I am having a little trouble wrapping my head around this. I understand
the Reflexive, Symmetric, and Transitive properties just fine but do not understand the Antisymmetric examples my textbook gives.
Consider the following relations on {1,2,3,4}
R5={(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3 ),(3,4),(4,4)}
R6={(3,4)}
If anybody could help explain why the above satisfy the Antisymmetric property I would really appreciate it.
R6 is vacuously antisymmetric, because the matrix of the statement defining the property is a material conditional, and the antecedent can not be satisfied.
You can show that R5 is antisymmetric by simply considering all those cases where the antecedent is satisfied,
and establishing that the consequent is also satisfied.
You can just grind it out, if you like.
E.g., if (1,1) in R5 and (1,1) in R5, then (1,1) in R5. ... .