Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion). However, the concept of formal correctness depends on time and on the context. Therefore, many notations in mathematics are qualified as abuse of notation in some context and are formally correct in other contexts; as many notations were introduced a long time before any formalization of the theory in which they are used, the qualification of abuse of notation is strongly time dependent. Moreover, many abuses of notation may be made formally correct by improving the theory. Abuse of notation should be contrasted with misuse of notation, which should be avoided.Many mathematical objects consist of a set, often called the underlying set, equipped with some additional structure, typically 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, Z {displaystyle mathbb {Z} }   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, and avoiding such an abuse of notation would make mathematical texts pedantic and difficult to read. When this abuse of notation may be confusing, one may distinguish between these structures by denoting ( Z , + ) {displaystyle (mathbb {Z} ,+)}   the group of integers with addition, and ( Z , + , ⋅ ) {displaystyle (mathbb {Z} ,+,cdot )}   the ring of integers.The terms 'abuse of language' and 'abuse of notation' depend on context. Writing 'f: A → B' for a partial function from A to B is almost always an abuse of notation, but not in a category theoretic context, where f can be seen as a morphism in the category of sets and partial functions.