eliminate() continue:

PR : #11485

Regarding issue #2720, It is something similar to
eliminate
present in wolfram. 
Right now it is working for real domain and not considering ImageSet, Intersection. Complement. If it find ImageSet, Intersection. Complement for the symbol to be eliminated, then just raises
NotImaplementedError
(in near future it can be implemented). 
It can take
N
nummber of equations andM
number ofsymbols
to be eliminated. It returnsFiniteSet
of equations which doesn’t contains thesesymbols
.
Continue  Diophantine in Solveset :
PR 11234
 In commit : returning
ConditionSet
when itdiophantine
doesn’t the diophantine equation type and not able to solve.
Idea for improved _solve_trig in Solveset :
In [1]: solveset(sin(x) + cos(y), x, S.Reals)
Out[1]: {x  x ∊ ℝ ∧ sin(x) + cos(y) = 0}
In [2]: solve(sin(x) + cos(y), x)
Out[2]: [asin(cos(y)) + π, asin(cos(y))]

This above examples is enough to tell that
_solve_trig
is not using inverse trigonometric function. We can have something, which can solve trig equations by making free the symbol in lhs and in rhs inverse trig function. 
solveset(sin(2*x)  sqrt(3)*cos(2*x), x, S.Reals)
for this right now_solve_trig
is converting it intoexp
form and solving it. But it is can be simply solved usingsin(x + y) == sin(x)*cos(y) + cos(x)*sin(y)
formula. 
First divide both side with
sqrt(a**2 + b**2)
wherea, b
is coeff ofsin(2*x)
,cos(2*x)
. 
sin(2*x)/2  (sqrt(3)/2)*cos(2*x)
==>sin(2*x)*cos(pi/3)  sin(pi/3)*cos(2*x)
==>sin(2*x  pi/3)
. 
Now
sin(2*x  pi/3)
is solvable usingsolve_decomposition
.
Meanwhile :
 Some analysis about
Abs
solver :
In [1]: solveset(Abs(x)  1, x)
ValueError:
Absolute values cannot be inverted in the complex domain.
In [2]: solveset(Abs(x)  (1 + I), x)
ValueError:
Absolute values cannot be inverted in the complex domain.
In [3]: solveset(Abs(x)  y , x)
ValueError:
Absolute values cannot be inverted in the complex domain.  In 1st case (for complex domain) ans should be http://www.wolframalpha.com/input/?i=Abs(x)++(1+)+%3D0+for+x

In 2nd case EmptySet. and in 3rd case (general solution when domain=S.Complexes) soln should be http://www.wolframalpha.com/input/?i=Abs(x)++y+%3D0+for+x

In general( 3rd case) it should print ConditionSet(x, Abs(re(y)) <= re(x) and re(x) <= Abs(re(y)) and re(y)>0, Eq(im(y), 0) , S.Complexes).
continue..
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