User’s manualfor the halloweenmath package

G. Mezzetti

November 1, 2019

Contents1 Package loading 2

2 Package usage 22.1 Ordinary symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Binary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 “Large” operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 “Fraction-like” symbols . . . . . . . . . . . . . . . . . . . . . . . . 32.5 “Arrow-like” symbols . . . . . . . . . . . . . . . . . . . . . . . . . 32.6 Extensible “arrow-like” symbols . . . . . . . . . . . . . . . . . . . 32.7 Extensible “over-/under-arrow-like” symbols . . . . . . . . . . . . 52.8 Script-style versions of amsmath’s over/under arrows . . . . . . . 5

3 Examples of use 83.1 Applying black magic . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Applications induced on power sets . . . . . . . . . . . . . . . . . 93.4 A comprehensive test . . . . . . . . . . . . . . . . . . . . . . . . . 10

List of Tables1 Ordinary symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Binary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 “Large” operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 “Fraction-like” symbols . . . . . . . . . . . . . . . . . . . . . . . . 35 “Arrow-like” symbols . . . . . . . . . . . . . . . . . . . . . . . . . 36 Extensible “arrow-like” symbols . . . . . . . . . . . . . . . . . . . 47 Extensible “over-/under-arrow-like” symbols . . . . . . . . . . . . 68 Over/under bats . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Extensible over/under arrows with reduced size . . . . . . . . . . 7

1

\mathleftghost \mathghost \mathrightghost\mathleftbat \mathbat \mathrightbat

Table 1: Ordinary symbols

\pumpkin \skull

Table 2: Binary operators

1 Package loadingLoad the halloweenmath package as any other LATEX2ε package, that is, via theusual \usepackage declaration:

\usepackage{halloweenmath}

Note that the halloweenmath package requires the amsmath package, and loadsit (without specifying any option) if it is not already loaded. If you want topass options to amsmath, load it before halloweenmath.

The halloweenmath package defines no options by itself; nevertheless, it doeshonor the [no]sumlimits options from the amsmath package.

2 Package usageThe halloweenmath package defines a handful of commands, all of which areintended for use in mathematical mode, where they yield some kind of symbolthat draws from the classic Halloween-related iconography (pumpkins, witches,ghosts, bats, and so on). Below, these symbols are grouped according to theirmathematical “rôle” (ordinary symbols, binary operators, arrows. . . ).

2.1 Ordinary symbolsTable 1 lists the ordinary symbols provided by the halloweenmath package.

2.2 Binary operatorsTable 2 lists the binary operators available. Note that each binary operator hasan associated “large” operator (see subsection 2.3).

2.3 “Large” operatorsTable 3 lists the “large” operators. Each of them is depicted in two variants:the variant used for in-line math and the variant used for displayed formulas.In the table, besides the “large” operators called \bigpumpkin1 and \bigskull,which are correlated to the binary operators \pumpkin and \skull, repectively,we find the commands \mathwitch and \reversemathwitch: note how thesetwo last command have a ∗-form that adds a black cat on the broomstick.

All the “large” operators listed in table 3 honor the [no]sumlimits optionsfrom the amsmath package.

1As a homage to Linus van Pelt, \greatpumpkin is defined as synonym of \bigpumpkin.

2

\mathwitch \reversemathwitch

\mathwitch* \reversemathwitch*

\bigpumpkin1 \bigskull

Table 3: “Large” operators

\mathcloud \reversemathcloud

Table 4: “Fraction-like” symbols

−−−< \leftbroom −>−− \rightbroom−−∈ \hmleftpitchfork 3−− \hmrightpitchfork

Table 5: “Arrow-like” symbols

2.4 “Fraction-like” symbolsThere are also two commands, listed on table 4, that yield symbols that aresomewhat similar to fractions, in that they grow in size when they are typesetin display style.2 They are intended to denote an unspecified subformula thatappears as a part of a larger one.

