Examples of sets

class sage.categories.examples.sets_cat.PrimeNumbers

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

An example of parent in the category of sets: the set of prime numbers.

The elements are represented as plain integers in \(\ZZ\) (facade implementation).

This is a minimal implementations. For more advanced examples of implementations, see also:

sage: P = Sets().example("facade")
sage: P = Sets().example("inherits")
sage: P = Sets().example("wrapper")

EXAMPLES:

sage: P = Sets().example()
sage: P(12)
Traceback (most recent call last):
...
AssertionError: 12 is not a prime number
sage: a = P.an_element()
sage: a.parent()
Integer Ring
sage: x = P(13); x
13
sage: type(x)
<type 'sage.rings.integer.Integer'>
sage: x.parent()
Integer Ring
sage: 13 in P
True
sage: 12 in P
False
sage: y = x+1; y
14
sage: type(y)
<type 'sage.rings.integer.Integer'>

sage: TestSuite(P).run(verbose=True)
running ._test_an_element() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_new() . . . pass
  running ._test_nonzero_equal() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . pass
running ._test_eq() . . . pass
running ._test_new() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
running ._test_some_elements() . . . pass
an_element()

Implements Sets.ParentMethods.an_element().

element_class

alias of sage.rings.integer.Integer

class sage.categories.examples.sets_cat.PrimeNumbers_Abstract

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

This class shows how to write a parent while keeping the choice of the datastructure for the children open. Different class with fixed datastructure will then be constructed by inheriting from PrimeNumbers_Abstract.

This is used by:

sage: P = Sets().example(“facade”) sage: P = Sets().example(“inherits”) sage: P = Sets().example(“wrapper”)
class Element

Bases: sage.structure.element.Element

is_prime()

Return whether self is a prime number.

EXAMPLES:

sage: P = Sets().example("inherits")
sage: x = P.an_element()
sage: P.an_element().is_prime()
True
next()

Return the next prime number.

EXAMPLES:

sage: P = Sets().example("inherits")
sage: p = P.an_element(); p
47
sage: p.next()
53

Note

This method is not meant to implement the protocol iterator, and thus not subject to Python 2 vs Python 3 incompatibilities.

an_element()

Implements Sets.ParentMethods.an_element().

next(i)

Return the next prime number.

EXAMPLES:

sage: P = Sets().example("inherits")
sage: x = P.next(P.an_element()); x
53
sage: x.parent()
Set of prime numbers
some_elements()

Return some prime numbers.

EXAMPLES:

sage: P = Sets().example("inherits")
sage: P.some_elements()
[47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
class sage.categories.examples.sets_cat.PrimeNumbers_Facade

Bases: sage.categories.examples.sets_cat.PrimeNumbers_Abstract

An example of parent in the category of sets: the set of prime numbers.

In this alternative implementation, the elements are represented as plain integers in \(\ZZ\) (facade implementation).

EXAMPLES:

sage: P = Sets().example("facade")
sage: P(12)
Traceback (most recent call last):
...
ValueError: 12 is not a prime number
sage: a = P.an_element()
sage: a.parent()
Integer Ring
sage: x = P(13); x
13
sage: type(x)
<type 'sage.rings.integer.Integer'>
sage: x.parent()
Integer Ring
sage: 13 in P
True
sage: 12 in P
False
sage: y = x+1; y
14
sage: type(y)
<type 'sage.rings.integer.Integer'>

sage: z = P.next(x); z
17
sage: type(z)
<type 'sage.rings.integer.Integer'>
sage: z.parent()
Integer Ring

The disadvantage of this implementation is that the elements do not know that they are prime, so that prime testing is slow:

sage: pf = Sets().example("facade").an_element()
sage: timeit("pf.is_prime()") #    random
625 loops, best of 3: 4.1 us per loop

compared to the other implementations where prime testing is only done if needed during the construction of the element, and later on the elements “know” that they are prime:

sage: pw = Sets().example("wrapper").an_element()
sage: timeit("pw.is_prime()")    # random
625 loops, best of 3: 859 ns per loop

sage: pi = Sets().example("inherits").an_element()
sage: timeit("pw.is_prime()")    # random
625 loops, best of 3: 854 ns per loop

