Non Negative Integers#

class sage.sets.non_negative_integers.NonNegativeIntegers(category=None)[source]#

Bases: UniqueRepresentation, Parent

The enumerated set of non negative integers.

This class implements the set of non negative integers, as an enumerated set (see InfiniteEnumeratedSets).

EXAMPLES:

sage: NN = NonNegativeIntegers()
sage: NN
Non negative integers
sage: NN.cardinality()
+Infinity
sage: TestSuite(NN).run()
sage: NN.list()
Traceback (most recent call last):
...
NotImplementedError: cannot list an infinite set
sage: NN.element_class
<... 'sage.rings.integer.Integer'>
sage: it = iter(NN)
sage: [next(it), next(it), next(it), next(it), next(it)]
[0, 1, 2, 3, 4]
sage: NN.first()
0

>>> from sage.all import *
>>> NN = NonNegativeIntegers()
>>> NN
Non negative integers
>>> NN.cardinality()
+Infinity
>>> TestSuite(NN).run()
>>> NN.list()
Traceback (most recent call last):
...
NotImplementedError: cannot list an infinite set
>>> NN.element_class
<... 'sage.rings.integer.Integer'>
>>> it = iter(NN)
>>> [next(it), next(it), next(it), next(it), next(it)]
[0, 1, 2, 3, 4]
>>> NN.first()
0

Currently, this is just a “facade” parent; namely its elements are plain Sage Integers with Integer Ring as parent:

sage: x = NN(15); type(x)
<... 'sage.rings.integer.Integer'>
sage: x.parent()
Integer Ring
sage: x+3
18

>>> from sage.all import *
>>> x = NN(Integer(15)); type(x)
<... 'sage.rings.integer.Integer'>
>>> x.parent()
Integer Ring
>>> x+Integer(3)
18

In a later version, there will be an option to specify whether the elements should have Integer Ring or Non negative integers as parent:

sage: NN = NonNegativeIntegers(facade = False) # todo: not implemented
sage: x = NN(5)                                # todo: not implemented
sage: x.parent()                               # todo: not implemented
Non negative integers

>>> from sage.all import *
>>> NN = NonNegativeIntegers(facade = False) # todo: not implemented
>>> x = NN(Integer(5))                                # todo: not implemented
>>> x.parent()                               # todo: not implemented
Non negative integers

This runs generic sanity checks on NN:

sage: TestSuite(NN).run()

>>> from sage.all import *
>>> TestSuite(NN).run()

TODO: do not use NN any more in the doctests for NonNegativeIntegers.

Element[source]#

alias of Integer

an_element()[source]#

EXAMPLES:

sage: NonNegativeIntegers().an_element()
42

>>> from sage.all import *
>>> NonNegativeIntegers().an_element()
42
from_integer[source]#

alias of Integer

next(o)[source]#

EXAMPLES:

sage: NN = NonNegativeIntegers()
sage: NN.next(3)
4

>>> from sage.all import *
>>> NN = NonNegativeIntegers()
>>> NN.next(Integer(3))
4
some_elements()[source]#

EXAMPLES:

sage: NonNegativeIntegers().some_elements()
[0, 1, 3, 42]

>>> from sage.all import *
>>> NonNegativeIntegers().some_elements()
[0, 1, 3, 42]
unrank(rnk)[source]#

EXAMPLES:

sage: NN = NonNegativeIntegers()
sage: NN.unrank(100)
100

>>> from sage.all import *
>>> NN = NonNegativeIntegers()
>>> NN.unrank(Integer(100))
100