modelparameters.sympy.sets package

Submodules

modelparameters.sympy.sets.conditionset module

class modelparameters.sympy.sets.conditionset.ConditionSet(sym, condition, base_set)[source]

Bases: Set

Set of elements which satisfies a given condition.

{x | condition(x) is True for x in S}

Examples

>>> from .. import Symbol, S, ConditionSet, Lambda, pi, Eq, sin, Interval
>>> x = Symbol('x')
>>> sin_sols = ConditionSet(x, Eq(sin(x), 0), Interval(0, 2*pi))
>>> 2*pi in sin_sols
True
>>> pi/2 in sin_sols
False
>>> 3*pi in sin_sols
False
>>> 5 in ConditionSet(x, x**2 > 4, S.Reals)
True
property base_set
property condition
contains(other)[source]

Returns True if ‘other’ is contained in ‘self’ as an element.

As a shortcut it is possible to use the ‘in’ operator:

Examples

>>> from .. import Interval
>>> Interval(0, 1).contains(0.5)
True
>>> 0.5 in Interval(0, 1)
True
default_assumptions = {}
property sym

modelparameters.sympy.sets.contains module

class modelparameters.sympy.sets.contains.Contains(x, S)[source]

Bases: BooleanFunction

Asserts that x is an element of the set S

Examples

>>> from .. import Symbol, Integer, S
>>> from .contains import Contains
>>> Contains(Integer(2), S.Integers)
True
>>> Contains(Integer(-2), S.Naturals)
False
>>> i = Symbol('i', integer=True)
>>> Contains(i, S.Naturals)
Contains(i, S.Naturals)

References

default_assumptions = {}
classmethod eval(x, S)[source]

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

modelparameters.sympy.sets.fancysets module

class modelparameters.sympy.sets.fancysets.ComplexRegion(sets, polar=False)[source]

Bases: Set

Represents the Set of all Complex Numbers. It can represent a region of Complex Plane in both the standard forms Polar and Rectangular coordinates.

  • Polar Form Input is in the form of the ProductSet or Union of ProductSets of the intervals of r and theta, & use the flag polar=True.

Z = {z in C | z = r*[cos(theta) + I*sin(theta)], r in [r], theta in [theta]}

  • Rectangular Form Input is in the form of the ProductSet or Union of ProductSets of interval of x and y the of the Complex numbers in a Plane. Default input type is in rectangular form.

Z = {z in C | z = x + I*y, x in [Re(z)], y in [Im(z)]}

Examples

>>> from .fancysets import ComplexRegion
>>> from ..sets import Interval
>>> from .. import S, I, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 6)
>>> c = Interval(1, 8)
>>> c1 = ComplexRegion(a*b)  # Rectangular Form
>>> c1
ComplexRegion(Interval(2, 3) x Interval(4, 6), False)

  • c1 represents the rectangular region in complex plane surrounded by the coordinates (2, 4), (3, 4), (3, 6) and (2, 6), of the four vertices.

>>> c2 = ComplexRegion(Union(a*b, b*c))
>>> c2
ComplexRegion(Union(Interval(2, 3) x Interval(4, 6), Interval(4, 6) x Interval(1, 8)), False)

  • c2 represents the Union of two rectangular regions in complex plane. One of them surrounded by the coordinates of c1 and other surrounded by the coordinates (4, 1), (6, 1), (6, 8) and (4, 8).

>>> 2.5 + 4.5*I in c1
True
>>> 2.5 + 6.5*I in c1
False

>>> r = Interval(0, 1)
>>> theta = Interval(0, 2*S.Pi)
>>> c2 = ComplexRegion(r*theta, polar=True)  # Polar Form
>>> c2  # unit Disk
ComplexRegion(Interval(0, 1) x Interval.Ropen(0, 2*pi), True)

  • c2 represents the region in complex plane inside the Unit Disk centered at the origin.

