# Most ambiguous and inconsistent phrases and notations in maths

What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts?

For instance, a function $f$:

$f^{-1}(x)$ can be an inverse and a preimage and sometimes even $\frac{1}{f(x)}$.

$f^2(x)$ can be $(f\circ f)(x)$ and $(f(x))^2$.

$f^{(2)}(x)$ on the other hand, is the second derivative, even though adding parentheses to a number usually does nothing.

And for some functions the parentheses for the argument are omitted: $f\:x = f(x)$.

So how should $f^{(2-3)}(x)$ be interpreted? $f^{(-1)}$, an integral of $f$? or a composition, $\left(f^{(2)}\circ f^{(-3)}\right)(x)$? Or just $f^{2-3}(x) = f^{-1}(x)$?

Another example is mathematicians notorious use of the word normal to describe… normal things?

Using similar symbols and expressions for different things is unavoidable, but it can create some ambiguity when first introduced to their other uses.

1. ‘The function $$f(x)f(x)$$‘. No, the function is $$ff$$.
2. Let $$ff$$ and $$gg$$ be real differentiable functions defined in $$R\mathbb R$$. Some people denote $$(f∘g)′(f\circ g)'$$ by $$df(g(x))dx\dfrac{\mathrm df(g(x))}{\mathrm dx}$$. Contrast with the above. I discuss this in greater detail here.
3. The differential equation $$y′=x2y+y3y'=x^2y+y^3$$. Just a minor variant of 1. Correct would be $$y′=fy+y3y'=fy+y^3$$ where $$f:I→R,x↦x2f\colon I\to \mathbb R, x\mapsto x^2$$, for some interval $$II$$.
4. This is one I find particularly disgusting. “If $$t(s)t(s)$$ is a function of $$ss$$ and it is invertible, then $$s(t)s(t)$$ is the inverse”, lol what? The concept of ‘function of a variable’ isn’t even definable in a satisfiable way in $$ZFC\sf ZFC$$. Also $$(dydx)−1=dxdy\left(\frac{\mathrm dy}{\mathrm dx}\right)^{-1}=\frac{\mathrm dx}{\mathrm dy}$$. Contrast with 1.
5. In algebra it’s common to denote the algebraic structure by the underlying set.
6. When $$⟨⋅⟩\langle \,\cdot\,\rangle$$ is a function which takes sets as their inputs, it’s common to abuse $$⟨{x}⟩\langle\{x\} \rangle$$ as $$⟨x⟩\langle x\rangle$$. More generally it’s common to look at a finite set $${x1,…,xn}\{x_1, \ldots ,x_n\}$$ as the finite sequence $$x1,…,xnx_1, \ldots ,x_n$$. This happens for instance in logic. Also in linear algebra and it’s usual to go even further and talk about ‘linearly independent vectors’ instead of ‘linearly independent set’ — this is only an abuse when linear (in)dependence is defined for sets instead of ‘lists’.
7. ‘Consider the set $$A={x∈R:P(x)}A=\{x\in \mathbb R\colon P(x)\}$$‘. I’m probably the only person who reads this as the set being the whole equality $$A={x∈R:P(x)}A=\{x\in \mathbb R\colon P(x)\}$$ instead of $$AA$$ or $${x∈R:P(x)}\{x\in \mathbb R\colon P(x)\}$$, in any case it is an abuse. Another example of this is ‘multiply by $$1=221=\frac 2 2$$‘.
8. Denoting by $$++$$ both scalar addition and function addition.
9. Instead of $$((φ∧ψ)→ρ)((\varphi\land \psi)\to \rho)$$ people first abandon the out parentheses and use $$(φ∧ψ)→ρ(\varphi\land \psi)\to \rho$$ and then $$∧\land$$ is given precedence over $$→\to$$, yielding the much more common (though formally incorrect) $$φ∧ψ→ρ\varphi\land \psi\to \rho$$.
10. Even ignoring the problem in 1., the symbol $$∫xdx=x22\int x\,\mathrm dx=\frac {x^2}2$$ is ambiguous as it can mean a number of things. Under one of the common interpretations the equal sign doesn’t even denote an equality. I allude to that meaning here, (it is the same issue as with $$f=O(g)f=O(g)$$).
11. There’s also the very common ‘$$…\ldots$$‘ mentioned by Lucian in the comments.
12. Lucian also mentions $$C=R2\mathbb C=\mathbb R^2$$ which is an abuse sometimes, but not all the time, depending on how you define things.
13. Given a linear map $$LL$$ and $$xx$$ on its domain, it’s not unusual to write $$LxLx$$ instead of $$L(x)L(x)$$. I’m not sure if this can even be considered an abuse of notation because $$LxLx$$ is meaningless and we should be free to define $$Lx:=L(x)Lx:=L(x)$$, there’s no ambiguity. Unless, of course, you equate linear maps with matrices and this is an abuse. On the topic of matrices, it’s common to look at $$1×11\times 1$$ matrices as scalars.
14. Geometers like to say $$R⊆R2⊆R3\mathbb R\subseteq \mathbb R^2\subseteq \mathbb R^3$$.
15. Using $$Mm×n(F)\mathcal M_{m\times n}(\mathbb F)$$ and $$Fm×n\mathbb F^{m\times n}$$ interchangeably. On the same note, $$Am+n=Am×AnA^{m+ n}=A^m\times A^n$$ and $$(Am)n=Am×n\left(A^m\right)^n=A^{m\times n}$$.
16. I don’t know how I forgot this one. The omission of quantifiers.
17. Calling ‘well formed formulas’ by ‘formulas’.
18. Saying $$∀x(P(x)→Q(x))\forall x(P(x)\to Q(x))$$ is a conditional statement instead of a universal conditional statement.
19. Stuff like $$∃yP(x,y)∀x\exists yP(x,y)\forall x$$ instead of (most likely, but not certainly) $$∃y∀xP(x,y)\exists y\forall xP(x,y)$$.
20. The classic $$u=x2⟹du=2xdxu=x^2\implies \mathrm du=2x\mathrm dx$$.
21. This one disturbs me deeply. Sometimes people want to say “If $$AA$$, then $$BB$$” or “$$A⟹BA\implies B$$” and they say “If $$A⟹BA\implies B$$“. “If $$A⟹BA\implies B$$” isn’t even a statement, it’s part of an incomplete conditional statement whose antecedent is $$A⟹BA\implies B$$. Again: mathematics is to be parsed with priority over natural language.
22. Saying that $$x=y⟹f(x)=f(y)x=y\implies f(x)=f(y)$$ proves that $$ff$$ is a function.
23. Using $$f(A)f(A)$$ to denote $${f(x):x∈A}\{f(x)\colon x\in A\}$$. Why not stick to $$f[A]f[A]$$ which is so standard? Another possibility is $$f→(A)f^\to(A)$$ (or should it be square brackets?) which I learned from egreg in this comment.