_may = sorted([y1[0], y1[-1]])
y2_min, y2_may = sorted([y2[0], y2[-1]])
else:
x1_min, x1_max = numpy.sort(x1)[[0, -1]]
x2_min, x2_max = numpy.sort(x2)[[0, -1]]
y1_min, y1_may = numpy.sort(y1)[[0, -1]]
y2_min, y2_may = numpy.sort(y2)[[0, -1]]
x1_interval = Interval(x1_min, x1_max)
x2_interval = Interval(x2_min, x2_max)
y1_interval = Interval(y1_min, y1_may)
y2_interval = Interval(y2_min, y2_may)
x_range = x1_interval.intersect(x2_interval)
y_range = y1_interval.intersect(y2_interval)
return (x_range, y_range)
def test_complement():
assert Interval(0, 1).complement == \
Union(Interval(-oo, 0, True, True), Interval(1, oo, True, True))
assert Interval(0, 1, True, False).complement == \
Union(Interval(-oo, 0, True, False), Interval(1, oo, True, True))
assert Interval(0, 1, False, True).complement == \
Union(Interval(-oo, 0, True, True), Interval(1, oo, False, True))
assert Interval(0, 1, True, True).complement == \
Union(Interval(-oo, 0, True, False), Interval(1, oo, False, True))
assert -S.EmptySet == S.EmptySet.complement
assert ~S.EmptySet == S.EmptySet.complement
assert S.EmptySet.complement == Interval(-oo, oo)
assert Union(Interval(0, 1), Interval(2, 3)).complement == \
Union(Interval(-oo, 0, True, True), Interval(1, 2, True, True),
Interval(3, oo, True, True))
assert FiniteSet(0).complement == Union(Interval(-oo,0, True,True) ,
Interval(0,oo, True, True))
assert (FiniteSet(5) + Interval(S.NegativeInfinity, 0)).complement == \
Interval(0, 5, True, True) + Interval(5, S.Infinity, True,True)
assert FiniteSet(1,2,3).complement == Interval(S.NegativeInfinity,1, True,True) + Interval(1,2, True,True) + Interval(2,3, True,True) + Interval(3,S.Infinity, True,True)
X = Interval(1,3)+FiniteSet(5)
assert X.intersect(X.complement) == S.EmptySet
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)
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)
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
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 == oo
assert (band * FiniteSet(1, 2, 3)).measure == nan
def test_interval_symbolic():
x = Symbol('x')
e = Interval(0, 1)
assert e.contains(x) == And(0 <= x, x <= 1)
raises(TypeError, lambda: x in e)
e = Interval(0, 1, True, True)
assert e.contains(x) == And(0 < x, x < 1)
def test_issue_10285():
assert FiniteSet(-x - 1).intersect(Interval.Ropen(1, 2)) == FiniteSet(x).intersect(Interval.Lopen(-3, -2))
eq = -x - 2 * (-x - y)
s = signsimp(eq)
ivl = Interval.open(0, 1)
assert FiniteSet(eq).intersect(ivl) == FiniteSet(s).intersect(ivl)
assert FiniteSet(-eq).intersect(ivl) == FiniteSet(s).intersect(Interval.open(-1, 0))
eq -= 1
ivl = Interval.Lopen(1, oo)
assert FiniteSet(eq).intersect(ivl) == FiniteSet(s).intersect(Interval.Lopen(2, oo))
def test_reduce_rational_inequalities_real_relational():
assert reduce_rational_inequalities([], x) == 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(S(3)/2, oo))
assert reduce_rational_inequalities([[(x - 8)/x <= 3 - x]], x,
relational=False) == \
Union(Interval.Lopen(-oo, -2), Interval.Lopen(0, 4))
# issue sympy/sympy#10237
assert reduce_rational_inequalities(
[[x < oo, x >= 0, -oo < x]], x, relational=False) == Interval(0, oo)
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)
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)
# 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 X.subset(XandY) and Y.subset(XandY)
raises(TypeError, "Union(1, 2, 3)")
class WordInterval(object):
SILENCE_WORD = '#'
def __init__(self, inf, sup, word):
self.word = word
self.interval = Interval(inf, sup)
@property