>>> 0.5 + 0.5*I in c2
True
>>> 1 + 2*I in c2
False

>>> unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True)
>>> upper_half_unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, S.Pi), polar=True)
>>> intersection = unit_disk.intersect(upper_half_unit_disk)
>>> intersection
ComplexRegion(Interval(0, 1) x Interval(0, pi), True)
>>> intersection == upper_half_unit_disk
True

See also

Reals

property a_interval

Return the union of intervals of x when, self is in rectangular form, or the union of intervals of r when self is in polar form.

Examples

>>> from .. import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.a_interval
Interval(2, 3)
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.a_interval
Union(Interval(2, 3), Interval(4, 5))
property args

Returns a tuple of arguments of ‘self’.

Examples

>>> from .. import cot
>>> from ..abc import x, y

>>> cot(x).args
(x,)

>>> cot(x).args[0]
x

>>> (x*y).args
(x, y)

>>> (x*y).args[1]
y

Notes

Never use self._args, always use self.args. Only use _args in __new__ when creating a new function. Don’t override .args() from Basic (so that it’s easy to change the interface in the future if needed).

property b_interval

Return the union of intervals of y when, self is in rectangular form, or the union of intervals of theta when self is in polar form.

Examples

>>> from .. import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.b_interval
Interval(4, 5)
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.b_interval
Interval(1, 7)
default_assumptions = {}
property expr
classmethod from_real(sets)[source]

Converts given subset of real numbers to a complex region.

Examples

>>> from .. import Interval, ComplexRegion
>>> unit = Interval(0,1)
>>> ComplexRegion.from_real(unit)
ComplexRegion(Interval(0, 1) x {0}, False)
is_ComplexRegion = True
property polar

Returns True if self is in polar form.

Examples

>>> from .. import Interval, ComplexRegion, Union, S
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> theta = Interval(0, 2*S.Pi)
>>> C1 = ComplexRegion(a*b)
>>> C1.polar
False
>>> C2 = ComplexRegion(a*theta, polar=True)
>>> C2.polar
True
property psets

Return a tuple of sets (ProductSets) input of the self.

Examples

>>> from .. import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.psets
(Interval(2, 3) x Interval(4, 5),)
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.psets
(Interval(2, 3) x Interval(4, 5), Interval(4, 5) x Interval(1, 7))
property sets

Return raw input sets to the self.

Examples

>>> from .. import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.sets
Interval(2, 3) x Interval(4, 5)
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.sets
Union(Interval(2, 3) x Interval(4, 5), Interval(4, 5) x Interval(1, 7))
property variables
class modelparameters.sympy.sets.fancysets.Complexes(*args, **kwargs)[source]

Bases: ComplexRegion

default_assumptions = {}
class modelparameters.sympy.sets.fancysets.ImageSet(lamda, base_set)[source]

Bases: Set

Image of a set under a mathematical function. The transformation must be given as a Lambda function which has as many arguments as the elements of the set upon which it operates, e.g. 1 argument when acting on the set of integers or 2 arguments when acting on a complex region.

This function is not normally called directly, but is called from imageset.

Examples

>>> from .. import Symbol, S, pi, Dummy, Lambda
>>> from .sets import FiniteSet, Interval
>>> from .fancysets import ImageSet

>>> x = Symbol('x')
>>> N = S.Naturals
>>> squares = ImageSet(Lambda(x, x**2), N) # {x**2 for x in N}
>>> 4 in squares
True
>>> 5 in squares
False

>>> FiniteSet(0, 1, 2, 3, 4, 5, 6, 7, 9, 10).intersect(squares)
{1, 4, 9}

>>> square_iterable = iter(squares)
>>> for i in range(4):
...     next(square_iterable)
1
4
9
16

If you want to get value for x = 2, 1/2 etc. (Please check whether the x value is in base_set or not before passing it as args)

