gsoc_trig_general_imageset_union_pr_11188.py

# master branch

In [1]: solveset(cos(x) + cos(3*x) + cos(5*x), x, S.Reals) 
Out[1]: 
⎧        π        ⎫   ⎧        3⋅π        ⎫   ⎧        4⋅π        ⎫   ⎧        2⋅π        ⎫   ⎧        5⋅π        ⎫   ⎧        π        ⎫   ⎧       
⎨2⋅n⋅π + ─ | n ∊ ℤ⎬ ∪ ⎨2⋅n⋅π + ─── | n ∊ ℤ⎬ ∪ ⎨2⋅n⋅π + ─── | n ∊ ℤ⎬ ∪ ⎨2⋅n⋅π + ─── | n ∊ ℤ⎬ ∪ ⎨2⋅n⋅π + ─── | n ∊ ℤ⎬ ∪ ⎨2⋅n⋅π + ─ | n ∊ ℤ⎬ ∪ ⎨2⋅n⋅π +
⎩        2        ⎭   ⎩         2         ⎭   ⎩         3         ⎭   ⎩         3         ⎭   ⎩         3         ⎭   ⎩        3        ⎭   ⎩       

 7⋅π        ⎫   ⎧        5⋅π        ⎫   ⎧        11⋅π        ⎫   ⎧        π        ⎫
 ─── | n ∊ ℤ⎬ ∪ ⎨2⋅n⋅π + ─── | n ∊ ℤ⎬ ∪ ⎨2⋅n⋅π + ──── | n ∊ ℤ⎬ ∪ ⎨2⋅n⋅π + ─ | n ∊ ℤ⎬
  6         ⎭   ⎩         6         ⎭   ⎩         6          ⎭   ⎩        6        ⎭

In [2]: print solveset(cos(x) + cos(3*x) + cos(5*x), x, S.Reals)
ImageSet(Lambda(_n, 2*_n*pi + pi/2), Integers()) U ImageSet(Lambda(_n, 2*_n*pi + 3*pi/2), Integers()) U ImageSet(Lambda(_n, 2*_n*pi + 4*pi/3), Integers()) U ImageSet(Lambda(_n, 2*_n*pi + 2*pi/3), Integers()) U ImageSet(Lambda(_n, 2*_n*pi + 5*pi/3), Integers()) U ImageSet(Lambda(_n, 2*_n*pi + pi/3), Integers()) U ImageSet(Lambda(_n, 2*_n*pi + 7*pi/6), Integers()) U ImageSet(Lambda(_n, 2*_n*pi + 5*pi/6), Integers()) U ImageSet(Lambda(_n, 2*_n*pi + 11*pi/6), Integers()) U ImageSet(Lambda(_n, 2*_n*pi + pi/6), Integers())

# soln is
img1 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)
img2 = ImageSet(Lambda(n, 2*n*pi + 3*pi/2), S.Integers)
img3 = ImageSet(Lambda(n, 2*n*pi + 4*pi/3), S.Integers)
img4 = ImageSet(Lambda(n, 2*n*pi + 2*pi/3), S.Integers)
img5 = ImageSet(Lambda(n, 2*n*pi + 5*pi/3), S.Integers)
img6 = ImageSet(Lambda(n, 2*n*pi + pi/3), S.Integers)
img7 = ImageSet(Lambda(n, 2*n*pi + 7*pi/6), S.Integers)
img8 = ImageSet(Lambda(n, 2*n*pi + 5*pi/6), S.Integers)
img9 = ImageSet(Lambda(n, 2*n*pi + 11*pi/6), S.Integers)
img10 = ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers)

#=====================================================================
# NEW IMPLEMENTATION (IMAGESET UNION IN SETS/FANCYSET.PY)
# PR : https://github.com/sympy/sympy/pull/11188
#----------------------------------------------------------
# img1 can be clubbed with img2
In [41]: Union(img1, img2)
Out[41]: 
⎧      π        ⎫
⎨π⋅n + ─ | n ∊ ℤ⎬
⎩      2        ⎭

newimg1 = ImageSet(Lambda(n, pi*n + pi/2), S.Integers)
#----------------------------------------------------------
# img8 can be clubbed with img9

