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 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_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 Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.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)).intersection(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersection(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
def test_halfcircle():
r, th = symbols('r, theta', extended_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 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', extended_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 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_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_fun():
assert (FiniteSet(*ImageSet(Lambda(x, sin(pi*x/4)), Range(-10, 11))) ==
FiniteSet(-1, -sqrt(2)/2, 0, sqrt(2)/2, 1))
def test_ImageSet_iterator_not_injetive():
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)
