Examples of sets#

class sage.categories.examples.sets_cat.PrimeNumbers[source]#

Bases: UniqueRepresentation, 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")

>>> from sage.all import *
>>> P = Sets().example("facade")
>>> P = Sets().example("inherits")
>>> 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)
<class '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)
<class '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_construction() . . . 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

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

>>> TestSuite(P).run(verbose=True)
running ._test_an_element() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_construction() . . . 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()[source]#

Implements Sets.ParentMethods.an_element().

element_class[source]#

alias of Integer

class sage.categories.examples.sets_cat.PrimeNumbers_Abstract[source]#

Bases: UniqueRepresentation, 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")

>>> from sage.all import *
>>> P = Sets().example("facade")
>>> P = Sets().example("inherits")
>>> P = Sets().example("wrapper")
class Element[source]#

Bases: Element

is_prime()[source]#

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

>>> from sage.all import *
>>> P = Sets().example("inherits")
>>> x = P.an_element()
>>> P.an_element().is_prime()
True
next()[source]#

Return the next prime number.

EXAMPLES:

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

>>> from sage.all import *
>>> P = Sets().example("inherits")
>>> p = P.an_element(); p
47
>>> 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()[source]#

Implements Sets.ParentMethods.an_element().

next(i)[source]#

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

>>> from sage.all import *
>>> P = Sets().example("inherits")
>>> x = P.next(P.an_element()); x
53
>>> x.parent()
Set of prime numbers
some_elements()[source]#

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]

>>> from sage.all import *
>>> P = Sets().example("inherits")
>>> P.some_elements()
[47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
class sage.categories.examples.sets_cat.PrimeNumbers_Facade[source]#

Bases: 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)
<class '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)
<class 'sage.rings.integer.Integer'>

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

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

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

>>> from sage.all import *
>>> pf = Sets().example("facade").an_element()
>>> 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

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

>>> pi = Sets().example("inherits").an_element()
>>> 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'...

>>> from sage.all import *
>>> 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

>>> from sage.all import *
>>> pw.next()
53
>>> pi.next()
53
element_class[source]#

alias of Integer

class sage.categories.examples.sets_cat.PrimeNumbers_Inherits[source]#

Bases: 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)
<class 'sage.rings.integer.Integer'>
sage: y.parent()
Integer Ring
sage: type(P(13)+P(17))
<class 'sage.rings.integer.Integer'>
sage: type(P(2)+P(3))
<class '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_construction() . . . 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

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

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

>>> TestSuite(P).run(verbose=True)
running ._test_an_element() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_construction() . . . 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")

>>> from sage.all import *
>>> P = Sets().example("facade")
>>> P = Sets().example("inherits")
>>> P = Sets().example("wrapper")
class Element(parent, p)[source]#

Bases: IntegerWrapper, Element

class sage.categories.examples.sets_cat.PrimeNumbers_Wrapper[source]#

Bases: 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)
<class '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)

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

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

Bases: ElementWrapper, Element

ElementWrapper[source]#

alias of ElementWrapper