>>> squares.lamda(2)
4
>>> squares.lamda(S(1)/2)
1/4

>>> n = Dummy('n')
>>> solutions = ImageSet(Lambda(n, n*pi), S.Integers) # solutions of sin(x) = 0
>>> dom = Interval(-1, 1)
>>> dom.intersect(solutions)
{0}

See also

sympy.sets.sets.imageset

property base_set
default_assumptions = {}
property is_iterable

bool(x) -> bool

Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.

property lamda
class modelparameters.sympy.sets.fancysets.Integers(*args, **kwargs)[source]

Bases: Set

Represents all integers: positive, negative and zero. This set is also available as the Singleton, S.Integers.

Examples

>>> from .. import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Integers)
>>> next(iterable)
0
>>> next(iterable)
1
>>> next(iterable)
-1
>>> next(iterable)
2

>>> pprint(S.Integers.intersect(Interval(-4, 4)))
{-4, -3, ..., 4}

See also

Naturals0

non-negative integers

Integers

positive and negative integers and zero

default_assumptions = {}
is_iterable = True
class modelparameters.sympy.sets.fancysets.Naturals(*args, **kwargs)[source]

Bases: Set

Represents the natural numbers (or counting numbers) which are all positive integers starting from 1. This set is also available as the Singleton, S.Naturals.

Examples

>>> from .. import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Naturals)
>>> next(iterable)
1
>>> next(iterable)
2
>>> next(iterable)
3
>>> pprint(S.Naturals.intersect(Interval(0, 10)))
{1, 2, ..., 10}

See also

Naturals0

non-negative integers (i.e. includes 0, too)

Integers

also includes negative integers

default_assumptions = {}
is_iterable = True
class modelparameters.sympy.sets.fancysets.Naturals0(*args, **kwargs)[source]

Bases: Naturals

Represents the whole numbers which are all the non-negative integers, inclusive of zero.

See also

Naturals

positive integers; does not include 0

Integers

also includes the negative integers

default_assumptions = {}
class modelparameters.sympy.sets.fancysets.Range(*args)[source]

Bases: Set

Represents a range of integers. Can be called as Range(stop), Range(start, stop), or Range(start, stop, step); when stop is not given it defaults to 1.

Range(stop) is the same as Range(0, stop, 1) and the stop value (juse as for Python ranges) is not included in the Range values.

>>> from .. import Range
>>> list(Range(3))
[0, 1, 2]

The step can also be negative:

>>> list(Range(10, 0, -2))
[10, 8, 6, 4, 2]

The stop value is made canonical so equivalent ranges always have the same args:

>>> Range(0, 10, 3)
Range(0, 12, 3)

Infinite ranges are allowed. If the starting point is infinite, then the final value is stop - step. To iterate such a range, it needs to be reversed:

>>> from .. import oo
>>> r = Range(-oo, 1)
>>> r[-1]
0
>>> next(iter(r))
Traceback (most recent call last):
...
ValueError: Cannot iterate over Range with infinite start
>>> next(iter(r.reversed))
0

Although Range is a set (and supports the normal set operations) it maintains the order of the elements and can be used in contexts where range would be used.

>>> from .. import Interval
>>> Range(0, 10, 2).intersect(Interval(3, 7))
Range(4, 8, 2)
>>> list(_)
[4, 6]

Athough slicing of a Range will always return a Range – possibly empty – an empty set will be returned from any intersection that is empty:

>>> Range(3)[:0]
Range(0, 0, 1)
>>> Range(3).intersect(Interval(4, oo))
EmptySet()
>>> Range(3).intersect(Range(4, oo))
EmptySet()
default_assumptions = {}
is_iterable = True
property reversed

Return an equivalent Range in the opposite order.

