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.
Read more about this topic: Equivalence Relation
Famous quotes containing the words comparing and/or relations:
“We cannot think of a legitimate argument why ... whites and blacks need be affected by the knowledge that an aggregate difference in measured intelligence is genetic instead of environmental.... Given a chance, each clan ... will encounter the world with confidence in its own worth and, most importantly, will be unconcerned about comparing its accomplishments line-by-line with those of any other clan. This is wise ethnocentricism.”
—Richard Herrnstein (1930–1994)
“I only desire sincere relations with the worthiest of my acquaintance, that they may give me an opportunity once in a year to speak the truth.”
—Henry David Thoreau (1817–1862)