The equivalence kernel of an injection is the identity relation.

* An equivalence relation ~ on X is the equivalence kernel of its surjective projection π: X → X /~.

Less formally, the equivalence relation ker on X, takes each function f: X → X to its kernel ker f. Likewise, ker ( ker ) is an equivalence relation on X ^ X.

This equivalence relation is known as the kernel of.

The kernel of f is the equivalence relation thus defined.

Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition:

equivalence classes induced by the kernel of the wqo.

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

The projection of ~ is the function defined by which maps elements of X into their respective equivalence classes by ~.

Each equivalence relation has a canonical projection map, the surjective function from to given by.

Any function itself defines an equivalence relation on according to which if and only if.

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

For such data, one must use a hash function that is compatible with the data equivalence criterion being used: that is, any two inputs that are considered equivalent must yield the same hash value.

* The electronvolt ( eV ) is primarily a unit of energy, but because of the mass – energy equivalence it can also function as a unit of mass.

In the equivalence of models of computability, a parallel is drawn between Turing machines which do not terminate for certain inputs and an undefined result for that input in the corresponding partial recursive function.

More precisely, then, a likelihood function is any representative from an equivalence class of functions,

where σ < sub > 0 </ sub >( n ) = d ( n ) is the divisor function, a multiplicative function that equals the number of positive divisors of the number n. To prove the equivalence of these sums, note that they all take the form of Lambert series and can thus be resummed as such.

The symbol deg ( D ) denotes the degree ( occasionally also called index ) of the divisor D, i. e. the sum of the coefficients occurring in D. It can be shown that the divisor of a global meromorphic function always has degree 0, so the degree of the divisor depends only on the linear equivalence class.

Up to birational equivalence, these are categorically equivalent to algebraic function fields.

There is a triple equivalence of categories between the category of smooth projective algebraic curves over the complex numbers, the category of compact Riemann surfaces, and the category of complex algebraic function fields, so that in studying these subjects we are in a sense studying the same thing.

are defined in the usual way ( for example, for a binary function +, ( a + b ) < sub > i </ sub > = a < sub > i </ sub > + b < sub > i </ sub > ), and an equivalence relation is defined by a ~ b if and only if

* the equivalence relation on the function's domain that roughly expresses the idea of " equivalent as far as the function f can tell ", or

If X and Y are algebraic structures of some fixed type ( such as groups, rings, or vector spaces ), and if the function f from X to Y is a homomorphism, then ker f will be a subalgebra of the direct product X × X. Subalgebras of X × X that are also equivalence relations ( called congruence relations ) are important in abstract algebra, because they define the most general notion of quotient algebra.

The start state of the automaton corresponds to the equivalence class containing the empty string, and the transition function from a state X on input symbol y takes the automaton to a new state, the state corresponding to the equivalence class containing string xy, where x is an arbitrarily chosen string in the equivalence class for X.

Such a definition can be formulated in terms of equivalence classes of smooth functions on M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x.

That is, it is the equivalence class of functions on M vanishing at x determined by g o f.

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.

A more formal definition: two subsets A and B of Euclidean space R < sup > n </ sup > are called congruent if there exists an isometry f: R < sup > n </ sup > → R < sup > n </ sup > ( an element of the Euclidean group E ( n )) with f ( A ) = B. Congruence is an equivalence relation.

* The notation || f ||< sub > p </ sup > with 1 ≤ p ≤ ∞ is a slight abuse, because in general it is only a norm of f if || f ||< sub > p </ sup > is finite and f is considered as equivalence class of μ-almost everywhere equal functions.

( f ) The superiority or equivalence of the investigational treatment has been shown in at least two independent research settings.

When using this notation, f is then intended as an entire equivalence class of maps, using the same letter f for any representative map.

Further the f < sub > ij </ sub > must satisfy conditions based on the reflexive, symmetric and transitive properties of an equivalence relation ( gluing conditions ).

Denote the equivalence class of ( p, f ) by.

From this follows that if f is a homotopy equivalence, then f < sub >*</ sub > is an isomorphism.

There is no formal equivalence to the supervisory ranks ; ;

Hence it is difficult to conceive of a packing of the atoms in this material in which the oxygen atoms are far from geometrical equivalence.

The end result of antimatter meeting matter is a release of energy proportional to the mass as the mass-energy equivalence equation, E = mc < sup > 2 </ sup > shows.

Titration involves the addition of a reactant to a solution being analyzed until some equivalence point is reached.

For the proof of the equivalence of the four approaches the reader is referred to mathematical expositions like or.

* the equivalence principle, whether or not Einstein's general theory of relativity is the correct theory of gravitation, and if the fundamental laws of physics are the same everywhere in the universe.

The tendency is to run lean, an equivalence ratio less than 1, to reduce the combustion temperature and thus reduce the NOx emissions ; however, running the combustion lean makes it very susceptible to combustion instability.

The major tool one employs to describe such a situation is called equivalence of categories, which is given by appropriate functors between two categories.

One can show that this map is an isomorphism, establishing the equivalence of the two definitions.

If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward.

For example, the Cyrillic letter Р is usually written as R in the Latin script, although in many cases it is not as simple as a one-for-one equivalence.

Normality is defined as the molar concentration divided by an equivalence factor.

But " having distance 0 " is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M. The original space is embedded in this space via the identification of an element x of M with the equivalence class of sequences converging to x ( i. e., the equivalence class containing the sequence with constant value x ).