Examples

>>> from .. import Range
>>> Range(10).reversed
Range(9, -1, -1)
property size
property start
property step
property stop
class modelparameters.sympy.sets.fancysets.Reals(*args, **kwargs)[source]

Bases: Interval

default_assumptions = {}
modelparameters.sympy.sets.fancysets.normalize_theta_set(theta)[source]

Normalize a Real Set theta in the Interval [0, 2*pi). It returns a normalized value of theta in the Set. For Interval, a maximum of one cycle [0, 2*pi], is returned i.e. for theta equal to [0, 10*pi], returned normalized value would be [0, 2*pi). As of now intervals with end points as non-multiples of pi is not supported.

Raises:

  • NotImplementedError – The algorithms for Normalizing theta Set are not yet implemented.

  • ValueError – The input is not valid, i.e. the input is not a real set.

  • RuntimeError – It is a bug, please report to the github issue tracker.

Examples

>>> from .fancysets import normalize_theta_set
>>> from .. import Interval, FiniteSet, pi
>>> normalize_theta_set(Interval(9*pi/2, 5*pi))
Interval(pi/2, pi)
>>> normalize_theta_set(Interval(-3*pi/2, pi/2))
Interval.Ropen(0, 2*pi)
>>> normalize_theta_set(Interval(-pi/2, pi/2))
Union(Interval(0, pi/2), Interval.Ropen(3*pi/2, 2*pi))
>>> normalize_theta_set(Interval(-4*pi, 3*pi))
Interval.Ropen(0, 2*pi)
>>> normalize_theta_set(Interval(-3*pi/2, -pi/2))
Interval(pi/2, 3*pi/2)
>>> normalize_theta_set(FiniteSet(0, pi, 3*pi))
{0, pi}

modelparameters.sympy.sets.sets module

class modelparameters.sympy.sets.sets.Complement(a, b, evaluate=True)[source]

Bases: Set, EvalfMixin

Represents the set difference or relative complement of a set with another set.

A - B = {x in A| x \notin B}

Examples

>>> from .. import Complement, FiniteSet
>>> Complement(FiniteSet(0, 1, 2), FiniteSet(1))
{0, 2}

See also

Intersection, Union

References

default_assumptions = {}
is_Complement = True
static reduce(A, B)[source]

Simplify a Complement.

class modelparameters.sympy.sets.sets.EmptySet(*args, **kwargs)[source]

Bases: Set

Represents the empty set. The empty set is available as a singleton as S.EmptySet.

Examples

>>> from .. import S, Interval
>>> S.EmptySet
EmptySet()

>>> Interval(1, 2).intersect(S.EmptySet)
EmptySet()

See also

UniversalSet

References

as_relational(symbol)[source]
default_assumptions = {}
is_EmptySet = True
is_FiniteSet = True
class modelparameters.sympy.sets.sets.FiniteSet(*args, **kwargs)[source]

Bases: Set, EvalfMixin

Represents a finite set of discrete numbers

Examples

>>> from .. import FiniteSet
>>> FiniteSet(1, 2, 3, 4)
{1, 2, 3, 4}
>>> 3 in FiniteSet(1, 2, 3, 4)
True

>>> members = [1, 2, 3, 4]
>>> f = FiniteSet(*members)
>>> f
{1, 2, 3, 4}
>>> f - FiniteSet(2)
{1, 3, 4}
>>> f + FiniteSet(2, 5)
{1, 2, 3, 4, 5}

References

as_relational(symbol)[source]

Rewrite a FiniteSet in terms of equalities and logic operators.

compare(other)[source]

Return -1, 0, 1 if the object is smaller, equal, or greater than other.

Not in the mathematical sense. If the object is of a different type from the “other” then their classes are ordered according to the sorted_classes list.

Examples

>>> from ..abc import x, y
>>> x.compare(y)
-1
>>> x.compare(x)
0
>>> y.compare(x)
1
default_assumptions = {}
is_FiniteSet = True
is_iterable = True
property measure

The (Lebesgue) measure of ‘self’

Examples

>>> from .. import Interval, Union
>>> Interval(0, 1).measure
1
>>> Union(Interval(0, 1), Interval(2, 3)).measure
2
class modelparameters.sympy.sets.sets.Intersection(*args, **kwargs)[source]

Bases: Set

Represents an intersection of sets as a Set.

