Symmetric Closure

How can we make a relation symmetric?

Mathematical Definition

$R$ is symmetric if $\forall \ (a_i, a_j) \in R \longrightarrow (a_j, a_i) \in R$.

Matrix Representation

We know that the symmetric closure of a relation is when the matrix and its transpose are equal.

Hence, the shortest way to make a relation symmetric is to fill the matrix with its transpose.

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

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

Graph Representation

In the graph representation, we make sure that for every edge $(a_i, a_j)$, there is an edge $(a_j, a_i)$.

So we make any and all edges bidirectional.

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

Once we add the missing edges, the graph will look like this: