Can you please help me understand Antisymmetric relation

Question

Hi ,

I am stuck with Antisymmetric relations. I know the formal definition . If A = {a,b}  , If aRb ^ bRa both true then a=b for all a,b belongs to A.

Now , while the formal definition is ok , for practical purpose I found out that diagonal elements and / or half of the diagonal elements are anti-symmetric.

 A={a,b,c}

so , relations

• { (a,a),(b,b),(c,c)} is anti-symmetric.
• {(a,b),(a,c),(b,c)} or {(b,a),(c,a),(c,b)} is anti-symmetric ( as one half of the diagonal ).
• {(a,b),(b,c),(c,c)} is also anti-symmetric ( one half of diagonal and (c,c) is diagonal element )

But , I can not understand how {(a,b) , (c,b)} is anti-symmetric .

Answer 1

See,

AntiSymmetric means that if aRb->bRa then a=b.Precisely if aRb exists then bRa should not exist.

Asymmetric is something which is both AntiSymmetric and irreflexive at the same time.

a b is antisymmetric because b a does not exist.

c b as b c does not exist.

Answer 2

 {(a,b) , (c,b)} this set is following the anti-symmetric properties

(A = {a,b}  , If aRb ^ bRa both true then a=b for all a,b belongs to A) .

and even {(a,b)} this set is anti-symmetric .

but this is not anti-symmetric {(a,b) , (b)} it`s violating the anti-symmetric properties . 

As a simple example, the divisibility order on the natural numbers is an antisymmetric relation .