Suggestions for good notation

Question

I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:

• Iverson introduced the notation [X] to mean 1 if X is true and 0 otherwise; so for example Σ_1≤n<x [n prime] is the number of primes less than x, and the unmemorable and confusing Kronecker delta function δ_n becomes [n=0]. (A similar convention is used in the C programming language.)

• The function taking x to x sin(x) can be denoted by x ↦ x sin(x). This has the same meaning as the lambda calculus notation λx.x sin(x) but seems easier to understand and use, and is less confusing than the usual convention of just writing x sin(x), which is ambiguous: it could also stand for a number.

• I find calculations with Homs and ⊗ easier to follow if I write Hom(A,B) as A→B. Similarly writing B^A for the set of functions from A to B is really confusing, and I find it much easier to write this set as A→B.

• Conway's notation for orbifolds almost trivializes the classification of wallpaper groups.

Has anyone come across any more similar examples of good notation that should be better known? (Excluding standard well known examples such as commutative diagrams, Hindu-Arabic numerals, etc.)

Answer 1

Since this one is on the front page again: In my personal notes, I have started writing sums/integrals over complicated index sets as $\sum \left( \text{summand} \mid \text{condition} \right)$, rather than a subscript. EG $$\sum {\large (} \log p \mid p \ \text{prime}, p \equiv 1 \bmod 4,\ p \leq N {\large )}$$ instead of $$\sum_{\substack{p \ \text{prime} \\ p \equiv 1 \bmod 4 \\ p \leq N}} \log p.$$

I find it a lot more readable. Additional benefits (1) it is reminiscent of set builder notation like $\{ \log p \mid p \ \text{prime}, p \equiv 1 \bmod 4,\ p \leq N \}$ (2) it means that the pairing between differential forms and cycles actually looks like a bilinear pairing.

Answer 2

Using $(a, b, ... )$ is handy to denote a column vector, which is the transpose of the row vector $[a, b, ... ]$, especially in linear text. Correspondingly, all displayed matrices should be written with brackets, not parentheses. This notation agrees with the usual identification of coordinates with column vectors.

Answer 3

The lack of a nice obviously symmetric notation for $\binom{a+b}{b}$ has bothered me; Dijkstra suggested in EWD 782 the notation $P(a,b)$, generalizing it also to $P(a_1,\ldots,a_k)$ for $\binom{a_1+\ldots+a_k}{a_1,\ldots,a_k}$. (Though I certainly disagree with him about $\binom{n}{k}$ being useless - you certainly do want to think about it that way a lot of the time.) I haven't actually had any reason to use this since I saw it but I can certainly think of times I would have.

Also the double-parentheses multichoose notation $\left(\!\binom{n}{k}\!\right)$ is nice because it lets you say "...and this is n multichoose k (which is equal to this binomial coefficient)" instead of just jumping directly to a binomial coefficient whose relevance may not be immediately obvious. But I suppose that's not really on the level of giving you a better way to look at things.

Answer 4

In the notation of Time scale calculus, the ordinary calculus derivative df/dt and the forward difference operator $\Delta f $ are both written as $f^\Delta$. Indefinite sums and indefinite integrals are both written as $\int{f(t)\Delta t}$ and called indefinite integrals. The context would say $\mathbb{T}=\mathbb{Z}, \mathbb{T}=\mathbb{R}$ or other $\mathbb{T}\subset\mathbb{R}$.

Answer 5

Multi-factorials are handy. Sometimes results can be expressed compactly by introducing a double factorial or possibly higher factorial. For example

$$\int_0^{\pi/2} \sin^{2n+1} \theta \:\: d\theta = \frac{(2n)!! }{ (2n+1)!!}$$

Answer 6

I think that Inuit numerals are cool. (http://en.wikipedia.org/wiki/Inuit_numerals) They are useful for vigesimal type things.

Answer 7

When I teach topology, I write $A\stackrel{\textrm{o}}{\subset} X$ to mean that A is an open subset of the topological space $X$, and I write $A \sqsubset X$ to mean that $A$ is a closed subset of $X$. I think it would drive me crazy to write the words open and closed constantly.

Answer 8

Cauchy-Binet as a generalized Pythagoras theorem.

