def test_imageset_intersect_real():
assert (imageset(Lambda(n, n + (n - 1)*(n + 1)*I),
S.Integers).intersect(S.Reals) ==
FiniteSet(-1, 1))
s = ImageSet(Lambda(n, -I*(I*(2*pi*n - pi/4) + log(Abs(sqrt(-I))))), S.Integers)
assert s.intersect(S.Reals) == imageset(Lambda(n, 2*n*pi - pi/4), S.Integers)
def test_imageset_intersection_real():
assert (imageset(Lambda(n, n + (n - 1)*(n + 1)*I),
S.Integers).intersection(S.Reals) ==
FiniteSet(-1, 1))
s = ImageSet(Lambda(n, -I*(I*(2*pi*n - pi/4) + log(Abs(sqrt(-I))))), S.Integers)
assert s.intersection(S.Reals) == imageset(Lambda(n, 2*n*pi - pi/4), S.Integers)
def _intersect(self, other):
from diofant import Dummy
from diofant.solvers.diophantine import diophantine
from diofant.sets.sets import imageset
if self.base_set is S.Integers:
if isinstance(other, ImageSet) and other.base_set is S.Integers:
f, g = self.lamda.expr, other.lamda.expr
n, m = self.lamda.variables[0], other.lamda.variables[0]
# Diophantine sorts the solutions according to the alphabetic
# order of the variable names, since the result should not depend
# on the variable name, they are replaced by the dummy variables
# below
a, b = Dummy('a'), Dummy('b')
f, g = f.subs(n, a), g.subs(m, b)
solns_set = diophantine(f - g)
if solns_set == set():
return EmptySet()
solns = list(diophantine(f - g))
if len(solns) == 1:
t = list(solns[0][0].free_symbols)[0]
else:
return
# since 'a' < 'b'
return imageset(Lambda(t, f.subs(a, solns[0][0])), S.Integers)
if other == S.Reals:
from diofant.solvers.diophantine import diophantine
from diofant.core.function import expand_complex
if len(self.lamda.variables
) > 1 or self.base_set is not S.Integers:
return
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, integer=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
sols = list(diophantine(im, n_))
if not sols:
return S.EmptySet
elif all(s[0].has(n_) is False for s in sols):
s = FiniteSet(*[s[0] for s in sols])
elif len(sols) == 1 and sols[0][0].has(n_):
s = imageset(Lambda(n_, sols[0][0]), S.Integers)
else:
return
return imageset(Lambda(n_, re), self.base_set.intersect(s))
def test_infinitely_indexed_set_2():
a = Symbol('a', integer=True)
assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(n, n + a), S.Integers)
assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(n, -n + a), S.Integers)
assert imageset(Lambda(n, -6*n), S.Integers) == ImageSet(Lambda(n, 6*n), S.Integers)
assert imageset(Lambda(n, 2*n + pi), S.Integers) == ImageSet(Lambda(n, 2*n + pi), S.Integers)
assert imageset(Lambda(n, pi*n + pi), S.Integers) == ImageSet(Lambda(n, pi*n + pi), S.Integers)
assert imageset(Lambda(n, exp(n)), S.Integers) != imageset(Lambda(n, n), S.Integers)
def test_infinitely_indexed_set_2():
a = Symbol('a', integer=True)
assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(n, n + a), S.Integers)
assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(n, -n + a), S.Integers)
assert imageset(Lambda(n, -6*n), S.Integers) == ImageSet(Lambda(n, 6*n), S.Integers)
assert imageset(Lambda(n, 2*n + pi), S.Integers) == ImageSet(Lambda(n, 2*n + pi), S.Integers)
assert imageset(Lambda(n, pi*n + pi), S.Integers) == ImageSet(Lambda(n, pi*n + pi), S.Integers)
assert imageset(Lambda(n, exp(n)), S.Integers) != imageset(Lambda(n, n), S.Integers)
def test_infinitely_indexed_set_1():
assert imageset(Lambda(n, n),
S.Integers) == imageset(Lambda(m, m), S.Integers)
