| from sympy.core.decorators import _sympifyit |
| from sympy.core.parameters import global_parameters |
| from sympy.core.logic import fuzzy_bool |
| from sympy.core.singleton import S |
| from sympy.core.sympify import _sympify |
|
|
| from .sets import Set, FiniteSet, SetKind |
|
|
|
|
| class PowerSet(Set): |
| r"""A symbolic object representing a power set. |
| |
| Parameters |
| ========== |
| |
| arg : Set |
| The set to take power of. |
| |
| evaluate : bool |
| The flag to control evaluation. |
| |
| If the evaluation is disabled for finite sets, it can take |
| advantage of using subset test as a membership test. |
| |
| Notes |
| ===== |
| |
| Power set `\mathcal{P}(S)` is defined as a set containing all the |
| subsets of `S`. |
| |
| If the set `S` is a finite set, its power set would have |
| `2^{\left| S \right|}` elements, where `\left| S \right|` denotes |
| the cardinality of `S`. |
| |
| Examples |
| ======== |
| |
| >>> from sympy import PowerSet, S, FiniteSet |
| |
| A power set of a finite set: |
| |
| >>> PowerSet(FiniteSet(1, 2, 3)) |
| PowerSet({1, 2, 3}) |
| |
| A power set of an empty set: |
| |
| >>> PowerSet(S.EmptySet) |
| PowerSet(EmptySet) |
| >>> PowerSet(PowerSet(S.EmptySet)) |
| PowerSet(PowerSet(EmptySet)) |
| |
| A power set of an infinite set: |
| |
| >>> PowerSet(S.Reals) |
| PowerSet(Reals) |
| |
| Evaluating the power set of a finite set to its explicit form: |
| |
| >>> PowerSet(FiniteSet(1, 2, 3)).rewrite(FiniteSet) |
| FiniteSet(EmptySet, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}) |
| |
| References |
| ========== |
| |
| .. [1] https://en.wikipedia.org/wiki/Power_set |
| |
| .. [2] https://en.wikipedia.org/wiki/Axiom_of_power_set |
| """ |
| def __new__(cls, arg, evaluate=None): |
| if evaluate is None: |
| evaluate=global_parameters.evaluate |
|
|
| arg = _sympify(arg) |
|
|
| if not isinstance(arg, Set): |
| raise ValueError('{} must be a set.'.format(arg)) |
|
|
| return super().__new__(cls, arg) |
|
|
| @property |
| def arg(self): |
| return self.args[0] |
|
|
| def _eval_rewrite_as_FiniteSet(self, *args, **kwargs): |
| arg = self.arg |
| if arg.is_FiniteSet: |
| return arg.powerset() |
| return None |
|
|
| @_sympifyit('other', NotImplemented) |
| def _contains(self, other): |
| if not isinstance(other, Set): |
| return None |
|
|
| return fuzzy_bool(self.arg.is_superset(other)) |
|
|
| def _eval_is_subset(self, other): |
| if isinstance(other, PowerSet): |
| return self.arg.is_subset(other.arg) |
|
|
| def __len__(self): |
| return 2 ** len(self.arg) |
|
|
| def __iter__(self): |
| found = [S.EmptySet] |
| yield S.EmptySet |
|
|
| for x in self.arg: |
| temp = [] |
| x = FiniteSet(x) |
| for y in found: |
| new = x + y |
| yield new |
| temp.append(new) |
| found.extend(temp) |
|
|
| @property |
| def kind(self): |
| return SetKind(self.arg.kind) |
|
|
