def test_ImageSet():
assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1)
assert ImageSet(Lambda(x, y), S.Integers) == FiniteSet(y)
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1/x), S.Naturals)
assert Rational(1, 5) in harmonics
assert Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.3) not in harmonics
assert harmonics.is_iterable
c = ComplexRegion(Interval(1, 3)*Interval(1, 3))
assert Tuple(2, 6) in ImageSet(Lambda((x, y), (x, 2*y)), c)
assert Tuple(2, S.Half) in ImageSet(Lambda((x, y), (x, 1/y)), c)
assert Tuple(2, -2) not in ImageSet(Lambda((x, y), (x, y**2)), c)
assert Tuple(2, -2) in ImageSet(Lambda((x, y), (x, -2)), c)
c3 = Interval(3, 7)*Interval(8, 11)*Interval(5, 9)
assert Tuple(8, 3, 9) in ImageSet(Lambda((t, y, x), (y, t, x)), c3)
assert Tuple(S(1)/8, 3, 9) in ImageSet(Lambda((t, y, x), (1/y, t, x)), c3)
assert 2/pi not in ImageSet(Lambda((x, y), 2/x), c)
assert 2/S(100) not in ImageSet(Lambda((x, y), 2/x), c)
assert 2/S(3) in ImageSet(Lambda((x, y), 2/x), c)
def test_imageset_intersect_real():
from sympy import I
from sympy.abc import n
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_infinitely_indexed_set_2():
from sympy import exp
from sympy.abc import n
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_issue_18050():
assert imageset(Lambda(x, I * x + 1),
S.Integers) == ImageSet(Lambda(x, I * x + 1), S.Integers)
assert imageset(Lambda(x, 3 * I * x + 4 + 8 * I),
S.Integers) == ImageSet(Lambda(x, 3 * I * x + 4 + 2 * I),
S.Integers)
# no 'Mod' for next 2 tests:
assert imageset(Lambda(x, 2 * x + 3 * I),
S.Integers) == ImageSet(Lambda(x, 2 * x + 3 * I),
S.Integers)
r = Symbol('r', positive=True)
assert imageset(Lambda(x, r * x + 10),
S.Integers) == ImageSet(Lambda(x, r * x + 10), S.Integers)
# reduce real part:
assert imageset(Lambda(x, 3 * x + 8 + 5 * I),
S.Integers) == ImageSet(Lambda(x, 3 * x + 2 + 5 * I),
S.Integers)
def test_infinitely_indexed_set_3():
from sympy.abc import n, m, t
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)
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_Integers_eval_imageset():
ans = ImageSet(Lambda(x, 2 * x + Rational(3, 7)), S.Integers)
im = imageset(Lambda(x, -2 * x + Rational(3, 7)), S.Integers)
assert im == ans
im = imageset(Lambda(x, -2 * x - Rational(11, 7)), S.Integers)
assert im == ans
y = Symbol('y')
L = imageset(x, 2 * x + y, S.Integers)
assert y + 4 in L
_x = symbols('x', negative=True)
eq = _x**2 - _x + 1
assert imageset(_x, eq, S.Integers).lamda.expr == _x**2 + _x + 1
eq = 3 * _x - 1
assert imageset(_x, eq, S.Integers).lamda.expr == 3 * _x + 2
assert imageset(x, (x, 1/x), S.Integers) == \
ImageSet(Lambda(x, (x, 1/x)), S.Integers)
def test_Integers_eval_imageset():
ans = ImageSet(Lambda(x, 2 * x + S(3) / 7), S.Integers)
im = imageset(Lambda(x, -2 * x + S(3) / 7), S.Integers)
assert im == ans
im = imageset(Lambda(x, -2 * x - S(11) / 7), S.Integers)
assert im == ans
y = Symbol('y')
assert imageset(x, 2*x + y, S.Integers) == \
imageset(x, 2*x + y % 2, S.Integers)
_x = symbols('x', negative=True)
eq = _x**2 - _x + 1
assert imageset(_x, eq, S.Integers).lamda.expr == _x**2 + _x + 1
eq = 3 * _x - 1
assert imageset(_x, eq, S.Integers).lamda.expr == 3 * _x + 2
assert imageset(x, (x, 1/x), S.Integers) == \