assert (imageset(Lambda(n, 2 * n), S.Integers).intersection(
imageset(Lambda(m, 2 * m + 1), S.Integers)) == EmptySet())
assert (imageset(Lambda(n, 2 * n), S.Integers).intersection(
imageset(Lambda(n, 2 * n + 1), S.Integers)) == EmptySet())
assert (imageset(Lambda(m, 2 * m), S.Integers).intersection(
imageset(Lambda(n, 3 * n),
S.Integers)) == ImageSet(Lambda(t, 6 * t), S.Integers))
def test_infinitely_indexed_set_1():
assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(m, m), S.Integers)
assert (imageset(Lambda(n, 2*n),
S.Integers).intersection(imageset(Lambda(m, 2*m + 1),
S.Integers)) ==
EmptySet())
assert (imageset(Lambda(n, 2*n),
S.Integers).intersection(imageset(Lambda(n, 2*n + 1),
S.Integers)) ==
EmptySet())
assert (imageset(Lambda(m, 2*m),
S.Integers).intersection(imageset(Lambda(n, 3*n),
S.Integers)) ==
ImageSet(Lambda(t, 6*t), S.Integers))
def test_integers():
Z = S.Integers
assert 5 in Z
assert -5 in Z
assert 5.5 not in Z
zi = iter(Z)
a, b, c, d = next(zi), next(zi), next(zi), next(zi)
assert (a, b, c, d) == (0, 1, -1, 2)
assert isinstance(a, Basic)
assert Z.intersection(Interval(-5, 5)) == Range(-5, 6)
assert Z.intersection(Interval(-5, 5, True, True)) == Range(-4, 5)
assert Z.intersection(Set(x)) == Intersection(Z, Set(x),
evaluate=False)
assert Z.inf == -oo
assert Z.sup == oo
assert Z.boundary == Z
assert imageset(Lambda((x, y), x*y), Z) == ImageSet(Lambda((x, y), x*y), Z)
def test_integers():
Z = S.Integers
assert 5 in Z
assert -5 in Z
assert 5.5 not in Z
zi = iter(Z)
a, b, c, d = next(zi), next(zi), next(zi), next(zi)
assert (a, b, c, d) == (0, 1, -1, 2)
assert isinstance(a, Basic)
assert Z.intersection(Interval(-5, 5)) == Range(-5, 6)
assert Z.intersection(Interval(-5, 5, True, True)) == Range(-4, 5)
assert Z.intersection(Set(x)) == Intersection(Z, Set(x), evaluate=False)
assert Z.inf == -oo
assert Z.sup == oo
assert Z.boundary == Z
assert imageset(Lambda((x, y), x * y),
Z) == ImageSet(Lambda((x, y), x * y), Z)
def test_ImageSet_simplification():
assert imageset(Lambda(n, n), S.Integers) == S.Integers
assert (imageset(Lambda(n, sin(n)),
imageset(Lambda(m, tan(m)), S.Integers)) ==
imageset(Lambda(m, sin(tan(m))), S.Integers))
def test_infinitely_indexed_set_3():
assert imageset(Lambda(n, 2*n + 1), S.Integers) == imageset(Lambda(n, 2*n - 1), S.Integers)
assert imageset(Lambda(n, 3*n + 2), S.Integers) == imageset(Lambda(n, 3*n - 1), S.Integers)
def test_infinitely_indexed_failed_diophantine():
assert (imageset(Lambda(m, 2*pi*m),
S.Integers).intersect(imageset(Lambda(n, 3*pi*n),
S.Integers)) ==
ImageSet(Lambda(t, -6*pi*t), S.Integers))
def test_infinitely_indexed_diophantine():
assert (imageset(Lambda(m, 2*pi*m),
S.Integers).intersection(imageset(Lambda(n, 3*pi*n),
S.Integers)) ==
ImageSet(Lambda(t, 6*pi*t), S.Integers))
def test_infinitely_indexed_set_3():
assert imageset(Lambda(n, 2*n + 1), S.Integers) == imageset(Lambda(n, 2*n - 1), S.Integers)
assert imageset(Lambda(n, 3*n + 2), S.Integers) == imageset(Lambda(n, 3*n - 1), S.Integers)
def test_image_is_ImageSet():
assert isinstance(imageset(x, sqrt(sin(x)), Range(5)), ImageSet)
def test_ImageSet_simplification():
assert imageset(Lambda(n, n), S.Integers) == S.Integers
assert (imageset(Lambda(n, sin(n)),
imageset(Lambda(m, tan(m)), S.Integers)) ==
imageset(Lambda(m, sin(tan(m))), S.Integers))
def test_image_is_ImageSet():
assert isinstance(imageset(x, sqrt(sin(x)), Range(5)), ImageSet)
