Establishing if a a binary relation is asymmetric by inspecting its matrix representation

A relation R is said to be asymmetric iff, whenever an element a is in the relation R with b, then b is not in the relation R with a:

\[R\quad is\quad asymmetric\quad \Leftrightarrow\forall a,b \in A, (aRb) \Rightarrow \neg (bRa)\]

A mathematical example of an asymmetric relation is «less than ( < )»

Practical tips: In a matrix, a relation is asymmetric if it is asymmetric in respect to its main diagonal and it has no elements in the main diagonal itself. Therefore, if i ≠ j, then cells i,j of the matrix contains an x, cells j,i is empty. If i=j then the cell i,j is empty.

An example of an asymmetric relation is:

cRa, aRb, bRd

In a matrix, this relation would be represented in this way:

matrix 1


Add as many crosses as possible to the matrix below to transform the depicted relation into an asymmetric relation on the set A = {a, b, c, d}.

  a b c d
a        X
b  X      
c    X    
d    X