Equivalence relation
Comparing equivalence relations

If ∼ {\displaystyle \sim } and ≈ {\displaystyle \approx } are two equivalence relations on the same set S {\displaystyle S}, and a ∼ b {\displaystyle a\sim b} implies a ≈ b {\displaystyle a\approx b} for all a, b ∈ S, {\displaystyle a,b\in S,} then ≈ {\displaystyle \approx } is said to be a coarser relation than ∼ {\displaystyle \sim }, and ∼ {\displaystyle \sim } is a finer relation than ≈ {\displaystyle \approx }. Equivalently,

The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest.

The relation " ∼ {\displaystyle \sim } is finer than ≈ {\displaystyle \approx } " on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection a geometric lattice.
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