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

GA relation R is said to be antisymmetric if (i.e., if and only if), for any pair a; b for which a is in the relation R with b and b is in the relation R with a, then a and b are equal:

\[R\quad is\quad antisymmetric\quad \Leftrightarrow\forall a,b \in A, (aRb\wedge bRa)\Rightarrow (a=b)\]

Practical tip: A relation is antisymmetric if it is asymmetric in respect to its main diagonal and it has some elements in its main diagonal. Therefore if the cell i,j of the matrix contains an X, then cell j,i should be empty.

Check the following relation:

aRa, bRb, cRc, dRd, aRd, bRd, cRb, dRc

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

matrix 1


Add the least possible amount of crosses to the matrix below to transform the depicted relation into an antisymmetric relation on the set A = {a, b, c, d}

  a b c d
a      X  
c    X    X
d    X