Let $X$ be an $ n \times k$ matrix with $n \ge k$. For any $k$-index $I=i_1...i_k, \; 1 \le i_1 < ... < i_k \le n$, there is some advantage to denote by $X_I$, the determinant of the $k \times k$ submatrix of $X$ with rows indexed by $I$. For any two such $X,Y$, we can state the Cauchy-Binet formula as a pairing $$ \det (X^TY)= \sum_{I} X_I Y_I $$ where the sum is over all $n \choose k$ $k$-indices. This is a Pythagoras theorem for $X=Y$ since it says that the the volume-squared of the parallelepiped spanned by the $k$ columns of $X$ in $\mathbb{R}^n$ is the sum of squares of the volume of the projections on the $n \choose k$ $k$-dimensional coordinates.

For any $n \times m$ matrix $A$ with $m,n \ge k$ and $k$ indices $I,J$, we also denote by $A_{IJ}$ the determinant of the $k \times k$ submatrix of $A$ with rows indexed by $I$ and column indexed by $J$. Then for $X(m \times k)$ and $Y(n \times k)$, we have by Cauchy-Binet twice, $$ \det(X^TAY)=\det(X^T(AY))=\sum_{I}X_I(AY)_I =\sum_I X_I \det(A^IY)=\sum_I X_I \sum_J A_{IJ} Y_J,$$ where $A^I$ is the $k \times n$ matrix given by the rows of $A$ indexed by $I$ and we note that $(AY)_I= \det(A^IY)$ and $(A^I)^T_J=A_{IJ}$. This notation thus allows us to view Cauchy-Binet (usually stated with $m=n,A=I$) as an extension of the usual $x^TAy=\sum_{ij}A_{ij}x_iy_j$ for $k=1$.

Answer 9

I use the notation

$V \oplus^{\perp}W$

to denote orthogonal direct sum [Edit: direct sum of, say, subspaces of a given inner-product space].

Or

$(M,g) \times^{\perp} (N,g')$, or simply $M \times^{\perp} N$, to denote (orthogonal) cartesian product of Riemannian manifolds.

Answer 10

Instead of writing $$|x-y|\le \varepsilon,$$ I used to write $$x\lessgtr y\pm \varepsilon.$$ You may read it as $x$ is more-or-less $y$ plus-minus $\varepsilon$.

One may also write something like $$x\lessgtr e^{\pm\varepsilon}\cdot y$$ which is much better than $$|\ln(y/x)|\le\varepsilon$$

It is easier to read, especially if instead of $x$ and $y$ you have long expressions.

Answer 11

UMBRAL NOTATION (courtesy of Blissard)

$$(q.)^n = q_n$$

Concise, elegant, and suggestive, it often allows for both brevity and comprehensibility of presentation and short derivations of operational results.

Examples of umbral variables:

$q.^n = q_n = d$, a constant quantity, maybe a single variable or polynomial,

$q.^n = q_n = d^n $, a constant quantity raised to powers,

$q.^n = q_n = x^n$, a variable raised to a power,

$q.^n = b.^n = b_n$; an element of a sequence of numbers, such as the Bernoulli,

$q.^n = B.(x)^n = B_n(x)$, an element of a sequence of polynomials, .e.g., Bernoulli,

$q.^n = (.)!^n = (n)!$, an expression containing an integer parameter,

$q.^n = \binom{x}{.}^n = \binom{x}{n} $, ditto,

$q.^n = M.^n = M_n$, a sequence of matrices, differential ops, ... whatever.

The umbral maneuver in more complex expressions:

1. reduce an expression that contains the umbral entity to an analytic series in monomial powers of $q.$ when possible

2. then lower the positive integer exponent to a subscript

3. when in doubt apply

$$e^{q.xD_y} \; f(y) \; |_{y=0} = f(q.x +y) \;|_{y=0} = f(q.x) \;$$

4. sensible results can often be obtained for functions not analytic at the origin from

$$f(q.x) := e^{-(1-q.)x D_y} \; f(y) \; |_{y=x} = f(x-(1-q.)x) = f((1-(1-q.))x) $$

(See below on Newton interpolation. E.g., with $f(x)=x^s$, then $f(q.x) = (q.x)^s = q_s\; x^s \; := (1-(1-q.))^sx^s \; $ when convergent.)

Basic umbral operations:

1. binomial convolution for commuting quantities:

$$c_n = c.^n = (a. + b.)^n = \sum_{k=0}^n \binom{n}{k} \; a.^k \; b.^{n-k} = \sum_{k=0}^n \binom{n}{k} \; a_k\; b_{n-k},$$

2. formation of e.g.f.s and o.g.f.s:

$$E(t) = e^{q.t} = \sum_{n \geq 0} \; q.^n \; \frac{t^n}{n!} = \sum_{n \geq 0} \; q_n \; \frac{t^n}{n!}$$

and, with $q_0 =1$,

$$O(t) = \frac{1}{1-q.t} = \sum_{n \geq 0} (q.t)^n = \sum_{n \geq 0} q_n \; t^n = e^{\bar{q}.t}= \bar{O}(t)$$

with $\bar{q}_n = n! \; q_n$.