2.5 “Arrow-like” symbolsAs we’ll see in subsection 2.6, the halloweenmath package provides a series ofcommands whose usage parallels that of “extensible arrows” like \xrightarrowor \xleftarrow; but the symbols that those commands yield when used withan empty argument turn out to be too short, and it is for this reason that thehalloweenmath package also offers you the four commands you can see in table 5:they produce brooms, or pitchforks, having fixed length, which is approximatelythe same size of a \longrightarrow (−→). All of these symbols are treated asrelations.

2.6 Extensible “arrow-like” symbols

You are probably already familiar with the “extensible arrows” like abc−−→ and abc←−−;for example, you probably know that the input

\[\bigoplus_{i=1}^{n} A_{i} \xrightarrow{f_{1}+\dots+f_{n}} B

\]

produces this result:n⊕i=1

Aif1+···+fn−−−−−−→ B

2Another TEXnical aspect of these commands is that they yield an atom of type Inner.

3

abc...z

−−−−−< \xleftwitchonbroom{abc\dots z}abc...z

−>−−−− \xrightwitchonbroom{abc\dots z}abc...z

−−−−−< \xleftwitchonbroom*{abc\dots z}abc...z

−>−−−− \xrightwitchonbroom*{abc\dots z}abc...z

−−−−−∈ \xleftwitchonpitchfork{abc\dots z}abc...z

3−−−−− \xrightwitchonpitchfork{abc\dots z}abc...z

−−−−−∈ \xleftwitchonpitchfork*{abc\dots z}abc...z

3−−−−− \xrightwitchonpitchfork*{abc\dots z}

abc...z−−−−−< \xleftbroom{abc\dots z}

abc...z−>−−−− \xrightbroom{abc\dots z}

abc...z−−−−−∈ \xleftpitchfork{abc\dots z}

abc...z3−−−−− \xrightpitchfork{abc\dots z}

abc...z\xleftswishingghost{abc\dots z}

abc...z\xrightswishingghost{abc\dots z}

abc...z\xleftflutteringbat{abc\dots z}

abc...z\xrightflutteringbat{abc\dots z}

Table 6: Extensible “arrow-like” symbols

The halloweenmath package features a whole assortment of extensible symbolsof this kind, which are listed in table 6. For example, you could say

\[G \xrightswishingghost{h_{1}+\dots+h_{n}}

\bigpumpkin_{t=1}^{n} S_{t}\]

to get the following in print:

Gh1+···+hn

n

t=1

St

More generally, exactly as the commands \xleftarrow and \xrightarrow,on which they are modeled, all the commands listed in table 6 take one optionalargument, in which you can specify a subscript, and one mandatory argument,where a—possibly empty—superscript must be indicated. For example,

\[A \xrightwitchonbroom*[abc\dots z]{f_{1}+\dots+f_{n}} B

\xrightwitchonbroom*{f_{1}+\dots+f_{n}} C\xrightwitchonbroom*[abc\dots z]{} D

\]

results in

Af1+···+fn−>−−−−−−−

abc...zB

f1+···+fn−>−−−−−−− C −>−−−−

abc...zD

Note that, also in this family of symbols, the commands that involve a witchall provide a ∗-form that adds a cat on the broom (or pitchfork).

The commands listed above should not be confused with those presented insubsection 2.7.

4

2.7 Extensible “over-/under-arrow-like” symbolsThe commands dealt with in subsection 2.6 typeset an extensible “arrow-like”symbol having some math above or below it. But the amsmath package alsoprovides commands that act the other way around, that is, they put an arrowover, or under, some math, as in the case of

\overrightarrow{x_{1}+\dots+x_{n}}

that yields −−−−−−−−−→x1 + · · ·+ xn. The halloweenmath package provides a whole bunchof commands like this, which are listed in table 7, and which all share the samesyntax as the \overrightarrow command.