ImageSet(Lambda(x, (x, 1/x)), S.Integers)
def test_ImageSet():
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1 / x), S.Naturals)
assert Rational(1, 5) in harmonics
assert .25 in harmonics
assert .3 not in harmonics
assert harmonics.is_iterable
def test_ImageSet():
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1/x), S.Naturals)
assert Rational(1, 5) in harmonics
assert .25 in harmonics
assert .3 not in harmonics
assert harmonics.is_iterable
def test_infinitely_indexed_set_1():
from sympy.abc import n, m, t
assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(m, m), S.Integers)
assert imageset(Lambda(n, 2*n), S.Integers).intersect(
imageset(Lambda(m, 2*m + 1), S.Integers)) is S.EmptySet
assert imageset(Lambda(n, 2*n), S.Integers).intersect(
imageset(Lambda(n, 2*n + 1), S.Integers)) is S.EmptySet
assert imageset(Lambda(m, 2*m), S.Integers).intersect(
imageset(Lambda(n, 3*n), S.Integers)).dummy_eq(
ImageSet(Lambda(t, 6*t), S.Integers))
assert imageset(x, x/2 + Rational(1, 3), S.Integers).intersect(S.Integers) is S.EmptySet
assert imageset(x, x/2 + S.Half, S.Integers).intersect(S.Integers) is S.Integers
# https://github.com/sympy/sympy/issues/17355
S53 = ImageSet(Lambda(n, 5*n + 3), S.Integers)
assert S53.intersect(S.Integers) == S53
def test_infinitely_indexed_set_1():
from sympy.abc import n, m, t
assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(m, m), S.Integers)
assert imageset(Lambda(n, 2*n), S.Integers).intersect(imageset(Lambda(m, 2*m + 1), S.Integers)) == \
EmptySet()
assert imageset(Lambda(n, 2*n), S.Integers).intersect(imageset(Lambda(n, 2*n + 1), S.Integers)) == \
EmptySet()
assert imageset(Lambda(m, 2*m), S.Integers).intersect(imageset(Lambda(n, 3*n), S.Integers)) == \
ImageSet(Lambda(t, 6*t), S.Integers)
def test_halfcircle():
# This test sometimes works and sometimes doesn't.
# It may be an issue with solve? Maybe with using Lambdas/dummys?
# I believe the code within fancysets is correct
r, th = symbols('r, theta', real=True)
L = Lambda((r, th), (r * cos(th), r * sin(th)))
halfcircle = ImageSet(L, Interval(0, 1) * Interval(0, pi))
assert (1, 0) in halfcircle
assert (0, -1) not in halfcircle
assert (0, 0) in halfcircle
assert not halfcircle.is_iterable
def imageset(*args):
r"""
Image of set under transformation ``f``.
If this function can't compute the image, it returns an
unevaluated ImageSet object.
.. math::
{ f(x) | x \in self }
Examples
========
>>> from sympy import Interval, Symbol, imageset, sin, Lambda
>>> x = Symbol('x')
>>> imageset(x, 2*x, Interval(0, 2))
[0, 4]
>>> imageset(lambda x: 2*x, Interval(0, 2))
[0, 4]
>>> imageset(Lambda(x, sin(x)), Interval(-2, 1))
ImageSet(Lambda(x, sin(x)), [-2, 1])
See Also
========
sympy.sets.fancysets.ImageSet
"""
from sympy.core import Dummy, Lambda
from sympy.sets.fancysets import ImageSet
if len(args) == 3:
f = Lambda(*args[:2])
else:
# var and expr are being defined this way to
# support Python lambda and not just sympy Lambda
f = args[0]
if not isinstance(f, Lambda):
var = Dummy()
expr = args[0](var)
f = Lambda(var, expr)
set = args[-1]
r = set._eval_imageset(f)
if r is not None:
return r
return ImageSet(f, set)
def intersection_sets(self, other): # noqa:F811
if other.is_ComplexRegion:
