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 simplify the exposition or suggest the correct intuition. However, since the concept of formal/syntactical correctness depends on both time and context, many notations in mathematics that are flagged as abuse of notation in some context could be formally correct in other contexts. An instance where abuses of notation is strongly time-dependent occurs when notations are introduced to a theory a long time before the theory is first formalized. In which case, many of such abuses of notation may be made formally correct by improving the theory. Abuse of notation should be contrasted with misuse of notation, which does not have the presentational benefits of the former and should be avoided. A related concept is abuse of language or abuse of terminology, where a term — rather than a notation — is misused. Abuse of language is an almost synonymous expression for abuses that are non-notational by nature. For example, while the wordrepresentation properly designates a group homomorphism from a groupG to GL, where V is a vector space, it is common to call V "a representation of G". Another common abuse of language consists in identifying two mathematical objects that are different, but canonically isomorphic. Other examples include identifying a constant function with its value, identifying a group with a binary operation with the name of its underlying set, or identifying to the Euclidean space of dimension three equipped with a Cartesian coordinate system.
Examples
Structured mathematical objects
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 notation for the underlying set and the structured object. For example, may denote the set of the integers, the group of integers together with addition, or the ring of integers with addition and multiplication. In general, there is no problem with this if the object under reference is well understood, and avoiding such an abuse of notation might even make mathematical texts more pedantic and more difficult to read. When this abuse of notation may be confusing, one may distinguish between these structures by denoting the group of integers with addition, and the ring of integers. Similarly, a topological space consists of a set and a topology which is characterized by a set of subsets of . Most frequently, one considers only one topology on, so there is usually no problem in referring as both the underlying set, and the pair consisting of and its topology — even though they are technically distinct mathematical objects. Nevertheless, it could occur on some occasions that two different topologies are considered simultaneously on the same set. In which case, one must exercise care and use notation such as and to distinguish between the different topological spaces.
Function notation
One may encounter, in many textbooks, sentences such as "Let be a function...". This is an abuse of notation, as the name of the function is, and usually denotes the value of the function for the element of its domain. The correct phrase would be "Let be a function of the variable ..." or "Let 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. A similar abuse of notation occurs in sentences such as "Let us consider the function...", when in fact is not a function. The function is the operation that associates to, often denoted as. Nevertheless, this abuse of notation is widely used, since it can help one avoid the pedantry while being generally not confusing.
Equality vs. isomorphism
Many mathematical structures are defined through a characterizing 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. 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: But this is strictly speaking not true: if, and, the identity would imply that and, and so 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 by x instead of is an abuse of notation. Formally, if a set X is partitioned by an equivalence relation ~, then for each x ∈ X, the equivalence class is denoted . 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 n can be formed by partitioning the integers via the equivalence relation "x ~ y if and only if x ≡ y ". The elements of that group would then be , , …, , but in practice they are usually denoted simply as 0, 1, …, n − 1. Another example is the space of measurable functions over a measure space, or classes of Lebesgue integrable functions, where the equivalence relation is equality "almost everywhere".
