Thread: Antisymmetric Relations

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.

Originally Posted by JazzGuitarist01

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.

If it's anti-symmetric then if and then .

So, does that happen here?

I'm not sure I follow you correctly.

So example R6 is Antisymmetric because it follows
if and then

But how does R6 satisfy that? In that example isn't x not equal to y?

Originally Posted by JazzGuitarist01

I'm not sure I follow you correctly.

So example R6 is Antisymmetric because it follows
if and then

But how does R6 satisfy that? In that example isn't x not equal to y?

R6 satisfies this vacuously. We only have that . There is no other info to check.

Oh ok.

I still feel a little unsure, so I am gonna ask about example R5.
In R5 there is (2,3)
2<=3 but 3 is not <=2
What would be the other info to check that R5 is antisymmetric?

Originally Posted by JazzGuitarist01

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. ... .