Transitive relation
- Author:
- jupa2011
Shows a transitive relation in a Cartesian plane. Also can show a degenerate/vacuous transitive relation.
Transitive relation is given as:
ordered pair A (x,y)
ordered pair B (y,z)
--------
ordered pair C (x,z)
Usually is not mentioned that x,y,z need not be different, aka.
they can be same, like:
ordered pair A (x,y)
ordered pair B (y,y)
--------
ordered pair C (x,y)
or even:
ordered pair A (x,x)
ordered pair B (x,x)
--------
ordered pair C (x,x)
These degenerate transitive relations are explained that they are like
a geometric 2D point is degenerate geometric 2D circle, https://en.wikipedia.org/wiki/Degeneracy_(mathematics)
or
A and B => C
A is false
------
C is vacuously true https://en.wikipedia.org/wiki/Vacuous_truth
Geometric point, is that really a geometric circle?
Degenerated transitive relation, is that really a transitive relation?
Wikipedia says that a 1 element set has 2 transitive relations https://en.wikipedia.org/wiki/Transitive_relation#Counting_transitive_relations
Even an 0 element set has 1 transitive relation!???
(I am not sure how transitive relation on a empty set is proved or explained)
This looks really weird and ugly, but if x,y,z can be same then this is possible:
ordered pair A ({},{})
ordered pair B ({},{})
-----
ordered pair C ({},{})