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 (while being unlikely to introduce errors or cause confusion). Abusing notation should be contrasted with "misusing" notation which should be avoided.

Common examples occur when speaking of compound mathematical objects. For example, a topological space consists of a set $$T$$ and a topology $$\mathcal{T}$$, and two topological spaces $$(T, \mathcal{T})$$ and $$(T, \mathcal{T'})$$ can be quite different if they have different topologies. Nevertheless, it is common to refer to such a space simply as $$T$$ when there is no danger of confusion or when it is implicitly clear what topology is being considered. Similarly, one often refers to a group $$(G, \star)$$ as simply $$G$$ when the group operation is clear from context. Another example is in the Leibniz notation for the derivative $$\frac{dy}{dx}$$. Although the derivative is not strictly a fraction, abusing this notation leads to the correct chain rule $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$$. Often good notation is judged by whether or not its abuses will lead to correct interpretations.

The new use may achieve clarity in the new area in an unexpected way, but it may borrow arguments from the old area that do not carry over, creating a false analogy.

Abuse of language is an almost synonymous expression that is usually used for non-notational abuses. 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 "a representation of G."

Examples
John Harrison cites "the use of f(x) to represent both application of a function f to an argument x, and the image under f of a subset, x, of f's domain".

The computation of the vector product as the determinant of the matrix
 * $$\mathbf{a}\times\mathbf{b}=\det \begin{bmatrix}

\mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{bmatrix}$$ is a significant abuse of notation as $$\mathbf{i},\mathbf{j},\mathbf{k}$$ are treated as scalars but are in fact vectors.

With Big O notation, we say that some function f "is" O(g(x)) (given some function g, where x is one of f ' s parameters). Intuitively this notation groups functions according to their growth respective to some parameter; as such, it would be appropriate to use the set membership notation and say that $$f \in O(g(x))$$. However, the usual notation is $$f = O(g(x))$$, despite the fact that the implied relationship is not symmetric (which the symbol "=" would imply). One reason for this is that, as an extension of the abuse, it is useful to overload relation symbols such as < and ≤, such that, for example, f < O(g(x)) means that f's real growth is less than g(x). But this further abuse is not necessary if, following Knuth one distinguishes between O and the closely related o and Θ notations.

Another common abuse of notation is to blur the distinction between equality and isomorphism. For instance, in the construction of the real numbers from Dedekind cuts of rational numbers, the rational number r is identified with the set of all rational numbers less than r, even though the two are obviously not the same thing (as one is a rational number and the other is a set of rational numbers). However, this ambiguity is tolerated, because the set of rational numbers and the set of Dedekind cuts of the form {x: x<r} have the same structure. It is though this abuse of notation that Q is regarded as a subset of R.

Quotation

 * "We will occasionally use this arrow notation unless there is no danger of confusion."

(Ronald L. Graham, Rudiments of Ramsey Theory)