Definition:Finer Equivalence Relation

Definition
Let $X$ be a set.

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

Then $\equiv$ is finer than $\sim$ :
 * $\forall x, y \in X : x \equiv y \implies x \sim y$

Also known as
If $\equiv$ is finer than $\sim$, then $\sim$ is said to be coarser than $\equiv$.

Also see

 * Equivalent Characterizations of Finer Equivalence Relation