Examples

>>> from .. import Intersection, Interval
>>> Intersection(Interval(1, 3), Interval(2, 4))
Interval(2, 3)

We often use the .intersect method

>>> Interval(1,3).intersect(Interval(2,4))
Interval(2, 3)

See also

Union

References

as_relational(symbol)[source]

Rewrite an Intersection in terms of equalities and logic operators

default_assumptions = {}
is_Intersection = True
property is_iterable

bool(x) -> bool

Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.

static reduce(args)[source]

Return a simplified intersection by applying rules.

We first start with global rules like ‘if any empty sets, return empty set’ and ‘distribute unions’.

Then we iterate through all pairs and ask the constituent sets if they can simplify themselves with any other constituent

class modelparameters.sympy.sets.sets.Interval(start, end, left_open=False, right_open=False)[source]

Bases: Set, EvalfMixin

Represents a real interval as a Set.

Usage:

Returns an interval with end points “start” and “end”.

For left_open=True (default left_open is False) the interval will be open on the left. Similarly, for right_open=True the interval will be open on the right.

Examples

>>> from .. import Symbol, Interval
>>> Interval(0, 1)
Interval(0, 1)
>>> Interval.Ropen(0, 1)
Interval.Ropen(0, 1)
>>> Interval.Ropen(0, 1)
Interval.Ropen(0, 1)
>>> Interval.Lopen(0, 1)
Interval.Lopen(0, 1)
>>> Interval.open(0, 1)
Interval.open(0, 1)

>>> a = Symbol('a', real=True)
>>> Interval(0, a)
Interval(0, a)

Notes

  • Only real end points are supported

  • Interval(a, b) with a > b will return the empty set

  • Use the evalf() method to turn an Interval into an mpmath ‘mpi’ interval instance

References

classmethod Lopen(a, b)[source]

Return an interval not including the left boundary.

classmethod Ropen(a, b)[source]

Return an interval not including the right boundary.

as_relational(x)[source]

Rewrite an interval in terms of inequalities and logic operators.

default_assumptions = {}
property end

The right end point of ‘self’.

This property takes the same value as the ‘sup’ property.

Examples

>>> from .. import Interval
>>> Interval(0, 1).end
1
is_Interval = True
property is_left_unbounded

Return True if the left endpoint is negative infinity.

property is_right_unbounded

Return True if the right endpoint is positive infinity.

property left

The left end point of ‘self’.

This property takes the same value as the ‘inf’ property.

Examples

>>> from .. import Interval
>>> Interval(0, 1).start
0
property left_open

True if ‘self’ is left-open.

Examples

>>> from .. import Interval
>>> Interval(0, 1, left_open=True).left_open
True
>>> Interval(0, 1, left_open=False).left_open
False
classmethod open(a, b)[source]

Return an interval including neither boundary.

property right

The right end point of ‘self’.

This property takes the same value as the ‘sup’ property.

Examples

>>> from .. import Interval
>>> Interval(0, 1).end
1
property right_open

True if ‘self’ is right-open.

Examples

>>> from .. import Interval
>>> Interval(0, 1, right_open=True).right_open
True
>>> Interval(0, 1, right_open=False).right_open
False
property start

The left end point of ‘self’.

This property takes the same value as the ‘inf’ property.

Examples

>>> from .. import Interval
>>> Interval(0, 1).start
0
to_mpi(prec=53)[source]
class modelparameters.sympy.sets.sets.ProductSet(*sets, **assumptions)[source]

Bases: Set

Represents a Cartesian Product of Sets.

Returns a Cartesian product given several sets as either an iterable or individual arguments.

