Question

Is the relation represented by the following matrix an equivalence relation? Is it a partial order? Explain why or why not. $\begin{bmatrix} 1 \ 1 \ 1 \0\\ 0 \ 1 \ 1 \0\\ 0 \ 0 \ 1 \0\\ 1 \ 1 \0 \1 \end{bmatrix}$

Answer 1

a relation can be represented using the matrix.

The given matrix is

$\begin{bmatrix} 1 1 1 0\\ 0 1 1 0\\ 0 0 1 0\\ 1 1 0 1 \end{bmatrix}$

To say a relation is equivalence, we have to prove the relation is

1. Reflexive
2. Symmetric
3. Transitive

Reflexive:

If a relation R is reflexive, all the values for m[i,i] in the corresponding matrix should be 1.

The given relation is clearly reflexive, because all m[i,i]=1. That is m[1,1]=m[2,2]=m[3,3]=m[4,4]=1.

Symmetric:

If a relation R is symmetric then m[i,j] should be equal to m[j,i]. i.e, m[i,j]=m[j,i].

The given relation is not symmetric, because m[2,1] not equal to m[1,2].

Also, m[3,1] not equal to m[1,3]

      m[4,1] not equal to m[1,4]

so on.

Transitivity:

If a relation is transitive, and if m[i,j]=1 and m[j,k]=1 then m[i,k] should be 1.

The given relation is not a transitive, because m[4,1]=1 and m[1,3]=1, but m[4,3] is not equal to 1.

To say a given relation is partial order, it should be reflexive, antisymmetric and transitive. The given relation is not a partial order relation , because the given relation is reflexive, but not anti-symmetric and transitive.