Then multiplication of formal series is

$$EA(t) \; EB(t) = e^{a.t} \; e^{b.t} = e^{(a.+b.)t} = e^{c.t} = EC(t) \; .$$

We can even disregard convergence of both series if we are interested only in formal compositional or multiplicative inverses or the formal Laplace transform of an e.g.f. to an o.g.f. since the production of the $n$-th coefficient of the target series depends only on the coefficients of order $n$ or less of the source series for these transformations.

Brevity and intelligibility in umbral versus conventional presentations:

The Euler transformation for e.g.f.s in umbral notation

1. $$ e^{a.x} = e^x \; e^{-(1-a. )x}$$

versus conventional notation

2. $$ \sum_{n \geq 0} a_n \; \frac{x^n}{n!} = e^x \; \sum_{n \geq 0} (-1)^n \; [\; \sum_{k=0}^n \; (-1)^k \; \binom{n}{k} \; a_k \; ] \; \frac{x^n}{n!} \; ,$$

and derivation of the conventional formula via the umbral,

3. $$ e^{a.x} = e^x \; e^{-x} e^{a.x} = e^x \; e^{-(1-a. )x}$$

$$ = e^x \; \sum_{n \geq 0} (-1)^n \; (1-a.)^n \; \frac{x^n}{n!} = e^x \; \sum_{n \geq 0} \; (-1)^n \; [\; \sum_{k=0}^n \; (-1)^k \; \binom{n}{k} a_k \; ] \; \frac{x^n}{n!} \;.$$

(23 keys tapped for eqn. 1 versus 102 keys for 2, not counting the appreciable difference in taps for blank spacing and latex spacing for readability. I could have saved a good fraction of a lifetime of math work by being able to use umbral notation in symbolic apps, study, and publications rather than sigma gymnastics.)

Natural generalizations via umbralization--monic to partition polynomials:

Stirling polynomials of the first kind, related to cyclic permutations, have the e.g.f.

$$e^{St1.(x)t} = e^{x \ln(1+t)} = (1+t)^x\; ,$$

and Stirling polynomials of the second kind, related to partitions of sets,

$$e^{St2.(x)t} = e^{x (e^t-1)} \, .$$

Flexibility and precision is afforded if some notation is used to indicate at what level the umbral exponent lowering is to occur. For example, with $\langle \cdots \rangle$ denoting evaluation of the expression between the angular brackets after reduction to a series in the umbral quantity and with $q_0=1$,

$$\langle \; e^{-\ln(1-q.t)} \;\rangle = \langle \frac{1}{1-q.t}\rangle = \langle \sum_{n \geq 0} q.^n \; t^n \; \rangle = \sum_{n\geq 0} \langle q.^n \rangle \; t^n =\sum_{n \geq 0} q_n \; t^n $$

and

$$e^{-\langle \ln(1-q.t)\rangle} = \exp[\langle \sum_{n \geq 1} q.^n \frac{t^n}{n}\rangle] = \exp(\sum_{n \geq 1} q_n \frac{t^n}{n}) ,$$

which is the e.g.f. for the cycle index partition polynomials of the symmetric groups, a.k.a. the Stirling partition polynomials of the first kind, OEIS A036039.

The natural generalization of the Stirling polynomials of the second kind is via the umbralization, for $q_0 = 1$,

$$e^{\langle e^{q.t}-1 \rangle} = \exp[ \; \sum_{n \ge 1} \; q_n \frac{t^n}{n!} \; ], $$

giving the complete Bell polynomials of the Faa di Bruno composition formula, a.k.a. the Stirling partition polynomials of the second kind, A036040.

Including an extra variable as in

$$e^{x\langle e^{q.t}-1 \rangle} = e^{Bell.(q_1,..,q.;x)t}$$

and

$$e^{-x\langle\ln(1-q.t)\rangle} = e^{Cyc.(q_1,..,q.;x)t}$$

allows better tracking of block sizes and combinatorics within each partition polynomial. (Note $q_n$ could be a commuting sequence of wedge products, curvature forms, or matrices, just as well as integers, giving basic formulas for the Chern and Pontryagin characteristic class polynomials.)

