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 ({},{})