Can use ‘*’ operator on any sets for convenient shorthand.

Examples

>>> from .. import Interval, FiniteSet, ProductSet
>>> I = Interval(0, 5); S = FiniteSet(1, 2, 3)
>>> ProductSet(I, S)
Interval(0, 5) x {1, 2, 3}

>>> (2, 2) in ProductSet(I, S)
True

>>> Interval(0, 1) * Interval(0, 1) # The unit square
Interval(0, 1) x Interval(0, 1)

>>> coin = FiniteSet('H', 'T')
>>> set(coin**2)
{(H, H), (H, T), (T, H), (T, T)}

Notes

  • Passes most operations down to the argument sets

  • Flattens Products of ProductSets

References

default_assumptions = {}
is_ProductSet = True
property is_iterable

A property method which tests whether a set is iterable or not. Returns True if set is iterable, otherwise returns False.

Examples

>>> from .. import FiniteSet, Interval, ProductSet
>>> I = Interval(0, 1)
>>> A = FiniteSet(1, 2, 3, 4, 5)
>>> I.is_iterable
False
>>> A.is_iterable
True
property sets
class modelparameters.sympy.sets.sets.Set(*args)[source]

Bases: Basic

The base class for any kind of set.

This is not meant to be used directly as a container of items. It does not behave like the builtin set; see FiniteSet for that.

Real intervals are represented by the Interval class and unions of sets by the Union class. The empty set is represented by the EmptySet class and available as a singleton as S.EmptySet.

property boundary

The boundary or frontier of a set

A point x is on the boundary of a set S if

  1. x is in the closure of S. I.e. Every neighborhood of x contains a point in S.

  2. x is not in the interior of S. I.e. There does not exist an open set centered on x contained entirely within S.

There are the points on the outer rim of S. If S is open then these points need not actually be contained within S.

For example, the boundary of an interval is its start and end points. This is true regardless of whether or not the interval is open.

Examples

>>> from .. import Interval
>>> Interval(0, 1).boundary
{0, 1}
>>> Interval(0, 1, True, False).boundary
{0, 1}
property closure

Property method which returns the closure of a set. The closure is defined as the union of the set itself and its boundary.

Examples

>>> from .. import S, Interval
>>> S.Reals.closure
S.Reals
>>> Interval(0, 1).closure
Interval(0, 1)
complement(universe)[source]

The complement of ‘self’ w.r.t the given the universe.

Examples

>>> from .. import Interval, S
>>> Interval(0, 1).complement(S.Reals)
Union(Interval.open(-oo, 0), Interval.open(1, oo))

>>> Interval(0, 1).complement(S.UniversalSet)
UniversalSet() \ Interval(0, 1)
contains(other)[source]

Returns True if ‘other’ is contained in ‘self’ as an element.

As a shortcut it is possible to use the ‘in’ operator:

Examples

>>> from .. import Interval
>>> Interval(0, 1).contains(0.5)
True
>>> 0.5 in Interval(0, 1)
True
default_assumptions = {}
property inf

The infimum of ‘self’

Examples

>>> from .. import Interval, Union
>>> Interval(0, 1).inf
0
>>> Union(Interval(0, 1), Interval(2, 3)).inf
0
property interior

Property method which returns the interior of a set. The interior of a set S consists all points of S that do not belong to the boundary of S.

Examples

>>> from .. import Interval
>>> Interval(0, 1).interior
Interval.open(0, 1)
>>> Interval(0, 1).boundary.interior
EmptySet()
intersect(other)[source]

Returns the intersection of ‘self’ and ‘other’.

