Graphs & Trees: Intro

See the idea? The number of rows (and columns) of the adjacency matrix is equal to $|S|$ (the size of the set $S$). Since $|S| = |\{1, 2, 3, 4\}| = 4$, this means that the adjacency matrix would consist of $4$ rows (and $4$ columns).

The way we fill out the matrix is based on the following $2$ rules:

1. If $(i, j) \in \text{GreaterThan}$ (that is, if $i > j$), then $M_{i,j} = 1$.
2. Otherwise, if $(i, j) \not\in \text{GreaterThan}$, then $M_{i,j} = 0$.

In other words, an adjacency matrix will tell us which pairs are 'lit up' and exist in the relation, and which pairs aren't.

Fun fact: the trend of the $1$s and $0$s in the adjacency matrix might show us some important patterns in the underlying relation. In the case of $M$ on the previous slide, for instance, we clearly see how all the $1$s are grouped below the main diagonal, which means that the relation is applied only to the pairs where $i > j$, which is the whole idea of $\text{GreaterThan}$!