Now for a little umbral mojo,

$$ e^{St1.(St2.(x)) \; t} = e^{St2.(x) \; \ln(1+t)}=e^{x \; (\exp(\ln(1+t))-1)} = e^{xt} \; ,$$

so the two sets of binomial Sheffer polynomial sequences are an inverse pair under umbral composition; i.e.,

$$St1_n(St2.(x)) = x^n = St2_n(St1.(x)) \; .$$

This is reflected in their lower triangular coefficient matrices being a matrix inverse pair. The same umbral compositional inverse and matrix relationships hold for any Sheffer polynomial sequence. For a pair of inverse/reciprocal Appell Sheffer sequences, this is due to their moment e.g.f.s being multiplicative inverses rather than compositional inverse. The generic Sheffer sequence has a mix of these conditions, belonging to the semidirect product of the Appell and binomial Sheffer subgroups.

Actions of diff op often become more transparent with umbral calculus and/or underlying combinatorics are revealed:

With $a.$ treated as independent of $x$ and $D = d/dx$ for $f(x)= e^{b.x}$,

$$e^{a.D_x} \; f(x) = f(a.+x) = e^{b.(a.+x)}.$$

For the Bernoulli number and polynomials (or any Appell Sheffer sequence),

$B_n(x) = e^{b.D} \; x^n = (b.+x)^n$,

$D \; B_n(x) = D \; (b.+x)^n = n \; (b.+x)^{n-1} = n \; B_{n-1}(x)$

$e^{a.D} \; B_n(x) = B(x + a.) = (x+b.+a.)^n = (B.(x)+a.)^n = (B.(a.)+x)^n \; .$

Adopt the additional notational convenience $(:AB:)^n := A^nB^n$. Alternatively, we could use $q.^n = q_n = A^nB^n$, but this would veil aspects/intuitions inherited from the properties of the derivative.

From our previous feat of umbral mojo,

$$St1_n(xD) = n! \; \binom{xD}{n} = ST1_n(ST2.(:xD:) = \; :xD:^n \; = x^nD^n,$$

consistent with the actions

$xD \; [x^n] = n x^n$,

$(1 + a.)^{xD} \; [x^n] = (1 + a.)^{n} \; x^n$,

and

$(1 + a.)^{xD} \; f(x) = \sum_{n \geq 0} a_n \; \binom{xD}{n} \; f(x) = e^{a.:xD:} \; f(x) = f((1+a.)x).$

The operational definition of the Stirling polynomials of the second kind is

$$(xD)^n = (St2.(:xD:))^n = St2_n(:xD:).$$

From this the e.g.f. given above for the polynomials can easily be derived, and, consequently,

$$e^{txD} \; f(x) = e^{t \; St2.(:xD:)} \; f(x) = e^{(e^t-1):xD:} \; f(x)$$

$$ =f((e^t-1)x + x) = f(e^t x),$$

a dilation, consistent with $e^{txD} \; x^n = e^{tn} \; x^n$.

The Lah polynomials $L_n(x)$ (the normalized, unsigned Laguerre polynomials of order -1, related to partitions of sets into ordered sets or lists) have several useful Stokes-Rodrigues diff op reps,

$$ x \; :Dx:^n \; x^{-1} = x \; D^n x^n x^{-1} = x^{-n+1} (xDx)^n x^{-1} = x^{-n} (x^2D)^n = L_n(:xD:) ,$$

and the e.g.f.

$$e^{L.(x)t} = e^{x \frac{t}{1-t}},$$

so

$$ e^{tx^2D} \; f(x) = e^{t \; :xL.(:xD:):} \; f(x) = e^{:\frac{tx^2}{1-tx}D:} \; f(x)$$

$$= f(\frac{tx^2}{1-tx} + x) = f(\frac{x}{1-tx}), $$

a special linear fractional, or Moebius, transformation.

(The inverse ops to the translation, $e^{tD}$; dilation, $e^{txD}$; and special linear fractional, $e^{tx^2D}$, transformations of SL2 are given by negating $t$, which is equivalent to additive ($t+(-t)=0$), multiplicative ($e^t \cdot 1/e^t =1$), and compositional inversion ($\frac{x}{1-tx}|_{x \to\frac{x}{1+tx}}=x$), respectively. SL2 becomes associated to the underlying combinatorics of the binomial Sheffer polynomials $St2_n(x)$ and $L_n(x)$ via the umbral symbolic calculus.)