def is_silent(self):
return self.word == WordInterval.SILENCE_WORD
@property
def inf(self):
return self.interval.inf
@property
def sup(self):
return self.interval.sup
def intersect(self, another_interval):
return self.interval.intersect(another_interval)
def __eq__(self, other):
return (self.interval == other.interval) and (self.word == other.word)
def __str__(self):
return "%s -> %s" % (self.interval, self.word)
def __repr__(self):
return self.__str__()
def test_complement():
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_solve_abs():
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x + 3) - 2 * Abs(x - 3), x) == FiniteSet(1, 9)
assert solveset_real(2 * Abs(x) - Abs(x - 1), x) == FiniteSet(-1, Rational(1, 3))
assert solveset_real(Abs(x - 7) - 8, x) == FiniteSet(-S(1), S(15))
# issue 9565. Note: solveset_real does not solve this as it is
# solveset's job to handle Relationals
assert solveset(Abs((x - 1) / (x - 5)) <= S(1) / 3, domain=S.Reals) == Interval(-1, 2)
# issue #10069
eq = abs(1 / (x - 1)) - 1 > 0
u = Union(Interval.open(0, 1), Interval.open(1, 2))
assert solveset_real(eq, x) == u
assert solveset(eq, x, domain=S.Reals) == u
raises(ValueError, lambda: solveset(abs(x) - 1, x))
def __init__(self):
self.set_str = ""
self.interval = FiniteSet()
self.val1 = None
self.val2 = None
self.left_border = True
self.right_border = True
self.interval_rep = 0
self.val_lst = []
def test_solve_abs():
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x + 3) - 2*Abs(x - 3), x) == \
FiniteSet(1, 9)
assert solveset_real(2*Abs(x) - Abs(x - 1), x) == \
FiniteSet(-1, Rational(1, 3))
assert solveset_real(Abs(x - 7) - 8, x) == FiniteSet(-S(1), S(15))
# issue 9565. Note: solveset_real does not solve this as it is
# solveset's job to handle Relationals
assert solveset(Abs((x - 1)/(x - 5)) <= S(1)/3, domain=S.Reals
) == Interval(-1, 2)
# issue #10069
assert solveset_real(abs(1/(x - 1)) - 1 > 0, x) == \
ConditionSet(x, Eq((1 - Abs(x - 1))/Abs(x - 1) > 0, 0),
S.Reals)
assert solveset(abs(1/(x - 1)) - 1 > 0, x, domain=S.Reals
) == Union(Interval.open(0, 1), Interval.open(1, 2))
def test_complement():
assert Interval(0, 1).complement == \
Union(Interval(-oo, 0, True, True), Interval(1, oo, True, True))
assert Interval(0, 1, True, False).complement == \
Union(Interval(-oo, 0, True, False), Interval(1, oo, True, True))
assert Interval(0, 1, False, True).complement == \
Union(Interval(-oo, 0, True, True), Interval(1, oo, False, True))
assert Interval(0, 1, True, True).complement == \
Union(Interval(-oo, 0, True, False), Interval(1, oo, False, True))
assert -S.EmptySet == S.EmptySet.complement
assert ~S.EmptySet == S.EmptySet.complement
assert S.EmptySet.complement == S.UniversalSet
assert S.UniversalSet.complement == S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).complement == \
Union(Interval(-oo, 0, True, True), Interval(1, 2, True, True),
Interval(3, oo, True, True))
assert FiniteSet(0).complement == Union(Interval(-oo, 0, True, True),
Interval(0, oo, True, True))
assert (FiniteSet(5) + Interval(S.NegativeInfinity, 0)).complement == \
Interval(0, 5, True, True) + Interval(5, S.Infinity, True, True)
assert FiniteSet(1, 2, 3).complement == \
Interval(S.NegativeInfinity, 1, True, True) + Interval(1, 2, True, True) + \
Interval(2, 3, True, True) + Interval(3, S.Infinity, True, True)
X = Interval(1, 3) + FiniteSet(5)
assert X.intersect(X.complement) == S.EmptySet
square = Interval(0, 1) * Interval(0, 1)
notsquare = square.complement
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_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)
