Determining matrix for relationship: reflexive, symmetric, transitive.

Question

I have two matrices below and need to determine if R is (a) reflexive, (b) symmetric, and (c) transitive.

$M_R = \begin{pmatrix} 1 & 0 & 1 & 0\\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0\\ 1 & 1 & 1 & 1\end{pmatrix}$ ; $M_R = \begin{pmatrix} 1 & 1 & 1 & 1\\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1\end{pmatrix}$

So, far I was able to figure out that for both it is reflexive because there is 1 diagonally, and not symmetric because $M_{21} \neq M_{12}$ and also $M_R \neq (M_R)^T$.

Can anyone please verify what I did is correct? And also how do I determine if it is transitive?

Answer 1

What you did is indeed correct.

For the last one, you need to check whether $$ M_{ij} = 1 \text{ and } M_{jk} = 1 \implies M_{ik} = 1 $$ This is not true for the first relation. In particular, $M_{21} = 1$ and $M_{13} = 1$, but $M_{23} = 0$.

This does, however, hold true for the second relation (in fact, $M_R$ is the matrix for the relation "$\leq$").

In determining transitivity, it helps to draw the digraph of the relation.