# Equivalent Characterizations of Finer Equivalence Relation

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## Theorem

Let $X$ be a set.

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

The following are equivalent:

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

- The graph of $\equiv$ is contained in the graph of $\sim$.
- Every $\equiv$-equivalence class is contained in a $\sim$-equivalence class.
- Every $\sim$-equivalence class is saturated under $\equiv$.

## Proof

## Sources

- 1986: Nicolas Bourbaki:
*Theory of Sets*... (previous) Chapter $\text I$: Description of Formal Mathematics: $\S 6$ Equivalence relations $7$. Quotients of equivalence relations