Equivalent Characterizations of Finer Equivalence Relation

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Let $X$ be a set.

Let $\equiv$ and $\sim$ be equivalence relations on $X$.

The following are equivalent:

  1. $\equiv$ is finer than $\sim$:
    $\forall x, y \in X : x \equiv y \implies x \sim y$
  2. The graph of $\equiv$ is contained in the graph of $\sim$.
  3. Every $\equiv$-equivalence class is contained in a $\sim$-equivalence class.
  4. Every $\sim$-equivalence class is saturated under $\equiv$.