Abuse of notation on arvau

Wikipedia / Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help to simplify the exposition or to suggest the correct intuition, while possibly minimizing errors and confusion at the same time.

A closely related concept is abuse of language or abuse of terminology, where a term — rather than a notation — is used informally. For example, while the word representation properly designates a group homomorphism from a group G to GL(V), where V is a vector space, it is common to call V itself a "representation of G."

Since both mathematical notation and terminology vary across time and context, what is flagged as abuse in one context could be formally correct in another. Furthermore, abuse of notation should not be conflated with misuse of notation, which does not have the presentational benefits of the former and should be avoided, such as the misuse of constants of integration.

Examples

Suppression of parameters

Many mathematical objects consist of a set, often called the underlying set, equipped with some additional structure, such as a mathematical operation or a topology. It is a common abuse of notation to use the same symbol for the underlying set and the structured object, a phenomenon known as suppression of parameters.

may denote the set of the integers, the group of integers together with addition, or the ring of integers with addition and multiplication is often used to mean 3D Euclidean space taken as a vector space, metric space, or topological space, depending on the properties relevant in context. This is because, for example, the Euclidean distance is understood to be the 'default' metric and is assumed when not stated otherwise, even though there are others.

Similarly, while a topological space and is ever considered — once that topology is clearly established.

to distinguish between different topological spaces with the same underlying set.

Function notation

be a function ...". This is an abuse of notation, as the name of the function be a function ..." This abuse of notation is widely used, as it simplifies the formulation, and the systematic use of a correct notation quickly becomes pedantic.

is a polynomial Similarly, a constant function." Nevertheless, these abuses of notation are widely used, since they are more concise and generally not confusing.

Equality vs. isomorphism

Many mathematical structures are defined through a characterizing property (often a universal property). Once this desired property is defined, there may be various ways to construct the structure, and the corresponding results are formally different objects, but which have exactly the same properties (i.e., isomorphic). As there is no way to distinguish these isomorphic objects through their properties, it is standard to consider them as equal, even if this is formally wrong.

One example of this is the Cartesian product, which is often seen as associative:

would mean nothing. However, these equalities can be legitimized and made rigorous in category theory—using the idea of a natural isomorphism.

Another example of similar abuses occurs in statements such as "there are two non-Abelian groups of order 8", which more strictly stated means "there are two isomorphism classes of non-Abelian groups of order 8".

Equivalence classes

Referring to an equivalence class of an equivalence relation is partitioned. But in practice, if the remainder of the discussion is focused on the equivalence classes rather than the individual elements of the underlying set, then it is common to drop the square brackets in the discussion.

For example, in modular arithmetic, a finite group of order.

Another example is the space of (classes of) measurable functions over a measure space, or classes of Lebesgue integrable functions, where the equivalence relation is equality "almost everywhere".

Subjectivity

The terms "abuse of language" and "abuse of notation" depend on context. Writing "" for a partial function from to is almost always an abuse of notation, but not in a category theoretic context, where can be seen as a morphism in the category of sets and partial functions.