# self in rectangular form
if (not self.polar) and (not other.polar):
return ComplexRegion(Intersection(self.sets, other.sets))
# self in polar form
elif self.polar and other.polar:
r1, theta1 = self.a_interval, self.b_interval
r2, theta2 = other.a_interval, other.b_interval
new_r_interval = Intersection(r1, r2)
new_theta_interval = Intersection(theta1, theta2)
# 0 and 2*Pi means the same
if ((2*S.Pi in theta1 and S.Zero in theta2) or
(2*S.Pi in theta2 and S.Zero in theta1)):
new_theta_interval = Union(new_theta_interval,
FiniteSet(0))
return ComplexRegion(new_r_interval*new_theta_interval,
polar=True)
if other.is_subset(S.Reals):
new_interval = []
x = symbols("x", cls=Dummy, real=True)
# self in rectangular form
if not self.polar:
for element in self.psets:
if S.Zero in element.args[1]:
new_interval.append(element.args[0])
new_interval = Union(*new_interval)
return Intersection(new_interval, other)
# self in polar form
elif self.polar:
for element in self.psets:
if S.Zero in element.args[1]:
new_interval.append(element.args[0])
if S.Pi in element.args[1]:
new_interval.append(ImageSet(Lambda(x, -x), element.args[0]))
if S.Zero in element.args[0]:
new_interval.append(FiniteSet(0))
new_interval = Union(*new_interval)
return Intersection(new_interval, other)
def test_infinitely_indexed_set_1():
from sympy.abc import n, m, t
assert imageset(Lambda(n, n),
S.Integers) == imageset(Lambda(m, m), S.Integers)
assert imageset(Lambda(n, 2 * n), S.Integers).intersect(
imageset(Lambda(m, 2 * m + 1), S.Integers)) is S.EmptySet
assert imageset(Lambda(n, 2 * n), S.Integers).intersect(
imageset(Lambda(n, 2 * n + 1), S.Integers)) is S.EmptySet
assert imageset(Lambda(m, 2*m), S.Integers).intersect(
imageset(Lambda(n, 3*n), S.Integers)) == \
ImageSet(Lambda(t, 6*t), S.Integers)
assert imageset(x, x / 2 + Rational(1, 3), S.Integers).intersect(
S.Integers) is S.EmptySet
assert imageset(x, x / 2 + S.Half, S.Integers).intersect(
S.Integers) is S.Integers
def test_ImageSet_iterator_not_injective():
L = Lambda(x, x - x % 2) # produces 0, 2, 2, 4, 4, 6, 6, ...
evens = ImageSet(L, S.Naturals)
i = iter(evens)
# No repeats here
assert (next(i), next(i), next(i), next(i)) == (0, 2, 4, 6)
def test_infinitely_indexed_failed_diophantine():
from sympy.abc import n, m, t
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_imageset_intersect_interval():
from sympy.abc import n
f1 = ImageSet(Lambda(n, n*pi), S.Integers)
f2 = ImageSet(Lambda(n, 2*n), Interval(0, pi))
f3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)
# complex expressions
f4 = ImageSet(Lambda(n, n*I*pi), S.Integers)
f5 = ImageSet(Lambda(n, 2*I*n*pi + pi/2), S.Integers)
# non-linear expressions
f6 = ImageSet(Lambda(n, log(n)), S.Integers)
f7 = ImageSet(Lambda(n, n**2), S.Integers)
f8 = ImageSet(Lambda(n, Abs(n)), S.Integers)
f9 = ImageSet(Lambda(n, exp(n)), S.Naturals0)
assert f1.intersect(Interval(-1, 1)) == FiniteSet(0)
assert f2.intersect(Interval(1, 2)) == Interval(1, 2)
assert f3.intersect(Interval(-1, 1)) == S.EmptySet
assert f3.intersect(Interval(-5, 5)) == FiniteSet(-3*pi/2, pi/2)
assert f4.intersect(Interval(-1, 1)) == FiniteSet(0)
assert f4.intersect(Interval(1, 2)) == S.EmptySet
assert f5.intersect(Interval(0, 1)) == S.EmptySet
assert f6.intersect(Interval(0, 1)) == FiniteSet(S.Zero, log(2))