Although they are not extensible, and are thus more similar to math accents,we have chosen to include in this subsection also the commands listed in table 8.They typeset a subformula either surmounted by the bat produced by \mathbat,or with that symbol underneath. Their normal (i.e., unstarred) form pretendsthat the bat has zero width (but some height), whereas the starred variant takesthe actual width of the bat be into account; for example, given the input

\begin{align*}&x+y+z && x+y+z \\&x+\overbat{y}+z && x+\overbat*{y}+z

\end{align*}

compare the spacing you get in the two columns of the output:

x+ y + z x+ y + z

x+ y + z x+ y + z

2.8 Script-style versions of amsmath’s over/under arrowsThe commands listed in table 9 all produce an output similar to that of thecorresponding amsmath’s command having the same name, but stripped of thescript substring, with the only difference that the size of the arrow is smaller.More precisely, they use for the arrow the relative script size of the current size(that is, of the size in which their argument is typeset). For example, whilst\overrightarrow{x+y+z} yields −−−−−−→x+ y + z, \overscriptrightarrow{x+y+z}results in −−−−−−−−→x+ y + z (do you see the difference?), which, in the author’s humbleopinion, looks much better.

5

−−−−−−−−<abc . . . z \overleftwitchonbroom{abc\dots z}

−>−−−−−−−abc . . . z \overrightwitchonbroom{abc\dots z}

−−−−−−−−<abc . . . z \overleftwitchonbroom*{abc\dots z}

−>−−−−−−−abc . . . z \overrightwitchonbroom*{abc\dots z}

−−−−−−−∈abc . . . z \overleftwitchonpitchfork{abc\dots z}

3−−−−−−−abc . . . z \overrightwitchonpitchfork{abc\dots z}

−−−−−−−∈abc . . . z \overleftwitchonpitchfork*{abc\dots z}

3−−−−−−−abc . . . z \overrightwitchonpitchfork*{abc\dots z}

−−−−−−<abc . . . z \overleftbroom{abc\dots z}

−>−−−−−abc . . . z \overrightbroom{abc\dots z}

−−−−−−−−<abc . . . z \overscriptleftbroom{abc\dots z}

−>−−−−−−−abc . . . z \overscriptrightbroom{abc\dots z}

−−−−−∈abc . . . z \overleftpitchfork{abc\dots z}

3−−−−−abc . . . z \overrightpitchfork{abc\dots z}

−−−−−−−∈abc . . . z \overscriptleftpitchfork{abc\dots z}

3−−−−−−−abc . . . z \overscriptrightpitchfork{abc\dots z}

abc . . . z \overleftswishingghost{abc\dots z} abc . . . z \overrightswishingghost{abc\dots z}

abc . . . z \overleftflutteringbat{abc\dots z} abc . . . z \overrightflutteringbat{abc\dots z}

abc . . . z−−−−−−−−<

\underleftwitchonbroom{abc\dots z} abc . . . z−>−−−−−−−

\underrightwitchonbroom{abc\dots z}

abc . . . z−−−−−−−−<

\underleftwitchonbroom*{abc\dots z} abc . . . z−>−−−−−−−

\underrightwitchonbroom*{abc\dots z}

abc . . . z−−−−−−−∈

\underleftwitchonpitchfork{abc\dots z} abc . . . z3−−−−−−−

\underrightwitchonpitchfork{abc\dots z}

abc . . . z−−−−−−−∈

\underleftwitchonpitchfork*{abc\dots z} abc . . . z3−−−−−−−

\underrightwitchonpitchfork*{abc\dots z}

abc . . . z−−−−−−<

\underleftbroom{abc\dots z} abc . . . z−>−−−−−

\underrightbroom{abc\dots z}

abc . . . z−−−−−−−−<

\underscriptleftbroom{abc\dots z} abc . . . z−>−−−−−−−

\underscriptrightbroom{abc\dots z}

abc . . . z−−−−−∈ \underleftpitchfork{abc\dots z} abc . . . z3−−−−− \underrightpitchfork{abc\dots z}

abc . . . z−−−−−−−∈

\underscriptleftpitchfork{abc\dots z} abc . . . z3−−−−−−−

\underscriptrightpitchfork{abc\dots z}

abc . . . z \underleftswishingghost{abc\dots z} abc . . . z \underrightswishingghost{abc\dots z}

abc . . . z \underleftflutteringbat{abc\dots z} abc . . . z \underrightflutteringbat{abc\dots z}

