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_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_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_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_Complement():
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 not 3 in Complement(Interval(0, 5), Interval(1, 4), evaluate=False)
assert -1 in Complement(S.Reals, S.Naturals, evaluate=False)
assert not 1 in Complement(S.Reals, S.Naturals, evaluate=False)
assert Complement(S.Integers, S.UniversalSet) == EmptySet()
assert S.UniversalSet.complement(S.Integers) == EmptySet()
assert (not 0 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))
def test_issue_20291():
from sympy import FiniteSet, Complement, Intersection, Reals, EmptySet
a = Symbol('a')
b = Symbol('b')
A = FiniteSet(a, b)
assert A.evalf(subs={a: 1, b: 2}) == FiniteSet(1.0, 2.0)
B = FiniteSet(a - b, 1)
assert B.evalf(subs={a: 1, b: 2}) == FiniteSet(-1.0, 1.0)
sol = Complement(
Intersection(FiniteSet(-b / 2 - sqrt(b**2 - 4 * pi) / 2), Reals),
FiniteSet(0))
assert sol.evalf(subs={b: 1}) == EmptySet
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 to_interval(relational):
if isinstance(relational, And):
return Intersection([to_interval(i) for i in relational.args])
elif isinstance(relational, Or):
return Union([to_interval(i) for i in relational.args])
elif isinstance(relational, Not):
return Complement([to_interval(i) for i in relational.args])
if relational == S.true:
return Interval(S.NegativeInfinity,
S.Infinity,
left_open=True,
right_open=True)
if len(relational.free_symbols) != 1:
raise ValueError('Relational must only have one free symbol')
if len(relational.args) != 2:
raise ValueError('Relational must only have two arguments')
free_symbol = list(relational.free_symbols)[0]
lhs = relational.args[0]
rhs = relational.args[1]
if isinstance(relational, GreaterThan):
if lhs == free_symbol:
return Interval(rhs, S.Infinity, left_open=False)
else:
return Interval(S.NegativeInfinity, rhs, right_open=False)
elif isinstance(relational, StrictGreaterThan):
if lhs == free_symbol:
return Interval(rhs, S.Infinity, left_open=True)
else:
return Interval(S.NegativeInfinity, rhs, right_open=True)
elif isinstance(relational, LessThan):
if lhs != free_symbol:
return Interval(rhs, S.Infinity, left_open=False)
else:
return Interval(S.NegativeInfinity, rhs, right_open=False)
elif isinstance(relational, StrictLessThan):
if lhs != free_symbol:
return Interval(rhs, S.Infinity, left_open=True)
else:
return Interval(S.NegativeInfinity, rhs, right_open=True)
else:
raise ValueError('Unsupported Relational: {}'.format(
relational.__class__.__name__))
def test_issue_9808():
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_Complement():
assert str(Complement(S.Reals, S.Naturals)) == 'Reals \\ Naturals'
def test_Complement():
assert str(Complement(S.Reals, S.Naturals)) == '(-oo, oo) \ Naturals()'
def test_Complement():
assert str(Complement(S.Reals, S.Naturals)) == 'Complement(Reals, Naturals)'
def _regular_point_ellipse(self, a, b, c, d, e, f):
D = 4 * a * c - b**2
ok = D
if not ok:
raise ValueError("Rational Point on the conic does not exist")
if a == 0 and c == 0:
K = -1
L = 4 * (d * e - b * f)
elif c != 0:
K = D
L = 4 * c**2 * d**2 - 4 * b * c * d * e + 4 * a * c * e**2 + 4 * b**2 * c * f - 16 * a * c**2 * f
else:
K = D
L = 4 * a**2 * e**2 - 4 * b * a * d * e + 4 * b**2 * a * f
ok = L != 0 and not (K > 0 and L < 0)
if not ok:
raise ValueError("Rational Point on the conic does not exist")
K = Rational(K).limit_denominator(10**12)
L = Rational(L).limit_denominator(10**12)
k1, k2 = K.p, K.q
l1, l2 = L.p, L.q
g = gcd(k2, l2)
a1 = (l2 * k2) / g
b1 = (k1 * l2) / g
c1 = -(l1 * k2) / g
a2 = sign(a1) * core(abs(a1), 2)
r1 = sqrt(a1 / a2)
b2 = sign(b1) * core(abs(b1), 2)
r2 = sqrt(b1 / b2)
c2 = sign(c1) * core(abs(c1), 2)
r3 = sqrt(c1 / c2)
g = gcd(gcd(a2, b2), c2)
a2 = a2 / g
b2 = b2 / g
c2 = c2 / g
g1 = gcd(a2, b2)
a2 = a2 / g1
b2 = b2 / g1
c2 = c2 * g1
g2 = gcd(a2, c2)
a2 = a2 / g2
b2 = b2 * g2
c2 = c2 / g2
g3 = gcd(b2, c2)
a2 = a2 * g3
b2 = b2 / g3
c2 = c2 / g3
x, y, z = symbols("x y z")
eq = a2 * x**2 + b2 * y**2 + c2 * z**2
solutions = diophantine(eq)
if len(solutions) == 0:
raise ValueError("Rational Point on the conic does not exist")
flag = False
for sol in solutions:
syms = Tuple(*sol).free_symbols
rep = {s: 3 for s in syms}
sol_z = sol[2]
if sol_z == 0:
flag = True
continue
if not (isinstance(sol_z, Integer) or isinstance(sol_z, int)):
syms_z = sol_z.free_symbols
if len(syms_z) == 1:
p = next(iter(syms_z))
p_values = Complement(
S.Integers, solveset(Eq(sol_z, 0), p, S.Integers))
rep[p] = next(iter(p_values))
if len(syms_z) == 2:
p, q = list(ordered(syms_z))
for i in S.Integers:
subs_sol_z = sol_z.subs(p, i)
q_values = Complement(
S.Integers,
solveset(Eq(subs_sol_z, 0), q, S.Integers))
if not q_values.is_empty:
rep[p] = i
rep[q] = next(iter(q_values))
break
if len(syms) != 0:
x, y, z = tuple(s.subs(rep) for s in sol)
else:
x, y, z = sol
flag = False
break
if flag:
raise ValueError("Rational Point on the conic does not exist")
x = (x * g3) / r1
y = (y * g2) / r2
z = (z * g1) / r3
x = x / z
y = y / z
if a == 0 and c == 0:
x_reg = (x + y - 2 * e) / (2 * b)
y_reg = (x - y - 2 * d) / (2 * b)
elif c != 0:
x_reg = (x - 2 * d * c + b * e) / K
y_reg = (y - b * x_reg - e) / (2 * c)
else:
y_reg = (x - 2 * e * a + b * d) / K
x_reg = (y - b * y_reg - d) / (2 * a)
return x_reg, y_reg
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))