>>> from .. import Interval

>>> Interval(1, 3).intersect(Interval(1, 2))
Interval(1, 2)

>>> from .. import imageset, Lambda, symbols, S
>>> n, m = symbols('n m')
>>> a = imageset(Lambda(n, 2*n), S.Integers)
>>> a.intersect(imageset(Lambda(m, 2*m + 1), S.Integers))
EmptySet()
intersection(other)[source]

Alias for intersect()

is_Complement = None
is_ComplexRegion = False
is_EmptySet = None
is_FiniteSet = False
is_Intersection = None
is_Interval = False
is_ProductSet = False
is_Union = False
is_UniversalSet = None
property is_closed

A property method to check whether a set is closed. A set is closed if it’s complement is an open set.

Examples

>>> from .. import Interval
>>> Interval(0, 1).is_closed
True
is_disjoint(other)[source]

Returns True if ‘self’ and ‘other’ are disjoint

Examples

>>> from .. import Interval
>>> Interval(0, 2).is_disjoint(Interval(1, 2))
False
>>> Interval(0, 2).is_disjoint(Interval(3, 4))
True

References

is_interval = False
is_iterable = False
is_number = False
property is_open

Property method to check whether a set is open. A set is open if and only if it has an empty intersection with its boundary.

Examples

>>> from .. import S
>>> S.Reals.is_open
True
is_proper_subset(other)[source]

Returns True if ‘self’ is a proper subset of ‘other’.

Examples

>>> from .. import Interval
>>> Interval(0, 0.5).is_proper_subset(Interval(0, 1))
True
>>> Interval(0, 1).is_proper_subset(Interval(0, 1))
False
is_proper_superset(other)[source]

Returns True if ‘self’ is a proper superset of ‘other’.

Examples

>>> from .. import Interval
>>> Interval(0, 1).is_proper_superset(Interval(0, 0.5))
True
>>> Interval(0, 1).is_proper_superset(Interval(0, 1))
False
is_subset(other)[source]

Returns True if ‘self’ is a subset of ‘other’.

Examples

>>> from .. import Interval
>>> Interval(0, 0.5).is_subset(Interval(0, 1))
True
>>> Interval(0, 1).is_subset(Interval(0, 1, left_open=True))
False
is_superset(other)[source]

Returns True if ‘self’ is a superset of ‘other’.

Examples

>>> from .. import Interval
>>> Interval(0, 0.5).is_superset(Interval(0, 1))
False
>>> Interval(0, 1).is_superset(Interval(0, 1, left_open=True))
True
isdisjoint(other)[source]

Alias for is_disjoint()

issubset(other)[source]

Alias for is_subset()

issuperset(other)[source]

Alias for is_superset()

property measure

The (Lebesgue) measure of ‘self’

Examples

>>> from .. import Interval, Union
>>> Interval(0, 1).measure
1
>>> Union(Interval(0, 1), Interval(2, 3)).measure
2
powerset()[source]

Find the Power set of ‘self’.

Examples

>>> from .. import FiniteSet, EmptySet
>>> A = EmptySet()
>>> A.powerset()
{EmptySet()}
>>> A = FiniteSet(1, 2)
>>> a, b, c = FiniteSet(1), FiniteSet(2), FiniteSet(1, 2)
>>> A.powerset() == FiniteSet(a, b, c, EmptySet())
True

References

property sup

The supremum of ‘self’

Examples

>>> from .. import Interval, Union
>>> Interval(0, 1).sup
1
>>> Union(Interval(0, 1), Interval(2, 3)).sup
3
symmetric_difference(other)[source]

Returns symmetric difference of self and other.

Examples

>>> from .. import Interval, S
>>> Interval(1, 3).symmetric_difference(S.Reals)
Union(Interval.open(-oo, 1), Interval.open(3, oo))
>>> Interval(1, 10).symmetric_difference(S.Reals)
Union(Interval.open(-oo, 1), Interval.open(10, oo))

>>> from .. import S, EmptySet
>>> S.Reals.symmetric_difference(EmptySet())
S.Reals

References

union(other)[source]

Returns the union of ‘self’ and ‘other’.

