Binary Relation

BinaryRelation can be used to store a mutable binary relation.

[1]:

from galactic.algebras.relational import BinaryRelation, SagittalDiagram

It can be initialized using an iterable of couples.

[2]:

relation = BinaryRelation[int, int]([(1, 2), (1, 3), (2, 3)])
SagittalDiagram(relation)

[2]:
../../../_images/notebooks_algebras_relational_notebook_3_0.png 
[3]:

list(relation)

[3]:

[(1, 2), (1, 3), (2, 3)]

Binary relation are considered as sets.

[4]:

len(relation)

[4]:

3

They have a domain and a co-domain also accessible by a tuple of universes.

[5]:

list(relation.domain)

[5]:

[1, 2]

[6]:

list(relation.universes[0])

[6]:

[1, 2]

[7]:

list(relation.co_domain)

[7]:

[2, 3]

[8]:

list(relation.universes[1])

[8]:

[2, 3]

[9]:

relation.add((2, 4))
SagittalDiagram(relation)

[9]:
../../../_images/notebooks_algebras_relational_notebook_12_0.png 
[10]:

list(relation)

[10]:

[(1, 2), (1, 3), (2, 3), (2, 4)]

[11]:

list(relation.domain)

[11]:

[1, 2]

[12]:

list(relation.co_domain)

[12]:

[2, 3, 4]

Successors and predecessors of an element are considered as sets. They are equivalent to a combination of a selection and a projection.

[13]:

relation.successors(1)

[13]:

<galactic.algebras.relational.Successors object at 0x7fd010080ec0>

[14]:

list(relation.successors(1))

[14]:

[2, 3]

[15]:

relation.successors(1).add(5)
SagittalDiagram(relation)

[15]:
../../../_images/notebooks_algebras_relational_notebook_19_0.png 
[16]:

list(relation)

[16]:

[(1, 2), (1, 3), (1, 5), (2, 3), (2, 4)]

[17]:

relation.successors(1).discard(3)
SagittalDiagram(relation)

[17]:
../../../_images/notebooks_algebras_relational_notebook_21_0.png 
[18]:

list(relation)

[18]:

[(1, 2), (1, 5), (2, 3), (2, 4)]

[19]:

list(relation.domain)

[19]:

[1, 2]

[20]:

list(relation.co_domain)

[20]:

[2, 3, 4, 5]

[21]:

relation.successors(2).clear()
SagittalDiagram(relation)

[21]:
../../../_images/notebooks_algebras_relational_notebook_25_0.png 
[22]:

list(relation)

[22]:

[(1, 2), (1, 5)]

[23]:

list(relation.predecessors(5))

[23]:

[1]

[24]:

predecessors = relation.predecessors(5)
predecessors |= {2, 3, 4}
SagittalDiagram(relation)

[24]:
../../../_images/notebooks_algebras_relational_notebook_28_0.png 
[25]:

list(relation)

[25]:

[(1, 2), (1, 5), (2, 5), (3, 5), (4, 5)]

[26]:

list(relation.domain)

[26]:

[1, 2, 3, 4]

[27]:

relation.domain.discard(1)
SagittalDiagram(relation)

[27]:
../../../_images/notebooks_algebras_relational_notebook_31_0.png 
[28]:

list(relation)

[28]:

[(2, 5), (3, 5), (4, 5)]