Reflexive Closure

How can we make a relation reflexive?

Mathematical Definition

$R$ is reflexive if $\forall \ a_i \in A, (a_i, a_i) \in R$.

Matrix Representation

We know that the diagonal filled with 1s represents the reflexive closure of a relation.

Hence, the shortest way to make a relation reflexive is to fill the matrix diagonal with 1s.

For example, here is how we can make some relation reflexive in matrix form:

$$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & · & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & · & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & · & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix} \sim \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & · & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & · & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & · & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}$$

Graph Representation

In the graph representation, we add a self-loop to each node.

For example, here is how we can make some relation reflexive in graph form:

Once we add the self-loops, the graph will look like this: