The Matrix of an Equivalence Relation

Question

Perhaps I'm missing something from just the definition of an equivalence relation but wouldn't a matrix representing an equivalence relation on any set be only ones and anything less than that is just an equivalence class? If someone can clarify this for me that'd be great.

Answer 1

For $n\gt1$, there is more than one possible equivalence relation on a set with $n$ elements. Obviously they can’t all be represented by the same matrix. In particular, the “minimal” equivalence relation on $S$ is $R=\{(x,x) \mid x\in S\}$. This relation is represented by the $n\times n$ identity matrix.

It’s a worthwhile exercise to work out what properties the representative matrix must have. The reflexive and symmetric properties are pretty easy to translate into statements about the matrix, but transitivity might be a bit tricky.