Comparing Equivalence Relations
If ~ and ≈ are two equivalence relations on the same set S, and a~b implies a≈b for all a,b ∈ S, then ≈ is said to be a coarser relation than ~, and ~ is a finer relation than ≈. Equivalently,
- ~ is finer than ≈ if every equivalence class of ~ is a subset of an equivalence class of ≈, and thus every equivalence class of ≈ is a union of equivalence classes of ~.
- ~ is finer than ≈ if the partition created by ~ is a refinement of the partition created by ≈.
The equality equivalence relation is the finest equivalence relation on any set, while the trivial relation that makes all pairs of elements related is the coarsest.
The relation "~ is finer than ≈" on the collection of all equivalence relations on a fixed set is itself a partial order relation.
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