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

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 ).

An equivalence relation partition of a set | partitions a set into several disjoint set | disjoint subsets, called equivalence class es.

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition.

Two elements of the set are considered equivalent ( with respect to the equivalence relation ) if and only if they are elements of the same cell.

Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are " a ~ b " and " a ≡ b ", which are used when R is the obvious relation being referenced, and variations of " a ~< sub > R </ sub > b ", " a ≡< sub > R </ sub > b ", or " aRb ".

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

This is an equivalence relation, which partitions the integers into two equivalence classes, the even and odd integers.

* The relation " is approximately equal to " between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change.

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.

* The relation " is a sibling of " ( used to connote pairs of distinct people who have the same parents ) on the set of all human beings is not an equivalence relation.

The small modification, " is a sibling of, or is the same person as ", is an equivalence relation.

* A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure.

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.

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

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.

Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders.

Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations.

Hence permutation groups ( also known as transformation groups ) and the related notion of orbit shed light on the mathematical structure of equivalence relations.

Dyson is best known for demonstrating in 1949 the equivalence of the formulations of quantum electrodynamics that existed by that time – Richard Feynman's diagrams, on the one hand, and, on the other, the operator method developed by Julian Schwinger and Sin-Itiro Tomonaga.

According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors ( see Eötvös experiment ), there is a universality of free fall ( also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass ): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties.

The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling ( and non-rotating ) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity.

When considered over a countable language, the completeness and compactness theorems are equivalent to each other and equivalent to a weak form of choice known as weak König's lemma, with the equivalence provable in RCA < sub > 0 </ sub > ( a second-order variant of Peano arithmetic restricted to induction over Σ < sup > 0 </ sup >< sub style =" margin-left :- 0. 6em "> 1 </ sub > formulas ).

This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vectors of M at x.

The corresponding addition and multiplication of equivalence classes is known as modular arithmetic.

Whether or not the story is apocryphal, it would only demonstrate the mathematical equivalence of a rotating earth to rotating spheres, as was well known to Khayyam's immediate predecessors, e. g. al-Biruni, and says nothing about heliocentrism, as a spinning earth can be made entirely consistent with geocentric models.

The Ricardian equivalence proposition ( also known as the Barro – Ricardo equivalence theorem ) is an economic theory holding that consumers internalize the government's budget constraint: as a result, the timing of any tax change does not affect their level of spending.

In programming language theory and proof theory, the Curry – Howard correspondence ( also known as the Curry – Howard isomorphism or equivalence, or the proofs-as-programs and propositions-or formulae-as-types interpretation ) is the direct relationship between computer programs and proofs.

An equivalence class of such metrics is known as a conformal metric or conformal class.

The problem of determining if two different strings belong to the same equivalence class is known as the word problem.

More generally, the entire collection of sets of elements of a Polish space X can be grouped into equivalence classes, known as Wadge degrees, that generalize the projective hierarchy.

Schusterman's emphasis is on the importance on a learning structure known as " equivalence classes.

Colloquially, the 5 mil coin was known as a " piastre " ( not an exact equivalence ) and the 50 mil coin as a " shilling " ( an exact equivalence ).

It is known that Hodge cycles are algebraic, and that algebraic equivalence coincides with homological equivalence, so that h < sup > 1, 1 </ sup > is an upper bound for ρ, the rank of the Néron-Severi group.

This equivalence is known as S-duality.

This equivalence is known as S-duality.