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