def test_boundary_ProductSet_line():
line_in_r2 = Interval(0, 1) * FiniteSet(0)
assert line_in_r2.boundary == line_in_r2
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_Union_of_ProductSets_shares():
line = Interval(0, 2)
points = FiniteSet(0, 1, 2)
assert Union(line * line, line * points) == line * line
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
assert (H, 3, 3) in coin * d6 * d6
HH, TT = sympify(H), sympify(T)
assert set(coin**2) == set(((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_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(S(1) / 25, oo))
assert imageset(x, f1, Interval(1, 2)) == FiniteSet(0, 1)
def test_is_closed():
assert Interval(0, 1, False, False).is_closed
assert not Interval(0, 1, True, False).is_closed
assert FiniteSet(1, 2, 3).is_closed
def test_solve_poly_inequality():
assert psolve(Poly(0, x), '==') == [S.Reals]
assert psolve(Poly(1, x), '==') == [S.EmptySet]
assert psolve(PurePoly(x + 1, x), ">") == [Interval(-1, oo, True, False)]
def generate_interval(self, A, OP):
if OP == '\setminus':
if len(A.interval) == 1:
self.interval_rep = 1
self.val1 = choice([A.val_lst[0], -oo, oo])
if self.val1 == -oo or self.val1 == oo:
self.val2 = A.val_lst[0]
else:
self.val2 = choice([-oo, oo])
else:
self.interval_rep = 0
self.val1 = choice(A.val_lst)
self.val2 = self.val1
self.left_border = choice([True, False])
self.left_border = choice([True, False])
elif OP == '\cap':
self.interval_rep = 0
if len(A.val_lst) == 1:
if randint(0, 1) == 0:
self.val1 = randint(-25, A.val_lst[0])
self.val2 = A.val_lst[0] + randint(0, 2)
else:
self.val1 = A.val_lst[-1]-randint(0, 2)
self.val2 = randint(A.val_lst[-1], 25)
else:
if randint(0, 1) == 0:
self.val1 = randint(-25, A.val_lst[1])
self.val2 = A.val_lst[1] + randint(0, 2)
else:
self.val1 = A.val_lst[-1] - randint(0, 2)
self.val2 = randint(A.val_lst[-1], 25)
else:
self.interval_rep = 2
tmp_lst = [-5, -4, -3, -2, -1, 1, 2, 3, 4, 5]
tmp_lst.extend(A.val_lst)
for i in range(0, randint(1, 5)):
self.val_lst.append(choice(tmp_lst))
if self.interval_rep == 0:
self.interval = Interval(self.val1, self.val2, self.left_border, self.right_border)
self.set_str += "\{r \ | \ r \in \mathbb{R} \wedge \ "+str(self.val1)
self.set_str += '<' if self.left_border == True else '\leq'
self.set_str += ' \ r'
self.set_str += '<' if self.right_border == True else '\leq'
self.set_str += str(self.val2)
self.set_str += '\}'
if isinstance(self.interval, EmptySet):
self.interval = FiniteSet(self.val1)
self.set_str = '\{'+str(self.val1)+'\}'
elif self.interval_rep == 1:
self.interval = Interval(self.val1, self.val2, self.left_border, self.right_border)
self.set_str += "\{r \ | \ r \in \mathbb{R} \wedge \ r"
if self.val1 == -oo or self.val1 == oo:
if self.val1 == -oo:
self.set_str += '<' if self.left_border == True else '\leq'
else:
self.set_str += '>' if self.left_border == True else '\geq'
#if self.val1 == oo:
# self.set_str += '<' if self.left_border == True else '\leq'
#else:
# self.set_str += '>' if self.left_border == True else '\geq'
self.set_str += str(self.val2)+'\}'
else:
if self.val2 == -oo:
self.set_str += '<' if self.left_border == True else '\leq'
else:
self.set_str += '>' if self.left_border == True else '\geq'
#if self.val2 == oo:
# self.set_str += '<' if self.left_border == True else '\leq'
#else:
# self.set_str += '>' if self.left_border == True else '\geq'
self.set_str += str(self.val1)+'\}'
else:
for i in range(len(self.val_lst)):
self.interval = self.interval.union(FiniteSet(self.val_lst[i]))
