Equivalence

Equivalence classes of numeric forms The numeric forms are placed in equivalence classes; each such equivalence class is a surreal number.

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation ) defined on them, then one may naturally split the set S into equivalence classes.

This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vectors of M at x. The equivalence class of the curve γ is written as γ'(0).

Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2. In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are the equivalence classes of A under ~.

A finer equivalence relation, Solovay equivalence, can be used to characterize the halting probabilities among the left-c.e. reals.

An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence : if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.

As a corollary of the theorem, any two simply connected open subsets of the Riemann sphere which both lack at least two points of the sphere can be conformally mapped into each other (because conformal equivalence is an equivalence relation).

Endpoint and equivalence point Though equivalence point and endpoint are used interchangeably, they are different terms.

Equivalence relations and mathematical logic Equivalence relations are a ready source of examples or counterexamples.

Equivalence relations The following are all equivalence relations: * "Has the same birthday as" on the set of all people.

Equivalences of categories If a functor F: C←D is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.

Examples * If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars.

Furthermore, is an equivalence relation because equality is an equivalence relation on outcome classes.

Homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can contain several homeomorphism classes.

In a quotient of a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup.

In the discretization stage, the space of signals is partitioned into equivalence classes and quantization is carried out by replacing the signal with representative signal of the corresponding equivalence class.

Maximal atlas The atlas containing all possible charts consistent with a given atlas is called the maximal atlas: i.e., an equivalence class containing that given atlas (under the already defined equivalence relation given in the previous paragraph).

More generally, a function may map equivalent arguments (under an equivalence relation ~ A ) to equivalent values (under an equivalence relation ~ B ).

Nevertheless, equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown to be a field that has the rational numbers as a subfield.

Rejections of Hempel's equivalence condition Some approaches for the resolution of the paradox reject Hempel's equivalence condition.