assert f7.intersect(Interval(0, 10)) == Intersection(f7, Interval(0, 10))
assert f8.intersect(Interval(0, 2)) == Intersection(f8, Interval(0, 2))
assert f9.intersect(Interval(1, 2)) == Intersection(f9, Interval(1, 2))
def test_ImageSet():
raises(ValueError, lambda: ImageSet(x, S.Integers))
assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1)
assert ImageSet(Lambda(x, y), S.Integers) == {y}
assert ImageSet(Lambda(x, 1), S.EmptySet) == S.EmptySet
empty = Intersection(FiniteSet(log(2) / pi), S.Integers)
assert unchanged(ImageSet, Lambda(x, 1), empty) # issue #17471
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1 / x), S.Naturals)
assert Rational(1, 5) in harmonics
assert Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.3) not in harmonics
assert (1, 2) not in harmonics
assert harmonics.is_iterable
assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0)
assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4)
assert ImageSet(Lambda((x, y), 2 * x), {4}, {3}).doit() == FiniteSet(8)
assert (ImageSet(Lambda((x, y), x + y), {1, 2, 3},
{10, 20, 30}).doit() == FiniteSet(11, 12, 13, 21, 22, 23,
31, 32, 33))
c = Interval(1, 3) * Interval(1, 3)
assert Tuple(2, 6) in ImageSet(Lambda(((x, y), ), (x, 2 * y)), c)
assert Tuple(2, S.Half) in ImageSet(Lambda(((x, y), ), (x, 1 / y)), c)
assert Tuple(2, -2) not in ImageSet(Lambda(((x, y), ), (x, y**2)), c)
assert Tuple(2, -2) in ImageSet(Lambda(((x, y), ), (x, -2)), c)
c3 = ProductSet(Interval(3, 7), Interval(8, 11), Interval(5, 9))
assert Tuple(8, 3, 9) in ImageSet(Lambda(((t, y, x), ), (y, t, x)), c3)
assert Tuple(Rational(1, 8), 3,
9) in ImageSet(Lambda(((t, y, x), ), (1 / y, t, x)), c3)
assert 2 / pi not in ImageSet(Lambda(((x, y), ), 2 / x), c)
assert 2 / S(100) not in ImageSet(Lambda(((x, y), ), 2 / x), c)
assert Rational(2, 3) in ImageSet(Lambda(((x, y), ), 2 / x), c)
S1 = imageset(lambda x, y: x + y, S.Integers, S.Naturals)
assert S1.base_pset == ProductSet(S.Integers, S.Naturals)
assert S1.base_sets == (S.Integers, S.Naturals)
# Passing a set instead of a FiniteSet shouldn't raise
assert unchanged(ImageSet, Lambda(x, x**2), {1, 2, 3})
S2 = ImageSet(Lambda(((x, y), ), x + y), {(1, 2), (3, 4)})
assert 3 in S2.doit()
# FIXME: This doesn't yet work:
#assert 3 in S2
assert S2._contains(3) is None
raises(TypeError, lambda: ImageSet(Lambda(x, x**2), 1))
def test_ImageSet():
raises(ValueError, lambda: ImageSet(x, S.Integers))
assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1)
assert ImageSet(Lambda(x, y), S.Integers) == {y}
assert ImageSet(Lambda(x, 1), S.EmptySet) == S.EmptySet
empty = Intersection(FiniteSet(log(2) / pi), S.Integers)
assert unchanged(ImageSet, Lambda(x, 1), empty) # issue #17471
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1 / x), S.Naturals)
assert Rational(1, 5) in harmonics
assert Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.3) not in harmonics
assert (1, 2) not in harmonics
assert harmonics.is_iterable
assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0)
assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4)
c = ComplexRegion(Interval(1, 3) * Interval(1, 3))
assert Tuple(2, 6) in ImageSet(Lambda((x, y), (x, 2 * y)), c)
assert Tuple(2, S.Half) in ImageSet(Lambda((x, y), (x, 1 / y)), c)