self.set_str = '\{'+str(self.interval)+'\}'
def test_factor_terms():
A = Symbol('A', commutative=False)
assert factor_terms(9*(x + x*y + 1) + (3*x + 3)**(2 + 2*x)) == \
9*x*y + 9*x + _keep_coeff(S(3), x + 1)**_keep_coeff(S(2), x + 1) + 9
assert factor_terms(9*(x + x*y + 1) + (3)**(2 + 2*x)) == \
_keep_coeff(S(9), 3**(2*x) + x*y + x + 1)
assert factor_terms(3**(2 + 2*x) + a*3**(2 + 2*x)) == \
9*3**(2*x)*(a + 1)
assert factor_terms(x + x*A) == \
x*(1 + A)
assert factor_terms(sin(x + x*A)) == \
sin(x*(1 + A))
assert factor_terms((3*x + 3)**((2 + 2*x)/3)) == \
_keep_coeff(S(3), x + 1)**_keep_coeff(S(2)/3, x + 1)
assert factor_terms(x + (x*y + x)**(3*x + 3)) == \
x + (x*(y + 1))**_keep_coeff(S(3), x + 1)
assert factor_terms(a*(x + x*y) + b*(x*2 + y*x*2)) == \
x*(a + 2*b)*(y + 1)
i = Integral(x, (x, 0, oo))
assert factor_terms(i) == i
assert factor_terms(x / 2 + y) == x / 2 + y
# fraction doesn't apply to integer denominators
assert factor_terms(x / 2 + y, fraction=True) == x / 2 + y
# clear *does* apply to the integer denominators
assert factor_terms(x / 2 + y, clear=True) == Mul(S.Half,
x + 2 * y,
evaluate=False)
# check radical extraction
eq = sqrt(2) + sqrt(10)
assert factor_terms(eq) == eq
assert factor_terms(eq, radical=True) == sqrt(2) * (1 + sqrt(5))
eq = root(-6, 3) + root(6, 3)
assert factor_terms(eq,
radical=True) == 6**(S(1) / 3) * (1 + (-1)**(S(1) / 3))
eq = [x + x * y]
ans = [x * (y + 1)]
for c in [list, tuple, set]:
assert factor_terms(c(eq)) == c(ans)
assert factor_terms(Tuple(x + x * y)) == Tuple(x * (y + 1))
assert factor_terms(Interval(0, 1)) == Interval(0, 1)
e = 1 / sqrt(a / 2 + 1)
assert factor_terms(e, clear=False) == 1 / sqrt(a / 2 + 1)
assert factor_terms(e, clear=True) == sqrt(2) / sqrt(a + 2)
eq = x / (x + 1 / x) + 1 / (x**2 + 1)
assert factor_terms(eq, fraction=False) == eq
assert factor_terms(eq, fraction=True) == 1
assert factor_terms((1/(x**3 + x**2) + 2/x**2)*y) == \
y*(2 + 1/(x + 1))/x**2
# if not True, then processesing for this in factor_terms is not necessary
assert gcd_terms(-x - y) == -x - y
assert factor_terms(-x - y) == Mul(-1, x + y, evaluate=False)
# if not True, then "special" processesing in factor_terms is not necessary
assert gcd_terms(exp(Mul(-1, x + 1))) == exp(-x - 1)
e = exp(-x - 2) + x
assert factor_terms(e) == exp(Mul(-1, x + 2, evaluate=False)) + x
assert factor_terms(e, sign=False) == e
assert factor_terms(exp(-4 * x - 2) -
x) == -x + exp(Mul(-2, 2 * x + 1, evaluate=False))
# sum/integral tests
for F in (Sum, Integral):
assert factor_terms(F(x, (y, 1, 10))) == x * F(1, (y, 1, 10))
assert factor_terms(F(x, (y, 1, 10)) + x) == x * (1 + F(1, (y, 1, 10)))
assert factor_terms(F(x * y + x * y**2,
(y, 1, 10))) == x * F(y * (y + 1), (y, 1, 10))
def test_reduce_rational_inequalities_real_relational():
assert reduce_rational_inequalities([], x) == 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(S(3)/2, oo))
assert reduce_rational_inequalities([[(x - 8)/x <= 3 - x]], x,
relational=False) == \
Union(Interval.Lopen(-oo, -2), Interval.Lopen(0, 4))
# issue sympy/sympy#10237
assert reduce_rational_inequalities([[x < oo, x >= 0, -oo < x]],
x,
relational=False) == Interval(0, oo)
def test_issue_10326():
assert Contains(oo, Interval(-oo, oo)) == False
assert Contains(-oo, Interval(-oo, oo)) == False
def test_interior():
assert Interval(0, 1, False, True).interior == Interval(0, 1, True, True)
def test_closure():
assert Interval(0, 1, False, True).closure == Interval(0, 1, False, False)
def create_ui_set(self, X):
result = None
nums = []
allowed = "-0123456789{},()[]o"
