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.

If ~ is an equivalence relation on X, and P ( x ) is a property of elements of X, such that whenever x ~ y, P ( x ) is true if P ( y ) is true, then the property P is said to be well-defined or a class invariant under the relation ~.

The set of all a and b for which a ~ b holds make up an equivalence class of X by ~.

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

The equivalence kernel of a function f is the equivalence relation ~ defined by.

* An equivalence relation ~ on a set X partitions X.

* Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.

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 ≈.

* ~ is finer than ≈ if every equivalence class of ~ is a subset of an equivalence class of ≈, and thus every equivalence class of ≈ is a union of equivalence classes of ~.

This yields a convenient way of generating an equivalence relation: given any binary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containing R. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub > in X such that a

Examples of such problems in electronic design automation ( EDA ) include formal equivalence checking, model checking, formal verification of pipelined microprocessors, automatic test pattern generation, routing of FPGAs, and so on.

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.

His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory.

Logical matrix | Logical matrices of the Bell number | 52 equivalence relations on a 5-element set ( Colored fields, including those in light gray, stand for ones ; white fields for zeros.

Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number B < sub > n </ sub >:

Thus there is a natural bijection from the set of all possible equivalence relations on X and the set of all partitions of X.

The equality equivalence relation is the finest equivalence relation on any set, while the trivial relation that makes all pairs of elements related is the coarsest.

The relation "~ is finer than ≈" on the collection of all equivalence relations on a fixed set is itself a partial order relation.

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

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

The set of all possible equivalence classes of X by ~, denoted, is the quotient set of X by ~.

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

Since each element of X belongs to a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by ~, each element of X belongs to a unique equivalence class of X by ~.

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.