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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.

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} imes\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 = usually implies). 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'').



QUOTATION


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

( Ronald L. Graham , ''Rudiments of Ramsey Theory '')


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