I am reading a paper by Carlsson on Topological pattern recognition for point cloud data. I was having a little trouble understanding the formal definition of an equivalence relation and equivalence class in relation to persistent homology. Could someone please clarify the definition below. I have my own interpretation, listed below, but please correct me if I am wrong.
A (binary) relation on a set $X$ is a subset of $X × X$. We will often denote relations by $\sim$, and write $x ∼ x$ to indicate that $(x, x)$ is in the relation.
Definition 2.1. A relation ∼ on a set X is an equivalence relation if the following three conditions hold:
- $x ∼ x$ for all $x \in X$,
- $x ∼ x'$ if and only if $x' ∼ x$,
- $x ∼ x'$ and $x' ∼ x''$ implies $x ∼ x''$
By the equivalence class of $x \in X$, denoted by $[x]$, we will mean the set $$\{x \mid x ∼ x \}$$
So as I interpret it, $\forall x \in X$, then $X \times X := \{(x,x)| x \in X\}$. If we denote $A = \{(x,x)| x \in X\}$, then there exists an equivalence relation between $x_1 \sim x_2$ if
- both $ x_1, x_2 \in X$
- $(x_1, x_2) \in A$
- $(x_2, x_1) \in A$
- if $(x_1, x_2) \in A$ and $(x_2, x3) \in A$, then $(x_1, x_3) \in A$.
I think this is the right interpretation, but just wanted to make sure I was on the right track.