In [45]: Union(img8, img9)
Out[45]: 
⎧      5⋅π        ⎫
⎨π⋅n + ─── | n ∊ ℤ⎬
⎩       6         ⎭

In [44]: print Union(img8, img9)
ImageSet(Lambda(n, pi*n + 5*pi/6), Integers())

newimg2 = ImageSet(Lambda(n, pi*n + 5*pi/6), S.Integers)
#----------------------------------------------------------
# img7 can be clubbed with img10
In [46]: Union(img7, img10)
Out[46]: 
⎧      π        ⎫
⎨π⋅n + ─ | n ∊ ℤ⎬
⎩      6        ⎭

In [47]: print Union(img7, img10)
ImageSet(Lambda(n, pi*n + pi/6), Integers())

newimg3 = ImageSet(Lambda(n, pi*n + pi/6), S.Integers)

#----------------------------------------------------------
# img3 can be clubbed with img6
In [48]: Union(img6, img3)
Out[48]: 
⎧      π        ⎫
⎨π⋅n + ─ | n ∊ ℤ⎬
⎩      3        ⎭

In [49]: print Union(img6, img3)
ImageSet(Lambda(n, pi*n + pi/3), Integers())

newimg4 = ImageSet(Lambda(n, pi*n + pi/3), S.Integers)

#----------------------------------------------------------
# img4 can be clubbed with img5
In [50]: print Union(img4, img5)
ImageSet(Lambda(n, pi*n + 2*pi/3), Integers())

In [51]: Union(img4, img5)
Out[51]: 
⎧      2⋅π        ⎫
⎨π⋅n + ─── | n ∊ ℤ⎬
⎩       3         ⎭

newimg5 = ImageSet(Lambda(n, pi*n + 2*pi/3), S.Integers)

#----------------------------------------------------------

# SO NOW REDUCED SOLN IS 
newimg1 = ImageSet(Lambda(n, pi*n + pi/2), S.Integers)
newimg2 = ImageSet(Lambda(n, pi*n + 5*pi/6), S.Integers)
newimg3 = ImageSet(Lambda(n, pi*n + pi/6), S.Integers)
newimg4 = ImageSet(Lambda(n, pi*n + pi/3), S.Integers)
newimg5 = ImageSet(Lambda(n, pi*n + 2*pi/3), S.Integers)

# we still can reduce these imageset

In [58]: Union(newimg5, newimg3)
Out[58]: 
⎧π⋅n   π        ⎫
⎨─── + ─ | n ∊ ℤ⎬
⎩ 2    6        ⎭

In [59]: print Union(newimg5, newimg3)
ImageSet(Lambda(n, pi*n/2 + pi/6), Integers())

final1 = ImageSet(Lambda(n, pi*n/2 + pi/6), S.Integers)
#------------------------------------

# similarly

In [60]: Union(newimg2, newimg4)
Out[60]: 
⎧π⋅n   π        ⎫
⎨─── + ─ | n ∊ ℤ⎬
⎩ 2    3        ⎭

In [61]: print Union(newimg2, newimg4)
ImageSet(Lambda(n, pi*n/2 + pi/3), Integers())

final2 = ImageSet(Lambda(n, pi*n/2 + pi/3), S.Integers)

#=================================
# So final reduce soln is = 

Union(final1, final2, newimg1)
⎧      π        ⎫   ⎧n⋅π   π        ⎫   ⎧n⋅π   π        ⎫
⎨n⋅π + ─ | n ∊ ℤ⎬ ∪ ⎨─── + ─ | n ∊ ℤ⎬ ∪ ⎨─── + ─ | n ∊ ℤ⎬
⎩      2        ⎭   ⎩ 2    6        ⎭   ⎩ 2    3        ⎭

#==================================

# One can check the answer using following lines
for i in range(0, 10):
....:     print newimg5.lamda(i) in final1
....:     print newimg3.lamda(i) in final1
....:     print newimg2.lamda(i) in final2
....:     print newimg4.lamda(i) in final2

# all true will be printed.