Note also that the next method for the elements does not exist:

sage: pf.next()
Traceback (most recent call last):
...
AttributeError: 'sage.rings.integer.Integer' object has no attribute 'next'

unlike in the other implementations:

sage: pw.next()
53
sage: pi.next()
53
element_class

alias of sage.rings.integer.Integer

class sage.categories.examples.sets_cat.PrimeNumbers_Inherits

Bases: sage.categories.examples.sets_cat.PrimeNumbers_Abstract

An example of parent in the category of sets: the set of prime numbers. In this implementation, the element are stored as object of a new class which inherits from the class Integer (technically IntegerWrapper).

EXAMPLES:

sage: P = Sets().example("inherits")
sage: P
Set of prime numbers
sage: P(12)
Traceback (most recent call last):
...
ValueError: 12 is not a prime number
sage: a = P.an_element()
sage: a.parent()
Set of prime numbers
sage: x = P(13); x
13
sage: x.is_prime()
True
sage: type(x)
<class 'sage.categories.examples.sets_cat.PrimeNumbers_Inherits_with_category.element_class'>
sage: x.parent()
Set of prime numbers
sage: P(13) in P
True
sage: y = x+1; y
14
sage: type(y)
<type 'sage.rings.integer.Integer'>
sage: y.parent()
Integer Ring
sage: type(P(13)+P(17))
<type 'sage.rings.integer.Integer'>
sage: type(P(2)+P(3))
<type 'sage.rings.integer.Integer'>

sage: z = P.next(x); z
17
sage: type(z)
<class 'sage.categories.examples.sets_cat.PrimeNumbers_Inherits_with_category.element_class'>
sage: z.parent()
Set of prime numbers

sage: TestSuite(P).run(verbose=True)
running ._test_an_element() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_new() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . pass
running ._test_eq() . . . pass
running ._test_new() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
running ._test_some_elements() . . . pass

See also:

sage: P = Sets().example("facade")
sage: P = Sets().example("inherits")
sage: P = Sets().example("wrapper")
class Element(parent, p)

Bases: sage.rings.integer.IntegerWrapper, sage.categories.examples.sets_cat.PrimeNumbers_Abstract.Element

class sage.categories.examples.sets_cat.PrimeNumbers_Wrapper

Bases: sage.categories.examples.sets_cat.PrimeNumbers_Abstract

An example of parent in the category of sets: the set of prime numbers.

In this second alternative implementation, the prime integer are stored as a attribute of a sage object by inheriting from ElementWrapper. In this case we need to ensure conversion and coercion from this parent and its element to ZZ and Integer.

EXAMPLES:

sage: P = Sets().example("wrapper")
sage: P(12)
Traceback (most recent call last):
...
ValueError: 12 is not a prime number
sage: a = P.an_element()
sage: a.parent()
Set of prime numbers (wrapper implementation)
sage: x = P(13); x
13
sage: type(x)
<class 'sage.categories.examples.sets_cat.PrimeNumbers_Wrapper_with_category.element_class'>
sage: x.parent()
Set of prime numbers (wrapper implementation)
sage: 13 in P
True
sage: 12 in P
False
sage: y = x+1; y
14
sage: type(y)
<type 'sage.rings.integer.Integer'>

sage: z = P.next(x); z
17
sage: type(z)
<class 'sage.categories.examples.sets_cat.PrimeNumbers_Wrapper_with_category.element_class'>
sage: z.parent()
Set of prime numbers (wrapper implementation)
class Element

Bases: sage.structure.element_wrapper.ElementWrapper, sage.categories.examples.sets_cat.PrimeNumbers_Abstract.Element

ElementWrapper

alias of sage.structure.element_wrapper.ElementWrapper