Most ambiguous and inconsistent phrases and notations in maths

Question

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.

Answer 1

1. 'The function $f(x)$'. No, the function is $f$.
2. Let $f$ and $g$ be real differentiable functions defined in $\mathbb R$. Some people denote $(f\circ g)'$ by $\dfrac{\mathrm df(g(x))}{\mathrm dx}$. Contrast with the above. I discuss this in greater detail here.
3. The differential equation $y'=x^2y+y^3$. Just a minor variant of 1. Correct would be $y'=fy+y^3$ where $f\colon I\to \mathbb R, x\mapsto x^2$, for some interval $I$.
4. This is one I find particularly disgusting. "If $t(s)$ is a function of $s$ and it is invertible, then $s(t)$ is the inverse", lol what? The concept of 'function of a variable' isn't even definable in a satisfiable way in $\sf ZFC$. Also $\left(\frac{\mathrm dy}{\mathrm dx}\right)^{-1}=\frac{\mathrm dx}{\mathrm dy}$. 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 $\langle\{x\} \rangle$ as $\langle x\rangle$. More generally it's common to look at a finite set $\{x_1, \ldots ,x_n\}$ as the finite sequence $x_1, \ldots ,x_n$. 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 $A=\{x\in \mathbb R\colon P(x)\}$'. I'm probably the only person who reads this as the set being the whole equality $A=\{x\in \mathbb R\colon P(x)\}$ instead of $A$ or $\{x\in \mathbb R\colon P(x)\}$, in any case it is an abuse. Another example of this is 'multiply by $1=\frac 2 2$'.
8. Denoting by $+$ both scalar addition and function addition.
9. Instead of $((\varphi\land \psi)\to \rho)$ people first abandon the out parentheses and use $(\varphi\land \psi)\to \rho$ and then $\land$ is given precedence over $\to$, yielding the much more common (though formally incorrect) $\varphi\land \psi\to \rho$.
10. Even ignoring the problem in 1., the symbol $\int x\,\mathrm dx=\frac {x^2}2$ 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 $f=O(g)$).
11. There's also the very common '$\ldots$' mentioned by Lucian in the comments.
12. Lucian also mentions $\mathbb C=\mathbb R^2$ which is an abuse sometimes, but not all the time, depending on how you define things.
13. Given a linear map $L$ and $x$ on its domain, it's not unusual to write $Lx$ instead of $L(x)$. I'm not sure if this can even be considered an abuse of notation because $Lx$ is meaningless and we should be free to define $Lx:=L(x)$, 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 $1\times 1$ matrices as scalars.
14. Geometers like to say $\mathbb R\subseteq \mathbb R^2\subseteq \mathbb R^3$.
15. Using $\mathcal M_{m\times n}(\mathbb F)$ and $\mathbb F^{m\times n}$ interchangeably. On the same note, $A^{m+ n}=A^m\times A^n$ and $\left(A^m\right)^n=A^{m\times n}$.
16. I don't know how I forgot this one. The omission of quantifiers.
17. Calling 'well formed formulas' by 'formulas'.
18. Saying $\forall x(P(x)\to Q(x))$ is a conditional statement instead of a universal conditional statement.
19. Stuff like $\exists yP(x,y)\forall x$ instead of (most likely, but not certainly) $\exists y\forall xP(x,y)$.
20. The classic $u=x^2\implies \mathrm du=2x\mathrm dx$.
21. This one disturbs me deeply. Sometimes people want to say "If $A$, then $B$" or "$A\implies B$" and they say "If $A\implies B$". "If $A\implies B$" isn't even a statement, it's part of an incomplete conditional statement whose antecedent is $A\implies B$. Again: mathematics is to be parsed with priority over natural language.
22. Saying that $x=y\implies f(x)=f(y)$ proves that $f$ is a function.
23. Using $f(A)$ to denote $\{f(x)\colon x\in A\}$. Why not stick to $f[A]$ which is so standard? Another possibility is $f^\to(A)$ (or should it be square brackets?) which I learned from egreg in this comment.