Interpolation methods and results of integral transforms are naturally suggested and expressed in umbral notation:

Laplace transform of e.g.f. to o.g.f.s with $q_0=1$:

$$\int_{0}^\infty \; e^{q.xt} \; e^{-t} dt = \int_{0}^\infty \; e^{-t(1-q.x)} dt = \frac{1}{1-q.x}$$

Mellin transform interpolation to continuous indices (related to Ramanujan's master formula, e.g., the Bernoulli polynomials are essentially discrete samples of the Hurwitz zeta function):

$$ q_{-s} = (q.)^{-s} := \int_{0}^\infty \; e^{-q.t} \; \frac{t^{s-1}}{(s-1)!} \; dt$$

Newton interpolation to continuous indices:

$$q_s = (q.)^s = (1-(1-q.))^s = \sum_{m=0}^\infty \; (-1)^m \; \binom{s}{m} \;\sum_{k=0}^m \; (-1)^k \; \binom{m}{k} \; q_k$$

$$= \int_{0}^\infty \; e^{-(1-(1-q.))t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt= \int_{0}^\infty \; e^{-t} \; e^{(1-q.)t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt$$

$$= \int_{0}^\infty \; e^{-q.t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt$$

via analytic continuation. If $q_n = D_x^n$, the action of the integral on $H(x) \frac{x^\alpha}{\alpha!}$, where $H(x)$ is the Heaviside step function, gives a classic fractional calculus in terms of an analytically-continued (AC) Mellin convolution (essentially that for the AC Euler beta function integral) for the interpretation of $D_x^{s}$ for real or complex $s$.

See Boole, Blissard, Sylvester, Cayley, Appell, (Jensen?), Heaviside, Steffensen, Sheffer, Pincherle, Bell, Riordan, Rota, Taylor, Roman, Ray, and Lenart for contributions to and influences on the development of the umbral symbolic calculus.

Answer 12

To resolve the ambiguity when using multiple instances of $\pm$ within a single expression, I use $\pm_n$ and $\mp_n$. $\pm_a$ must be equal to $\pm_b$ iff $a=b$.

Answer 13

I recently saw the following notation in the context of divisors on algebraic varieties, and I liked it very much.

Suppose that $D$ and $E$ are reduced divisors on a normal algebraic variety $X$. One can use $D \vee E$ to denote the reduced divisor with support equal to $D + E$ and $D \wedge E$ to denote $(D + E) - (D \vee E)$. I could imagine variants on this if $D$ and $E$ are non-reduced (involving taking max's, respecitvely mins, of the divisors component-wise).

EDIT: I'm slightly curious as to why this was downvoted. I guess it's too common to be interesting?

Answer 14

Has anyone come across any more similar examples of good notation that should be better known?

Some interesting glyphs:

1. Combinatorial Principles in Set Theory:

   • ♣ (Clubsuit)
   • ◊ (Diamondsuit)

2. Bisimulation:

   • Given two states p and q in S, p is bisimilar to q, written p ~ q, if there is a bisimulation R such that (p,q) is in R.

3. Boxplus operator in Coding Theory

   • ⊞

Answer 15

In algebra, it is very useful to write $J\cap A$ for the inverse image of an ideal $J$ in a ring $B$ under a homomorphism $f:A\to B$, rather than $f^{-1}(J)$. I normally also omit the morphism and write $IB$ for the ideal generated by the image in $B$ of an ideal $I$ in $A$, rather than $f(I)B$.

Answer 16

I like the notation $A\mathrel{\in\in}\mathcal C$ for objects (rather than morphisms) in a category (not my own invention).

Answer 17

The notation of $u \mathbin{[\wedge]} v$ for the product of Lie-algebra valued differential forms (I prefer this to another variant $[u \wedge v]$).

Writing the tangent natural tranformation as $\tau: T \rightarrow \operatorname{Id}$ which expands to $\tau_M : TM \rightarrow M$.

I've seen some category theorists write $\operatorname{Hom}_A[a,b]$ for the homset for a category $A$, which gets a bit long-winded when $A$ is a category with a long name like $\mathrm{Ring}$ or $R\text-\mathrm{Mod}$. It seems much more natural to write $A[a,b]$.

Personally, I like $\bar{\mathbb{R}}$ to write the order completions of the reals. It's less clumsy than $\mathbb{R} \cup \infty$.