Table 7: Extensible “over-/under-arrow-like” symbols

xyz \overbat{xyz} xyz \underbat{xyz}

Table 8: Over/under bats

6

←−−−−−−abc . . . z \overscriptleftarrow{abc\dots z} abc . . . z←−−−−−− \underscriptleftarrow{abc\dots z}

−−−−−−→abc . . . z \overscriptrightarrow{abc\dots z} abc . . . z−−−−−−→ \underscriptrightarrow{abc\dots z}

←−−−−−→abc . . . z \overscriptleftrightarrow{abc\dots z} abc . . . z←−−−−−→ \underscriptleftrightarrow{abc\dots z}

Table 9: Extensible over/under arrows with reduced size

7

3 Examples of useThis section illustrates the use of the commands provided by the halloweenmathpackage: by reading the source code for this document, you can see how theoutput presented below can be obtained.

3.1 Applying black magicThe symbol was invented with the intent to provide a notation for theoperation of applying black magic to a formula. Its applications range fromsimple reductions sometimes made by certain undergraduate freshmen, as in

2 sinx

2= sinx

to key steps that permit to simplify greatly the proof of an otherwise totallyimpenetrable theorem, for example(

sup { p ∈ N | p and p+ 2 are both prime })=∞

Another way of denoting the same operation is to place the broom and the witchover the relevant subformula:

−>−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−sup { p ∈ N | p and p+ 2 are both prime } =∞

Different types of magic, that you might want to apply to a given formula,can be distinguished by adding a black cat on the broom: for example, a studentcould claim that

2x sinx = 2 sinx2

whereas, for another student,

2x sinx = sin 3x

3.2 MonoidsLet X be a non-empty set, and suppose there exists a map

X ×X −→ X, (x, y) 7−→ P (x, y) = x y (1)

Suppose furthermore that this map satisfies the associative property

∀x ∈ X, ∀y ∈ X, ∀z ∈ X x (y z) = (x y) z (2)

Then, the pair (X, ) is called a semigroup, and denotes its operation.If, in addition, there exists in X an element with the property that

∀x ∈ X x = x = x (3)

8

the triple (X, , ) is called a monoid, and the element is called its unit.It is immediate to prove that the unit of a monoid is unique: indeed, if ′ isanother element of X having the property (3), then

′ = ′ =

(the first equality holds because ′ ∈ X and satisfies (3), and the secondbecause ∈ X and ′ satisfies (3)).

Let (X, , ) be a monoid. Since its operation is associative, we may set,for x, y, z ∈ X,

x y z =def (x y) z = x (y z)

More generally, since the order in which the operations are performed doesn’tmatter, given n elements x1, . . . , xn ∈ X, with n ∈ N, the result of

n

i=1

xi = x1 · · · xn

is unambiguously defined (it being if n = 0).A monoid (X, , ) is said to be commutative if

∀x ∈ X, ∀y ∈ X x y = y x (4)

In this case, even the order of the operands becomes irrelevant, so that, for anyfinite (possibly empty) set F , the notation i∈F xi also acquires a meaning.

3.3 Applications induced on power setsIf X is a set, we’ll denote by ℘(X) the set of all subsets of X, that is

℘(X) = {S : S ⊆ X }

Let f : A −→ B a function. Starting from f , we can define two other func-tions f : ℘(A) −→ ℘(B) and f : ℘(B) −→ ℘(A) in the following way:

for X ⊆ A, f (X) = { f(x) : x ∈ X } (5)for Y ⊆ B, f (Y ) = {x ∈ A : f(x) ∈ Y } (6)

In the case of functions with long names, or with long descriptions, we’ll alsouse a notation like f1 + · · ·+ fn to mean the same thing as (f1 + · · ·+ fn) .

