Equivalence Relation in a sentence

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).

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

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

The advantages of regarding an equivalence relation as a special case of a groupoid include: *Whereas the notion of "free equivalence relation" does not exist, that of a free groupoid on a directed graph does.

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.

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).

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.

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).

The characteristic feature of a Mal'cev algebra is that we can recover the entire equivalence relation ker f from the equivalence class of the neutral element.

The set of all equivalence classes in X with respect to an equivalence relation R is denoted as X/R and called X modulo R (or the quotient set of X by R ).

Definition A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive.

That is, the equality relation may now be interpreted by an arbitrary equivalence relation on the domain of discourse that is congruent with respect to the functions and relations of the interpretation.

A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph.

A semigroup congruence is an equivalence relation that is compatible with the semigroup operation.

Explicitly, the Brauer group of the reals consists of two classes, represented by the reals and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a matrix ring over another.

Field of fractions main The field of fractions K of an integral domain R is the set of fractions a/b with a and b in R and b ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations.

However, if the approximation is defined asymptotically, for example by saying that two functions f and g are approximately equal near some point if the limit of f − g is 0 at that point, then this defines an equivalence relation.

Interpretability main Given a mathematical structure, there are very often associated structures which can be constructed as a quotient of part of the original structure via an equivalence relation.