Answer 2

The inconsistent treatment of raising trig functions to powers: $$ \sin^n x \,.$$

Seriously, starting ab inito $$\sin^2 x$$ could mean either $$\sin( \sin(x) )$$ if you are a quantum mechanic and like to see everything as an operator or as $$(\sin x)^2$$ which is the conventional meaning.

So why is $$\sin^{-1} x$$ used for $$\arcsin x$$ (which is vaguely consistent with the former) instead of $$(\sin x)^{-1} $$ in keeping with the latter.

Answer 3

Double factorial $n!!=n(n-2)(n-4)\cdots$, where the product run through positive integers.
At the first time this notation confused me a lot because it looks the same as $(n!)!$ .
Similar argument about multifactorial.

Answer 4

In the first year at college, I was really confused with the notion of a sequence

$$\{ x_n : n \in \mathbb N \}$$

because this could also be a set! Then I discovered

$$(x_n)_{n \in \mathbb N}$$

And now I am fine with sequences.

Answer 5

$\mathbb N\;\;\;\;\;\;\;\;\;\;\;\;$

Answer 6

We often write $f(n) = O(g(n))$, when in fact $O(g(n))$ is a set, and should be written as $f(n) \in O(g(n))$. Similarly for other asymptotic notation, such as $\Theta$ and $\Omega$.

Answer 7

In financial mathematics one encounters index numbers, such as consumer price indexes, where a base year (such as 1992) is commonly specified by truly ghastly expressions such as

1992=100.

And no, they're not working modulo a divisor of 1892: they're referring to the fact that the index number for the base year is 100. It feels wrong to even bring this up in polite company.

Answer 8

Einstein summation convention is a self-explanatory example.

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Fourier transforms

I feel the majority of people (myself included) abuse notation when describing Fourier transforms.

For example, it's common to see:

$$\mathcal{F}\{{f(x)}\} = F(\omega)$$

to which my natural response is: uhm, no, I believe you mean

$$\mathcal{F_x}\{{f}\} = F$$

or perhaps

$$\mathcal{F}\{f(\,\cdot\,)\} = F(\,\cdot\,)$$

The original is clearly incorrect because $f(x)$ is the value of $f$ at some point $x$, and its Fourier transform $F$ is defined as $$F(\omega) = f(x) \delta(\omega)$$ which is clearly not what is intended.

Answer 9

Don't know whether these count as notational abuses as such, but a few common causes of confusion I have come across are

\begin{align} &\log(x)\text{ and }\ln(x)\\ &\sin^2(x)\text{ and }\sin(x)^2\\ &\sin^{-1}(x)\text{ and }\arcsin(x)\\ &\log_2(x)\text{ meaning }\log(\log(x)),\text{ & }\log_2(x)\text{ meaning base }2\\ &\log(\log(x))\text{ and }\log\log x \text{ etc. }\\ &\mathbb{S}^n\text{ in topology & }\mathbb{S}^n\text{ in geometry }\\ \end{align}

re last one, from MathWorld:

... A geometer would therefore regard the object described b $ x_1^2+x_2^2=R^2 $ as a $2$-sphere, while a topologist would consider it a $1$-sphere and denote it $\mathbb{S}^1$. Similarly, a geometer would regard the object described by $x_1^2+x_2^2+x_3^2=R^2 $ as a $3$-sphere, while a topologist would call it a $2$-sphere and denote it $\mathbb{S}^2$. Extreme caution is therefore advised when consulting the literature. Following the literature, both conventions are used in this work, depending on context, which is stated explicitly wherever it might be ambiguous.

This last one was included as a curiosity, but generally though, subscript and superscript is often abused / ambiguous unless explicitly stated.

... more interesting anecdotes and opinions.

Answer 10

Calculus I,II,III: The '$dx$'s in integrals and derivatives are just notation to help keep track of the important variables in a given problem, and they're otherwise meaningless in isolation.

Real Analysis: In fact, when the integration variable is unambiguous we may as well dispense with the differentials altogether and just denote the integral of $f$ over a region $R\subset \operatorname{dom}{f}$ as,

$$\int_{R}f.$$

Differential Geometry: Wait nevermind, $dx$ is a differential 1-form, $d$ is itself an operator, and the placement of '$dx$'s and so forth in notation for integrals couldn't be less optional, case and in point being the general Stokes' theorem:

$$\int_{R}d\omega=\oint_{\partial R}\omega.$$

Sorry we lied.