if not all(c in allowed for c in X):
return False
if len(X) < 3 and X != '{}':
return False
if (X[0] == '(' or X[0] == '[') and (X[-1] == ')' or X[-1] == ']'):
nums = re.findall("[-\d]+", X)
if '-' in nums:
nums.remove('-')
if len(nums) == 1:
if '-oo' in X or 'oo' in X:
if '-oo' in X:
nums.append(nums[0])
nums[0] = -oo
elif 'oo' in X:
nums.append(oo)
else:
return False
if len(nums) != 2:
return False
commata = re.findall(",", X)
if len(commata) != 1:
return False
for i in range(len(X)):
if X[i] == ',' and not (((X[i-1].isdigit() or X[i+1].isdigit()) or (X[i-1] == 'o' or X[i+1] == 'o'))):
return False
left = True if X[0] == '(' else False
right = True if X[-1] == ')' else False
if oo in nums or -oo in nums:
if oo in nums:
result = Interval(int(nums[0]), oo, left, right)
return result
else:
result = Interval(-oo, int(nums[1]), left, right)
return result
else:
if nums[0] <= nums[1]:
result = Interval(int(nums[0]), int(nums[1]), left, right)
return result
else:
return False
elif X[0] == '{' and X[-1] == '}':
if len(X) == 2:
return EmptySet()
else:
nums = re.findall("[-\d]+", X)
if len(re.findall(",", X)) != len(nums)-1:
return False
num_count = 0
for i in range(len(nums)):
if X[i] == ',' and not X.index(nums[num_count]) < i and X[i+1] == ',':
return False
num_count += 1
result = FiniteSet()
for i in range(len(nums)):
result = result.union(FiniteSet(int(nums[i])))
return result
else:
return False
def test_image_interval():
from sympy.core.numbers import Rational
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_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 __init__(self, inf, sup, word):
self.word = word
self.interval = Interval(inf, sup)
def test_reduce_poly_inequalities_real_interval():
assert reduce_rational_inequalities([[Eq(x**2, 0)]], x,
relational=False) == FiniteSet(0)
assert reduce_rational_inequalities([[Le(x**2, 0)]], x,
relational=False) == FiniteSet(0)
assert reduce_rational_inequalities([[Lt(x**2, 0)]], x,
relational=False) == S.EmptySet
assert reduce_rational_inequalities(
[[Ge(x**2, 0)]], x, relational=False) == \
S.Reals if x.is_real else Interval(-oo, oo)
assert reduce_rational_inequalities(
[[Gt(x**2, 0)]], x, relational=False) == \
FiniteSet(0).complement(S.Reals)
assert reduce_rational_inequalities(
[[Ne(x**2, 0)]], x, relational=False) == \
FiniteSet(0).complement(S.Reals)
assert reduce_rational_inequalities([[Eq(x**2, 1)]], x,
relational=False) == FiniteSet(-1, 1)
assert reduce_rational_inequalities([[Le(x**2, 1)]], x,
relational=False) == Interval(-1, 1)
assert reduce_rational_inequalities([[Lt(x**2, 1)]], x,
relational=False) == Interval(
-1, 1, True, True)
assert reduce_rational_inequalities(
[[Ge(x**2, 1)]], x, relational=False) == \
Union(Interval(-oo, -1), Interval(1, oo))
assert reduce_rational_inequalities(
[[Gt(x**2, 1)]], x, relational=False) == \
Interval(-1, 1).complement(S.Reals)
assert reduce_rational_inequalities(
[[Ne(x**2, 1)]], x, relational=False) == \
FiniteSet(-1, 1).complement(S.Reals)
assert reduce_rational_inequalities([[Eq(x**2, 1.0)]], x,
relational=False) == FiniteSet(
-1.0, 1.0).evalf()
assert reduce_rational_inequalities([[Le(x**2, 1.0)]], x,
relational=False) == Interval(
-1.0, 1.0)
assert reduce_rational_inequalities([[Lt(x**2, 1.0)]], x,
relational=False) == Interval(
-1.0, 1.0, True, True)
assert reduce_rational_inequalities(
[[Ge(x**2, 1.0)]], x, relational=False) == \
Union(Interval(-inf, -1.0), Interval(1.0, inf))
assert reduce_rational_inequalities(
[[Gt(x**2, 1.0)]], x, relational=False) == \
