0 votes 0 votes 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 . Set Theory & Algebra set-theory&algebra + – worst_engineer asked Jul 3, 2015 worst_engineer 695 views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
Best answer 1 votes 1 votes 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. Jarvis answered Jul 6, 2015 • selected Jul 7, 2015 by worst_engineer Jarvis comment Share Follow See all 0 reply Please log in or register to add a comment.
1 votes 1 votes {(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 . Pranay Datta 1 answered Jul 3, 2015 Pranay Datta 1 comment Share Follow See all 0 reply Please log in or register to add a comment.