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:
“There is no comparing the brutality and cynicism of today’s pop culture with that of forty years ago: from High Noon to Robocop is a long descent.”
—Charles Krauthammer (b. 1950)
“I want relations which are not purely personal, based on purely personal qualities; but relations based upon some unanimous accord in truth or belief, and a harmony of purpose, rather than of personality. I am weary of personality.... Let us be easy and impersonal, not forever fingering over our own souls, and the souls of our acquaintances, but trying to create a new life, a new common life, a new complete tree of life from the roots that are within us.”
—D.H. (David Herbert)