Union(Interval(-inf, -1.0, right_open=True),
Interval(1.0, inf, left_open=True))
assert reduce_rational_inequalities([[Ne(
x**2, 1.0)]], x, relational=False) == \
FiniteSet(-1.0, 1.0).complement(S.Reals)
s = sqrt(2)
assert reduce_rational_inequalities(
[[Lt(x**2 - 1, 0), Gt(x**2 - 1, 0)]], x,
relational=False) == S.EmptySet
assert reduce_rational_inequalities(
[[Le(x**2 - 1, 0), Ge(x**2 - 1, 0)]], x,
relational=False) == FiniteSet(-1, 1)
assert reduce_rational_inequalities(
[[Le(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, False, False),
Interval(1, s, False, False))
assert reduce_rational_inequalities(
[[Le(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, False, True),
Interval(1, s, True, False))
assert reduce_rational_inequalities(
[[Lt(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, True, False),
Interval(1, s, False, True))
assert reduce_rational_inequalities(
[[Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, True, True),
Interval(1, s, True, True))
assert reduce_rational_inequalities(
[[Lt(x**2 - 2, 0), Ne(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, True, True),
Interval(-1, 1, True, True),
Interval(1, s, True, True))
def test_is_open():
assert not Interval(0, 1, False, False).is_open
assert not Interval(0, 1, True, False).is_open
assert Interval(0, 1, True, True).is_open
assert not FiniteSet(1, 2, 3).is_open
#!/usr/bin/env python from sympy import Interval from sympy.abc import a, b, x from sympy import factor e = Interval.map(Interval(10, 12), Interval(-1, 1), x) print(e.subs(x, 10)) print(e.subs(x, 11)) print(e.subs(x, 12)) e = Interval.map(Interval(a, b), Interval(-1, 1), x) print(factor(e))
def set(self):
if self.is_symbolic:
return Intersection(S.Naturals0, Interval(0, self.n))
return set(self.dict.keys())
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 == set([x])
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))
f = FiniteSet(s, hello)
print f
for a in f:
print a, isinstance(a,String)
print hello + world
print hello + ' ' + Rational(3,4)
t = Tuple(hello, f)
print t
print t[0]
Doubles = Interval(-1e300,1e300)
print Doubles
v = S(1e300)
print type(v),v,Doubles.contains(v)
print type(Doubles * Doubles)
x = Symbol('x', real = True, bounded = True)
print ( Interval(x+2,100) | Interval(x+3,100) ).contains(x+4.0) & (x < 90)
print Interval(x+4.0,100) & Interval(x+3,100)
print x.is_bounded
print Interval(sin(x),100)
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))
#!/usr/bin/python
from sympy.solvers import solveset
from sympy import Symbol, Interval, pprint
x = Symbol('x')
sol = solveset(x**2 - 1, x, Interval(0, 100))
print(sol)
def test_piecewise_integrate():
x, y = symbols('x y', real=True, finite=True)
# XXX Use '<=' here! '>=' is not yet implemented ..
f = Piecewise(((x - 2)**2, 0 <= x), (1, True))
assert integrate(f, (x, -2, 2)) == Rational(14, 3)
g = Piecewise(((x - 5)**5, 4 <= x), (f, True))
assert integrate(g, (x, -2, 2)) == Rational(14, 3)
assert integrate(g, (x, -2, 5)) == Rational(43, 6)
g = Piecewise(((x - 5)**5, 4 <= x), (f, x < 4))
assert integrate(g, (x, -2, 2)) == Rational(14, 3)
assert integrate(g, (x, -2, 5)) == Rational(43, 6)
g = Piecewise(((x - 5)**5, 2 <= x), (f, x < 2))
assert integrate(g, (x, -2, 2)) == Rational(14, 3)
assert integrate(g, (x, -2, 5)) == -Rational(701, 6)
g = Piecewise(((x - 5)**5, 2 <= x), (f, True))
assert integrate(g, (x, -2, 2)) == Rational(14, 3)
assert integrate(g, (x, -2, 5)) == -Rational(701, 6)
g = Piecewise(((x - 5)**5, 2 <= x), (2 * f, True))
assert integrate(g, (x, -2, 2)) == 2 * Rational(14, 3)