For example,

sin (R) = [−1, 1]

sin([0, π]

)= [0, 1]

arcsin([

0, π2])

= [0, 1]

sin+ cos(R) =[−√2,√2]

log(]−∞, 0]

)= ]0, 1]

9

3.4 A comprehensive testA comparison between the “standard” and the “script” extensible over/underarrows:

−−−−−−−−−→f1 + · · ·+ fn 6=

−−−−−−−−−−−→f1 + · · ·+ fn

←−−−−−−−−−f1 + · · ·+ fn 6=

←−−−−−−−−−−−f1 + · · ·+ fn

←−−−−−−−→f1 + · · ·+ fn 6=

←−−−−−−−−−−→f1 + · · ·+ fn

f1 + · · ·+ fn−−−−−−−−−→6= f1 + · · ·+ fn−−−−−−−−−−−→

f1 + · · ·+ fn←−−−−−−−−−6= f1 + · · ·+ fn←−−−−−−−−−−−

f1 + · · ·+ fn←−−−−−−−→6= f1 + · · ·+ fn←−−−−−−−−−−→

A reduction my students are likely to make:

sinx

s= x in

The same reduction as an in-line formula: sin xs = x in.

Now with limits:n

i=1

i-th magic term2i-th wizardry

And repeated in-line: ni=1 xiyi.

The bold math version is honored:⟨something terribly

complicated

⟩= 0

Compare it with normal math:⟨something terribly

complicated

⟩= 0

In-line math comparison: f(x) versus f(x).There is also a left-facing witch:

sinx

s= x in

And here is the in-line version: sin xs = x in.

Test for \dots:

n1

i1=1

· · ·np

ip=1

i1-th magic factor2i1-th wizardry

· · · ip-th magic factor2ip -th wizardry

And repeated in-line: · · · ni=1 xiyi.

10

Now the pumpkins. First the bold math version::m⊕

h=1

n

k=1

Ph,k

Then the normal one:m⊕h=1

n

k=1

Ph,k

In-line math comparison: ni=1 Pi 6=

⊕ni=1 Pi versus

ni=1 Pi 6=

⊕ni=1 Pi.

Close test:⊕⊕

. And against the pumpkins:⊕⊕

.

In-line, but with \limits:m⊕h=1

n

k=1

Ph,k.

Binary: x y 6= x⊕ y. And in display:

ax y

x⊕ y⊗ b

Close test: ⊕⊕. And with the pumpkins too: ⊕⊕.In general,

n

i=1

Pi = P1 · · · Pn

The same in bold:n

i=1

Pi = P1 · · · Pn

Other styles: x y2 , exponent Z , subscript Wx y, double script 2tx y .

Clouds. A hypothetical identity: sin2 x+cos2 xcos2 x = . Now the same identity

set in display:sin2 x+ cos2 x

cos2 x=

Now in smaller size: sin x+cos x = 1.Specular clouds, bold. . .

←→

. . . and in normal math.←→

In-line math comparison: ↔ versus ↔ . Abutting: .

Ghosts: . Now with letters: H H h ab f wxy , and also2 3 + 5 2 − 3 i = 12 4

j . Then, what about x2 and z +1 = z2 + z ?In subscripts:

F +2 = F +1 + F

F +2 = F +1 + F

Another test: | | | | | | | | . We should also try this: .Let us now compare ghosts set in normal math with (a few words

to push the bold ghosts to the right) ghosts like these , which areset in bold math.

11

Extensible arrows:

Ax1+···+xn

−>−−−−−−−a

Bx+z

−>−−− C −>−− D

Ax1+···+xn

−>−−−−−−−a

Bx+z

−>−−− C −>−− D

Ax1+···+xn

−−−−−−−−<a

Bx+z

−−−−< C −−−< D

Ax1+···+xn

−−−−−−−−<a

Bx+z

−−−−< C −−−< D

And−>−−−−−−−−−−−−x1 + · · ·+ xn = 0 versus

−>−−−−−−−−−−−−x1 + · · ·+ xn = 0; or

−−−−−−−−−−−−−<x1 + · · ·+ xn = 0 versus

−−−−−−−−−−−−−<x1 + · · ·+ xn = 0.