Answer 18

Many textbooks say that the normal distribution is the one whose density is $$ x\mapsto \frac1{\sqrt{2\pi}} \times \frac1\sigma \exp\left( -\frac12 \left( \frac{x-\mu}\sigma \right)^2 \right) $$ and the gamma distribution is the one whose density is $$ x\mapsto \frac1{\Gamma(\alpha)} \times \lambda^\alpha x^{\alpha-1} e^{-\lambda x} \text{ for } x>0$$ and the Pareto distribution is the one whose density is $$ x\mapsto \frac{\alpha\kappa^\alpha}{x^{\alpha+1}} \text{ for } x>\kappa $$ etc.

I prefer to say that the normal distribution is $$ \frac1{\sqrt{2\pi}} \times \exp\left( -\frac12\left( \frac{x-\mu}\sigma\right)^2 \right)\, \frac{dx}\sigma $$ and the gamma distribution is $$ \frac1{\Gamma(\alpha)} \times (\lambda x)^{\alpha-1} e^{-\lambda x} (\lambda \, dx) \text{ for } x>0 $$ and the Pareto distribution is $$ \frac\alpha{(x/\kappa)^{\alpha+1}} \, \frac{dx}\kappa \text{ for } x>\kappa, $$ and so on.

This shows that each scale parameter, or its reciprocal, the rate parameter or intensity parameter, goes with an $\text{“} x\text{,”}$ and suggests a substitution to be used when thinking about integrating.

Answer 19

Many years ago, I invented the notation $\mathcal{Lex}_{T}(X)$ as the function equal to $1$ if and only if the consideration of $X$ in the theory $T$ entails no contradiction and $0$ otherwise. Lex is both "law" in Latin and a shortcut for "Logical existence", which in some sense is the only law of mathematics.

Answer 20

I like the notation $f:A\cong\subseteq B$ for "$f$ is an embedding of $A$ into $B$." The idea is that the relation of embeddability is obtained by composing the relations "isomorphic to" and "substructure of."

Answer 21

$x\gtrless 0$ to denote that $x\in\mathbb{R}\setminus\{0\}$. Or, if $x\notin\mathbb{R}$ but rather $x\in\mathbb{Z}\setminus\{0\}$ for example, then we write $x\in\mathbb{Z}_{\gtrless 0}$. I prefer the $\gtrless$ notation because it is simpler and more understandable when first met with the eye, and I hope it is set in stone sooner or later.

I also came across the following notation: $$\sum_p'f(x).$$ The little dash $'$ on top of the sigma $\Sigma$ denotes that $p$ is prime. I like this, but I don't feel this is necessary because we can write $$\sum_{p \ \text{prime}}f(x)$$ but when looking at both conventions, the former looks a lot neater.

Also, we have notation $>$, $\gg$ and $\ggg$ with $<$, $\ll$, and $\lll$ but I was thinking, is there a notation to denote that a value $a$ is not much less than a value $b$? I came up with $$a\overline{<} b\tag*{$a$ is not much less than $b$}$$ $$a\overline{>}b\tag*{$a$ is not much greater than $b$}$$

I have also seen that the symbol for concatenation is sometimes $||$, but if I was to see $$A \ || \ B$$ then my first thought would be $A$ is parallel to $B$, but that is just me. With some research, I discovered that there is however some kind of official notation, such that $A^\frown B$ but I don't like it.

I also went extreme and invented this: $$\mathop{\LARGE\Omega}_{\substack{x=k \\ \\ R}}^{x_0} (x, y)$$ to define a set of coordinates; a relation. Here, $x\to x_0$ and $R$ simply denotes the rule, $y = \cdots$. This way, we can write stuff like $$\mathop{\LARGE\Omega}_{\substack{x=36 \\ \\ y = 2x+1}}^{144}(x, y)$$ And then if we don't have a rule, but something like $y\geq0$ for example, then we can write a double index to refer to what value $y$ tends towards, namely $y_0$.

Oh and I almost forgot, I thought that maybe we can symbolise contradiction? For instance, I want to prove that $\sqrt{2}$ is irrational. I would first suppose it is rational, then come to a conclusion that contradicts this statement. Since most of us use $\Box$ in notation of completing a proof, I decided that I could use $\Diamond$ in notation of a contradiction. I thought about it because it is like a titled box $-$ a bit like approaching the proof $\Box$ on a different angle $\Diamond$, if you get what I am saying.

I am not taking this too seriously, for I was just being creative, but I believe there's nothing wrong in trying new things out. Any thoughts?