Examples

As a shortcut it is possible to use the ‘+’ operator:

>>> from .. import Interval, FiniteSet
>>> Interval(0, 1).union(Interval(2, 3))
Union(Interval(0, 1), Interval(2, 3))
>>> Interval(0, 1) + Interval(2, 3)
Union(Interval(0, 1), Interval(2, 3))
>>> Interval(1, 2, True, True) + FiniteSet(2, 3)
Union(Interval.Lopen(1, 2), {3})

Similarly it is possible to use the ‘-’ operator for set differences:

>>> Interval(0, 2) - Interval(0, 1)
Interval.Lopen(1, 2)
>>> Interval(1, 3) - FiniteSet(2)
Union(Interval.Ropen(1, 2), Interval.Lopen(2, 3))
class modelparameters.sympy.sets.sets.SymmetricDifference(a, b, evaluate=True)[source]

Bases: Set

Represents the set of elements which are in either of the sets and not in their intersection.

Examples

>>> from .. import SymmetricDifference, FiniteSet
>>> SymmetricDifference(FiniteSet(1, 2, 3), FiniteSet(3, 4, 5))
{1, 2, 4, 5}

See also

Complement, Union

References

default_assumptions = {}
is_SymmetricDifference = True
static reduce(A, B)[source]
class modelparameters.sympy.sets.sets.Union(*args, **kwargs)[source]

Bases: Set, EvalfMixin

Represents a union of sets as a Set.

Examples

>>> from .. import Union, Interval
>>> Union(Interval(1, 2), Interval(3, 4))
Union(Interval(1, 2), Interval(3, 4))

The Union constructor will always try to merge overlapping intervals, if possible. For example:

>>> Union(Interval(1, 2), Interval(2, 3))
Interval(1, 3)

See also

Intersection

References

as_relational(symbol)[source]

Rewrite a Union in terms of equalities and logic operators.

default_assumptions = {}
is_Union = True
property is_iterable

bool(x) -> bool

Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.

static reduce(args)[source]

Simplify a Union using known rules

We first start with global rules like ‘Merge all FiniteSets’

Then we iterate through all pairs and ask the constituent sets if they can simplify themselves with any other constituent

class modelparameters.sympy.sets.sets.UniversalSet(*args, **kwargs)[source]

Bases: Set

Represents the set of all things. The universal set is available as a singleton as S.UniversalSet

Examples

>>> from .. import S, Interval
>>> S.UniversalSet
UniversalSet()

>>> Interval(1, 2).intersect(S.UniversalSet)
Interval(1, 2)

See also

EmptySet

References

as_relational(symbol)[source]
default_assumptions = {}
is_UniversalSet = True
modelparameters.sympy.sets.sets.imageset(*args)[source]

Return an image of the set under transformation f.

If this function can’t compute the image, it returns an unevaluated ImageSet object.

\[{ f(x) | x \in self }\]

Examples

>>> from .. import S, Interval, Symbol, imageset, sin, Lambda
>>> from ..abc import x, y

>>> imageset(x, 2*x, Interval(0, 2))
Interval(0, 4)

>>> imageset(lambda x: 2*x, Interval(0, 2))
Interval(0, 4)

>>> imageset(Lambda(x, sin(x)), Interval(-2, 1))
ImageSet(Lambda(x, sin(x)), Interval(-2, 1))

>>> imageset(sin, Interval(-2, 1))
ImageSet(Lambda(x, sin(x)), Interval(-2, 1))
>>> imageset(lambda y: x + y, Interval(-2, 1))
ImageSet(Lambda(_x, _x + x), Interval(-2, 1))

Expressions applied to the set of Integers are simplified to show as few negatives as possible and linear expressions are converted to a canonical form. If this is not desirable then the unevaluated ImageSet should be used.

>>> imageset(x, -2*x + 5, S.Integers)
ImageSet(Lambda(x, 2*x + 1), S.Integers)

See also

sympy.sets.fancysets.ImageSet

Module contents