Edit: correctified for grammars.

Answer 11

The word "trivial." The many uses of this word include:

1.) The colloquial usage as a synonym for "easy."

2.) The trivial group which consists only of the identity element.

3.) The trivial ring which consists only of the multiplicative and additive identities.

4.) A trivial solution to an equation, often when a variable equals 0 (or constant in the cases of differential equations).

5.) Similar to #2 and #3, any object which satisfies the bare minimum of some particular definition but has no further structure. Often some sort of identity or null element.

6.) A trivial application of a theorem can refer to a special case where the truth of the theorem is more or less self-evident, e.g. a theorem in set theory which is obviously true when applied to the empty set.

This list could go on forever, but I'll stop here.

Note that the converse, of course, is the word "non-trivial," which is just as ambiguous.

I'll also note that although you may consider this a trivial answer, but I have had students get tripped up by my usage of the word "trivial" in lecture, so the ambiguity must be somewhat non-trivial.

Answer 12

My favorite has always been with Fourier transforms.

Suppose a particle is in a potential $V$ given by $$V(x) = V_0\cos(x/a).$$ Then $$V(k) = V_0\sqrt{\dfrac{\pi}{2}} \left( \delta(k-a) + \delta(k+a) \right).$$

I know physicists do this a lot. I am not sure about mathematicians.

Answer 13

Here is one which I think goes back to Euler: $$i^i = e^{-\frac{\pi}{2}}$$

Answer 14

This may be a regional thing, but when I started studying at a British university, so many of the lecturers wrote multiplication as a single . (full-stop). This got really confusing when, after having studied in the states I was used to $$ 0.5 + 0.5 = 1$$ whereas here it meant $$0.5 + 0.5 = 0 + 0 = 0$$

Answer 15

Outer measures aren't measures.

Answer 16

The inconsistency between the reading of "Negative" versus "Minus" has, in my opinion, been a thorn in the side of every teacher and student since their acceptance.

Answer 17

Lebesgue and (proper or improper) Riemann integrals written in the same way, ex.: $$\int_a^b f(x)dx.$$ I have even found "mixed integrals" where $\int_{A\cup B}f(x)+g(x)dx$ means (by writing the measure in the Lebesgue integral and marking the Riemann integral with $\mathscr{R}$) $\int_Af+gd\mu+\int_Afd\mu+\mathscr{R}\int_B g(x)dx$ (p. 422 here).

Lebesgue-Stieltjes and Riemann-Stieltjes integrals written both$$\int_a^bf(x)d\Phi(x).$$

Limit point can be found as meaning accumulation point, or adherent point, or point that is the limit of a subsequence of a sequence. I have found them used for all the three things even in the same textbook. I am not aware of other possible usages of the term.

Hilbert spaces sometimes intended to be separable, sometimes not necessarily separable.

Writing $A=B$ for isomorphisms $A\simeq B$ is something that can be confusing for less advanced students, especially if done to prove something without explaining why the isomorphism proves the desired result.

I find it particularly confusing when books do not specify the scalar field associated to a vector space which is beeing discussed, and I find it even more confusing when texts do not specify domains and codomains of maps. Such omissions are not too rare in engineering-oriented or older textbooks in general.

Answer 18

1. $\mathbb N \subset \mathbb Z \subset \mathbb Q \subset \mathbb R \subset \mathbb C \subset \mathbb H$

2. If $f \colon X \rightarrow Y$ is a map and $A \subseteq X$, then people often write $f(A)$ to denote $f'' A := \{ y \in Y \mid \exists x \in X : f(x) = y \}$, which can be quite confusing in cases where $A \in X$.

3. "Canonical" ...

4. $(a,b,c) = ((a,b),c) = (a,(b,c)) = f,$ where $f \colon \{0,1,2\} \rightarrow \{a,b,c\}$ is such that $f(0) = a$, $f(1) = b$ and $f(2) = c$.

5. "regular", "dense", "dimension","rank", "computable", "recursive", "closed", "compatible", "compactification"... and other notions which, in a given context, may have several different meanings.

