Question

Let R be a reflexive relation on a finite set A, and let MR be the bit matrix representing R. Specify the value of the entries on the main diagonal.

Answer 1

If $R$ is a binary relation between the finite sets X and Y, that is $R ⊆ X×Y$, then $R$ can be represented by the logical matrix $M_R$ whose row and column indices index the elements of $X$ and $Y$, respectively, such that the entries of $M_R$ are defined by:

${\displaystyle M_{i,j}={\begin{cases}1&(x_{i},y_{j})\in R\\0&(x_{i},y_{j})\not \in R\end{cases}}}$

Since $R\subset A\times A$ is reflexive relation, $(x,x)\in R$ for all $x\in A$. Therefore, $M_{i,i}=1$ for all $i\in\{1,...,|A|\}$. Consequently, the value of all entries on the main diagonal is 1.