assert integrate(g, (x, -2, 5)) == -Rational(673, 6)
g = Piecewise((1, x > 0), (0, Eq(x, 0)), (-1, x < 0))
assert integrate(g, (x, -1, 1)) == 0
g = Piecewise((1, x - y < 0), (0, True))
assert integrate(g, (y, -oo, 0)) == -Min(0, x)
assert integrate(g, (y, 0, oo)) == oo - Max(0, x)
assert integrate(g, (y, -oo, oo)) == oo - x
g = Piecewise((0, x < 0), (x, x <= 1), (1, True))
assert integrate(g, (x, -5, 1)) == Rational(1, 2)
assert integrate(g, (x, -5, y)).subs(y, 1) == Rational(1, 2)
assert integrate(g, (x, y, 1)).subs(y, -5) == Rational(1, 2)
assert integrate(g, (x, 1, -5)) == -Rational(1, 2)
assert integrate(g, (x, 1, y)).subs(y, -5) == -Rational(1, 2)
assert integrate(g, (x, y, -5)).subs(y, 1) == -Rational(1, 2)
assert integrate(g, (x, -5, y)) == Piecewise(
(0, y < 0), (y**2 / 2, y <= 1), (y - 0.5, True))
assert integrate(g, (x, y, 1)) == Piecewise(
(0.5, y < 0), (0.5 - y**2 / 2, y <= 1), (1 - y, True))
g = Piecewise((1 - x, Interval(0, 1).contains(x)),
(1 + x, Interval(-1, 0).contains(x)), (0, True))
assert integrate(g, (x, -5, 1)) == 1
assert integrate(g, (x, -5, y)).subs(y, 1) == 1
assert integrate(g, (x, y, 1)).subs(y, -5) == 1
assert integrate(g, (x, 1, -5)) == -1
assert integrate(g, (x, 1, y)).subs(y, -5) == -1
assert integrate(g, (x, y, -5)).subs(y, 1) == -1
assert integrate(g, (x, -5, y)) == Piecewise(
(-y**2 / 2 + y + 0.5, Interval(0, 1).contains(y)),
(y**2 / 2 + y + 0.5, Interval(-1, 0).contains(y)), (0, y <= -1),
(1, True))
assert integrate(g, (x, y, 1)) == Piecewise(
(y**2 / 2 - y + 0.5, Interval(0, 1).contains(y)),
(-y**2 / 2 - y + 0.5, Interval(-1, 0).contains(y)), (1, y <= -1),
(0, True))
g = Piecewise((0, Or(x <= -1, x >= 1)), (1 - x, x > 0), (1 + x, True))
assert integrate(g, (x, -5, 1)) == 1
assert integrate(g, (x, -5, y)).subs(y, 1) == 1
assert integrate(g, (x, y, 1)).subs(y, -5) == 1
assert integrate(g, (x, 1, -5)) == -1
assert integrate(g, (x, 1, y)).subs(y, -5) == -1
assert integrate(g, (x, y, -5)).subs(y, 1) == -1
assert integrate(g, (x, -5, y)) == Piecewise((0, y <= -1), (1, y >= 1),
(-y**2 / 2 + y + 0.5, y > 0),
(y**2 / 2 + y + 0.5, True))
assert integrate(g, (x, y, 1)) == Piecewise((1, y <= -1), (0, y >= 1),
(y**2 / 2 - y + 0.5, y > 0),
(-y**2 / 2 - y + 0.5, True))
def test_issue_8715():
eq = x + 1 / x > -2 + 1 / x
assert solveset(eq, x, S.Reals) == (Interval.open(-2, oo) - FiniteSet(0))
assert solveset(eq.subs(x, log(x)), x, S.Reals) == Interval.open(exp(-2), oo) - FiniteSet(1)
def test_piecewise_solve2():
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
assert solve(f, x) == [2, Interval(0, oo, True, True)]
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_as_expr_set_pairs():
assert Piecewise((x, x > 0), (-x, x <= 0)).as_expr_set_pairs() == \
[(x, Interval(0, oo, True, True)), (-x, Interval(-oo, 0))]
assert Piecewise(((x - 2)**2, x >= 0), (0, True)).as_expr_set_pairs() == \
[((x - 2)**2, Interval(0, oo)), (0, Interval(-oo, 0, True, True))]
def set(self):
if self.is_symbolic:
return Intersection(S.Naturals0, Interval(0, self.n))
return set(map(Integer, list(range(0, self.n + 1))))
def test_issue_5724_7680():
assert I not in S.Reals # issue 7680
assert Interval(-oo, oo).contains(I) is S.false
def set(self):
N, m, n = self.N, self.m, self.n
if self.is_symbolic:
return Intersection(S.Naturals0, Interval(self.low, self.high))
return set([i for i in range(max(0, n + m - N), min(n, m) + 1)])
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)
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)
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))