assert Tuple(2, -2) not in ImageSet(Lambda((x, y), (x, y**2)), c)
assert Tuple(2, -2) in ImageSet(Lambda((x, y), (x, -2)), c)
c3 = Interval(3, 7) * Interval(8, 11) * Interval(5, 9)
assert Tuple(8, 3, 9) in ImageSet(Lambda((t, y, x), (y, t, x)), c3)
assert Tuple(S(1) / 8, 3, 9) in ImageSet(Lambda((t, y, x), (1 / y, t, x)),
c3)
assert 2 / pi not in ImageSet(Lambda((x, y), 2 / x), c)
assert 2 / S(100) not in ImageSet(Lambda((x, y), 2 / x), c)
assert 2 / S(3) in ImageSet(Lambda((x, y), 2 / x), c)
assert imageset(lambda x, y: x + y, S.Integers,
S.Naturals).base_set == ProductSet(S.Integers, S.Naturals)
def test_imageset_intersection():
n = Dummy()
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*pi*n + pi*Rational(7, 4)), S.Integers)
def test_imageset_intersect_diophantine():
from sympy.abc import m, n
# Check that same lambda variable for both ImageSets is handled correctly
img1 = ImageSet(Lambda(n, 2 * n + 1), S.Integers)
img2 = ImageSet(Lambda(n, 4 * n + 1), S.Integers)
assert img1.intersect(img2) == img2
# Empty solution set returned by diophantine:
assert ImageSet(Lambda(n, 2 * n), S.Integers).intersect(
ImageSet(Lambda(n, 2 * n + 1), S.Integers)) == S.EmptySet
# Check intersection with S.Integers:
assert ImageSet(Lambda(n, 9 / n + 20 * n / 3),
S.Integers).intersect(S.Integers) == FiniteSet(
-61, -23, 23, 61)
# Single solution (2, 3) for diophantine solution:
assert ImageSet(Lambda(n, (n - 2)**2), S.Integers).intersect(
ImageSet(Lambda(n, -(n - 3)**2), S.Integers)) == FiniteSet(0)
# Single parametric solution for diophantine solution:
assert ImageSet(Lambda(n, n**2 + 5), S.Integers).intersect(
ImageSet(Lambda(m, 2 * m),
S.Integers)) == ImageSet(Lambda(n, 4 * n**2 + 4 * n + 6),
S.Integers)
# 4 non-parametric solution couples for dioph. equation:
assert ImageSet(Lambda(n, n**2 - 9), S.Integers).intersect(
ImageSet(Lambda(m, -m**2), S.Integers)) == FiniteSet(-9, 0)
# Double parametric solution for diophantine solution:
assert ImageSet(Lambda(m, m**2 + 40), S.Integers).intersect(
ImageSet(Lambda(n, 41 * n), S.Integers)) == Intersection(
ImageSet(Lambda(m, m**2 + 40), S.Integers),
ImageSet(Lambda(n, 41 * n), S.Integers))
# Check that diophantine returns *all* (8) solutions (permute=True)
assert ImageSet(Lambda(n, n**4 - 2**4), S.Integers).intersect(
ImageSet(Lambda(m, -m**4 + 3**4), S.Integers)) == FiniteSet(0, 65)
assert ImageSet(Lambda(n, pi / 12 + n * 5 * pi / 12),
S.Integers).intersect(
ImageSet(Lambda(n, 7 * pi / 12 + n * 11 * pi / 12),
S.Integers)) == ImageSet(
Lambda(n, 55 * pi * n / 12 + 17 * pi / 4),
S.Integers)
# TypeError raised by diophantine (#18081)
assert ImageSet(Lambda(n, n*log(2)), S.Integers).intersection(S.Integers) \
== Intersection(ImageSet(Lambda(n, n*log(2)), S.Integers), S.Integers)
# NotImplementedError raised by diophantine (no solver for cubic_thue)
assert ImageSet(Lambda(n, n**3 + 1), S.Integers).intersect(
ImageSet(Lambda(n, n**3), S.Integers)) == Intersection(
ImageSet(Lambda(n, n**3 + 1), S.Integers),
ImageSet(Lambda(n, n**3), S.Integers))
from sympy import Symbol, S, pi, Dummy, Lambda
from sympy.sets.sets import FiniteSet, Interval
from sympy.sets.fancysets import ImageSet
x = Symbol('x')
N = S.Naturals
squares = ImageSet(Lambda(x, x**2), N) # {x**2 for x in N}