6. "$f \colon X \rightarrow Y$ smooth" or similar expressions, where "smooth" may refer to $f$, $X$ or $Y$.

7. "pictures" and "diagrams" can sometimes be ambiguous to an extent where they don't mean anything to anyone - or maybe that's just me.

8. the "constant" $c$.

9. $a < b < c \in d$

10. $1 = \left( \frac 2 7 \right) \neq \frac 2 7$

11. $\prod_{i \in I} (X_i, \le_i) \subseteq \prod_{i \in J} (Y_j, \le_j)$

Answer 19

"$\subset$" is called proper subset (http://mathworld.wolfram.com/ProperSubset.html) and for example if $A = \{1,2,3\}$ then by definition, $\{1,2,3\}$ is not a proper subset of $A$ so we cannot write $\{1,2,3\} \subset A$ or $A \subset A$ simply. Instead we use $\subseteq$ but for some of the people, use of $\subset$ is ambiguous and it can include the equality of the sets. In other words, if $A \subset B$ and $B \subset A$ then $A = B$ rather than a contradiction.

Here is an example of an ambiguity from MSE: Proof verification: prove $A\subseteq B$ if and only if $A\cap B=A$.

Answer 20

"X is dense in itself" is not equivalent to "X is dense in X".

An ellipse is not an elliptical curve.

I absolutely detest the "infinity is not a number, it's just a concept".

I believe it's already been addressed - natural numbers, whole numbers, counting numbers.

Answer 21

It is true that 5 is a divisor of 0, and yet it is true that there are no divisors of 0.

Answer 22

$L/K$ for field extensions.

First, this is not a quotient, and second, this is not even some other kind of operation, just the statement that $K \subseteq L$ as fields.

Answer 23

When we list seasons it goes, summer, autumn, winter, spring, summer, autumn, and one says seasons occur cyclically.

In the group denoted (by those without broken pieces of chalk) as $\mathbf{Z}$ the elements are (half of them) go like this 1,2,3,4, etc without ever repeating and yet it is called the infinite cyclic group!

Answer 24

• $\sqrt{2}$ is usually read as "root two". The degree of root should be mentioned though (like "square root two").
• $\tan^{-1}$ is used for $\arctan$ mostly in Physics and Electronics.
• $j$ is used for $\sqrt{-1}$ in Electronics, while $i$ is used for it in Mathematics.
• Dots are used for time derivative in Pyhsics and Control Engineering (e.g.; $\ddot{y}+4\dot{y}+3y = 3\dot{x} + 2x + 8$)
• Vectors are sometimes denoted in bold ($\mathbf{x}$), with bar on ($\bar{x}$) or with arrow on ($\overset\rightarrow x$).
• Maybe the most frustrating things is when someone uses the word "integral" in the meaning of "anti-derivative" (e.g.; "Integral is the inverse of derivation.").
• The unit scalers are used in the place of units themselves (e.g.; "I bought two kilos of potatoes.")
• The $\nabla$ (del) operator can be very confusing sometimes.
• I have never understood the purpose of omitting the preceding zero in a decimal number (e.g.; writing $.5$ instead of $0.5$).
• In C++ language, log() is used for ln() and log10() is used for log().
• I still don't know what $2^{3^4}$ does equal to. $2^{(3^4)}=2^{81}$ or ${(2^3)}^4=8^4$?
• The division operator $/$ is misused. Example: $N/A\cdot m$ is written for $N/(A\cdot m)$.

Answer 25

$$\bigcup \{ A ,B, C\} = A \cup B \cup C $$ or $$\bigcup \mathcal A $$ where $\mathcal A $ is a set of sets. It should be

$$\bigcup_{X \in \mathcal A} X. $$ This can come up in topology sometimes (unioning over a collection of open sets), and can get rather confusing if you start doing things (edit:) like $\left(\bigcup \mathcal A\right) \cup U \cup V$ where $U$ and $V$ are just sets (eg. when adding open sets to a cover).

Edit: It's like defining $\sum S = \sum_{x\in S} x$ where $S \subset \mathbb R$ is finite (or ordered). Then usually you expect in something like $(\sum S) + x + y$ for $S$, $x$, and, $y$ to be the same sort of thing.