By demonstrating that burning diamond and graphite releases the same amount of gas he established the chemical equivalence of these substances.

Functions equivalent in this manner form an equivalence class on, and these equivalence classes partition.

In other words, if is an equivalence relation on a set, and and are two elements of, then these statements are equivalent:

The equivalence of these definitions can be proved using the Dandelin spheres.

In these units, both distances and times are expressed in inverse energy units ( while energy and mass are expressed in the same units, see Mass – energy equivalence ).

Here ( Z / 2Z ) is the polynomial ring of Z / 2Z and ( Z / 2Z )/( T < sup > 2 </ sup >+ T + 1 ) are the equivalence classes of these polynomials modulo T < sup > 2 </ sup >+ T + 1.

However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.

In general terms, a calculus is a formal system that consists of a set of syntactic expressions ( well-formed formulæ or wffs ), a distinguished subset of these expressions ( axioms ), plus a set of formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions.

However these negotiations may prove difficult in that both parties in a conflict believe that they are morally superior to the other, and are unwilling to negotiate on basis of moral equivalence.

* Define a b as " not b < a " ( i. e., take the inverse complement of the relation ), which corresponds to defining a ~ b as " neither a < b nor b < a "; these relations and ~ are in general not transitive ; however, if they are, ~ is an equivalence ; in that case "<" is a strict weak order.

The dimensional equivalence of these units, of course, is not simply a coincidence: A torque of 1 N · m applied through a full revolution will require an energy of exactly 2π joules.

where v, v < sub > 1 </ sub > and v < sub > 2 </ sub > are vectors from V, while w, w < sub > 1 </ sub >, and w < sub > 2 </ sub > are vectors from W, and c is from the underlying field K. Denoting by R the space generated by these four equivalence relations, the tensor product of the two vector spaces V and W is then the quotient space

The first three of these characterizations can be proven equivalent in Zermelo – Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.

They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms ( e. g., a weak equivalence of spaces passes to an isomorphism of homology groups ), verified that all existing ( co ) homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory.

Together, these equivalence classes are the elements of a quotient group.

Since the meaning of the existential quantifier in formal notation is usually explained in terms of its equivalence to English expressions such as " there is " and " there exist ", and since these English expressions are not reliably ontologically committing, it comes to seem that we cannot be sure of our theory's ontological commitments even after it has been regimented into a canonical formulation.

Of these criteria, " equivalence " is the most controversial.

While imprecise, comparison of the results of trials of sertraline with separate trials of other anti-panic agents ( clomipramine ( Anafranil ), imipramine ( Tofranil ), clonazepam ( Klonopin ), alprazolam, fluvoxamine ( Luvox ) and paroxetine ) indicates approximate equivalence of these medications.

We define an equivalence relation ~ upon these pairs with the following rule:

Also use these KPIs to measure the effectiveness of your in-office staff to maintain an equivalence among the distinct employee categories.

However, these symbols are also used for material equivalence ; the proper interpretation depends on the context.

Unicly which is optimized for creation of UUIDs ( v3, v4, and v5 ) and offers an extended interface for converting among different representations and interrogating UUID equivalence across these different representations.

An equivalence between the conserved quantities and corresponding symmetry groups implies the same level of equivalence for both these two ways of specifying the testable information in the maximum entropy method.

Intuitively, an algebraic space is a scheme modulo a " nice " equivalence relation ; the resulting category of algebraic spaces extends the category of schemes and allows to carry out several natural constructions that are needed for example in deformation theory or in the construction of moduli spaces but are not possible in the smaller category of schemes.

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 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.

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.

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.

All the elements in a given equivalence class are equivalent among themselves, and no element is equivalent with any element from a different class.

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 ".

The following are all equivalence relations:

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.

Then all elements of X equivalent to each other are also elements of the same equivalence class.

In both cases, the cells of the partition of X are the equivalence classes of X by ~.

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 ≈.

Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are three equivalent ways of specifying the same thing.

:* Any subset of the identity relation on X has equivalence classes that are the singletons of X.

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.

Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalence relations are grounded in partitioned sets, which are sets closed under bijections and preserve partition structure.

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

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.