print 4 in squares
print squares
print FiniteSet(0, 1, 2, 3, 4, 5, 6, 7, 9, 10).intersect(squares)
print FiniteSet(9, 16, 25).intersect(squares)
def test_issue_16871b():
assert ImageSet(Lambda(x, x - 3), S.Integers).is_subset(S.Integers)
def _set_pow(x, y): # noqa:F811
return ImageSet(Lambda((_x, _y), (_x**_y)), x, y)
def _eval_imageset(self, f):
from sympy.sets.fancysets import ImageSet
return ImageSet(f, self)
def test_no_mod_on_imaginary():
assert imageset(Lambda(x, 2*x + 3*I), S.Integers
) == ImageSet(Lambda(x, 2*x + I), S.Integers)
def test_fun():
assert (FiniteSet(
*ImageSet(Lambda(x, sin(pi * x / 4)), Range(-10, 11))) == FiniteSet(
-1, -sqrt(2) / 2, 0,
sqrt(2) / 2, 1))
def _(x, y):
return ImageSet(Lambda((_x, _y), (_x**_y)), x, y)
def test_ImageSet():
assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1)
assert ImageSet(Lambda(x, y), S.Integers) == FiniteSet(y)
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1 / x), S.Naturals)
assert Rational(1, 5) in harmonics
assert Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.3) not in harmonics
assert harmonics.is_iterable
c = ComplexRegion(Interval(1, 3) * Interval(1, 3))
assert Tuple(2, 6) in ImageSet(Lambda((x, y), (x, 2 * y)), c)
assert Tuple(2, S.Half) in ImageSet(Lambda((x, y), (x, 1 / y)), c)
assert Tuple(2, -2) not in ImageSet(Lambda((x, y), (x, y**2)), c)
assert Tuple(2, -2) in ImageSet(Lambda((x, y), (x, -2)), c)
c3 = Interval(3, 7) * Interval(8, 11) * Interval(5, 9)
assert Tuple(8, 3, 9) in ImageSet(Lambda((t, y, x), (y, t, x)), c3)
assert Tuple(S(1) / 8, 3, 9) in ImageSet(Lambda((t, y, x), (1 / y, t, x)),
c3)
assert 2 / pi not in ImageSet(Lambda((x, y), 2 / x), c)
assert 2 / S(100) not in ImageSet(Lambda((x, y), 2 / x), c)
assert 2 / S(3) in ImageSet(Lambda((x, y), 2 / x), c)
def test_issue_9543():
assert ImageSet(Lambda(x, x**2), S.Naturals).is_subset(S.Reals)
def test_issue_16871():
assert ImageSet(Lambda(x, x), FiniteSet(1)) == {1}
assert ImageSet(Lambda(x, x - 3), S.Integers).intersection(
S.Integers) is S.Integers
latex)
from sympy.sets.sets import FiniteSet, Interval
from sympy.sets.fancysets import ImageSet
t = Symbol('t')
T = Symbol(r'\mathbb{T}')
# Iterables
print(latex(T))
TS0 = tsc(S.EmptySet, latex(T) + '=' + latex(S.EmptySet))
TS1 = tsc(Range(-2, 1), latex(T) + '=' + latex(Range(-2, 1)))
TS2 = tsc(Range(-oo, 5, 5), latex(T) + '=' + latex(Range(-oo, 5, 5)))
TS3 = tsc(Range(0, oo, 7), latex(T) + '=' + latex(Range(0, oo, 7)))
TS4 = tsc(S.Naturals, latex(T) + '=' + latex(S.Naturals))
TS5 = tsc(S.Naturals0, latex(T) + '=' + latex(S.Naturals0))
TS6 = tsc(S.Integers, latex(T) + '=' + latex(S.Integers))
TS7 = tsc(ImageSet(Lambda(t, -t), S.Naturals),
latex(T) + '=' + latex(ImageSet(Lambda(t, -t), S.Naturals)))
# TODO: Create an instance of rational numbers in some subset like Interval(0,1s).
# TODO: Recorrer las fracciones de la siguiente manera:
# https://mathoverflow.net/questions/200656/is-there-a-natural-bijection-from-mathbbn-to-mathbbq
# Non-interables
TS9 = tsc(S.Reals, latex(T) + '=' + latex(S.Reals)) # Means Interval(-oo,+oo).
TS10 = tsc(S.UniversalSet, latex(T) + '=' + latex(S.UniversalSet))
# print(list(TS1.ts))
# TODO: Implementar raise
try:
iterator = iter(TS9.ts)
except TypeError: