Transformations on sets/intervals/unions

[Ben Lucato]

Howdy, say I do:

domain = Interval(-pi, 0, True, True) + Interval(0, pi/2, True, True)

I could hope to do a transformation to dilate this domain by a factor 2 from the origin, like so:

new_domain = domain.replace(lambda expr: expr.is_Real, lambda expr: 2*expr)

OR:

new_domain = TransformationSet(Lambda(x, 2*x), domain)

in either case, checking:

3*pi/4 in new_domain

returns False.

I may be using TransformationSet incorrectly - I just looked in the docs and it looked like something relevant. Perhaps it only works for sets? 

[Matthew Rocklin]

This is what I get.  Can you verify that things don't work for you?  This was done in sympy/master.

In [1]: domain = Interval(-pi, 0, True, True) + Interval(0, pi/2, True, True)

In [2]: new_domain = TransformationSet(Lambda(x, 2*x), domain)

In [3]: domain

Out[3]: 

⎛   π⎞          

⎜0, ─⎟ ∪ (-π, 0)

⎝   2⎠          

In [4]: new_domain

Out[4]: 

⎧          ⎛   π⎞          ⎫

⎨2⋅x | x ∊ ⎜0, ─⎟ ∪ (-π, 0)⎬

⎩          ⎝   2⎠          ⎭

In [5]: 3*pi/4 in new_domain

Out[5]: True

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[Ben Lucato]

Thanks - I updated to the git master and now the TransformationSet works. Is there some way to take this new TransformationSet object and rewrite it as a Union/Interval?

i.e. 

get this:

Interval(-2*pi, 0, True, True) + Interval(0, pi, True, True)

[Matthew Rocklin]

At some point we might do a simplify sort of treatment for sets.

In the meantime something like the following could work in the case of monotonic functions.  In general we know how to transform Intervals and FiniteSets.  We know that unions and intersections will compose well with these transformations.  That's all you really need.

This should eventually be broken out into methods on the various classes:

def transform_set(x, expr, set):

    """ Transform a set by an expression

    >>> domain = Interval(-pi, 0, True, True) + Interval(0, pi/2, True, True)

    (0, pi/2) U (-pi, 0)

    >>> transform(x, 2*x, domain)

    (0, pi) U (-2*pi, 0)

    >>> transform(x, x**2, domain)

    (0, pi**2)

    """

    if isinstance(set, Union):

        return Union(transform_set(x, expr, arg) for arg in set.args)

    if isinstance(set, Intersection):

        return Intersection(transform_set(x, expr, arg) for arg in set.args)

    z = Dummy('z', real=True)

    f = Lambda(x, expr)

    if isinstance(set, Interval):

        # TODO: manage left_open and right_open better

        left, right = f(set.left), f(set.right)

        return Interval(Min(left, right), Max(left, right),

                        set.left_open, set.right_open)

    if isinstance(set, FiniteSet):

        return FiniteSet(map(f, set))

[Ben Lucato]

Awesome! You are a gentleman and a scholar, Matthew Rocklin.

Thanks for letting me know there is no way to do this in the current sympy.

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Ben Lucato

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[Matthew Rocklin]

Really I just like working on sets.  They're fairly simple and fun to play with.  Here is a pull request adding a `transform` method to sets

https://github.com/sympy/sympy/pull/2340

[Matthew Rocklin]

I'm renaming TransformationSet to ImageSet (I think this is a more precise name).  

https://github.com/sympy/sympy/pull/2341

2340 is in by the way but the interface will change shortly to 

>>> Interval(0, 1).image(x, 2*x)

[0, 2]

[Matthew Rocklin]

The API has changed in a recent PR

https://github.com/sympy/sympy/pull/2341

>>> imageset(x, 2*x, Interval(0, 1)

[0, 2]