Now repeat in bold:

Ax1+···+xn

−>−−−−−−−a

Bx+z

−>−−− C −>−− D

Ax1+···+xn

−>−−−−−−−a

Bx+z

−>−−− C −>−− D

Ax1+···+xn

−−−−−−−−<a

Bx+z

−−−−< C −−−< D

Ax1+···+xn

−−−−−−−−<a

Bx+z

−−−−< C −−−< D

And−>−−−−−−−−−−−−x1 + · · ·+ xn = 0 versus

−>−−−−−−−−−−−−x1 + · · ·+ xn = 0; or

−−−−−−−−−−−−−<x1 + · · ·+ xn = 0

versus−−−−−−−−−−−−−<x1 + · · ·+ xn = 0.

Hovering ghosts: x1 + · · ·+ xn = 0. I wonder whether there is enough space

left for the swishing ghost; let’s try again: (x1 + · · ·+ xn)y = 0! Yes, it lookslike there is enough room, although, of course, we cannot help the line spacing

going awry. Also try .

Ax1+···+xn

aB

x+zC D

Ax1+···+xn

aB

x+zC D

Another hovering ghost: x1 + · · ·+ xn = 0. Lorem ipsum dolor sit amet con-sectetur adipisci elit. Ulla rutrum, vel sivi sit anismus oret, rubi sitiunt silvae.Let’s see how it looks like when the ghost hovers on a taller formula, as inH1 ⊕ · · · ⊕Hk. Mmm, it’s suboptimal, to say the least.3

Under “arrow-like” symbols: x1 + · · ·+ xn = 0 and x+ y + z. There are

x1 + · · ·+ xn−−−−−−−−−−−−−<

= 0 and x+ y + z−>−−−−−−−−

as well.

3We’d better try y1 + · · ·+ yn, too; well, this one looks good!

12

Compare Ax1+···+xn

B with (add a few words to push it to the next line)

its bold version Ax1+···+xn

B.

Bats: . We are interested in seeing whether a bat affixed to a letteras an exponent causes the lines of a paragraph to be further apart than usual.Therefore, we now try f , also in bold f , then we type a few more words(just enough to obtain another typeset line or two) in order to see what happens.We need to look at the transcript file, to check the outcome of the followingtracing commands.

Asymmetric bats: , and also . Exponents: this is normal mathx y , while this is bold math x y . Do you note the difference?Let’s try subscripts, too: f g versus bold f g . Now, keep onrepeating some silly text, just in order to fill up the paragraph with a sufficientnumber of lines. Now, keep on repeating some silly text, just in order to fill upthe paragraph with a sufficient number of lines. Now, keep on repeating somesilly text, just in order to fill up the paragraph with a sufficient number of lines.That’s enough!

Hovering bats: x1 + · · ·+ xn = 0. I wonder whether there is enough spaceleft for the swishing bat; let’s try again: (x1 + · · ·+ xn)y = 0! Yes, it looks likethere is enough room (with the usual remark abut line spacing). Also try .

Ax1+···+xn

aB

x+zC D

Ax1+···+xn

aB

x+zC D

Another hovering bat: x1 + · · ·+ xn = 0.Under “arrow-like” bats: x1 + · · ·+ xn = 0 and x+ y + z.

Compare Ax1+···+xn

B with (add a few words to push it to the next line)

its bold version Ax1+···+xn

B.Test for checking the placement of the formulas that go over or under the

fluttering bat:

Aa long superscript

a long subscriptB

|

a long subscriptC

|D E

Aa long superscript

a long subscriptB

|

a long subscriptC

|D E

I’d say it’s now OK. . .

Extensible arrows with pitchfork:

Ax1+···+xn

3−−−−−−−a

Bx+z

3−−− C 3−− D

Ax1+···+xn

3−−−−−−−a

Bx+z

3−−− C 3−− D

Ax1+···+xn

−−−−−−−∈a

Bx+z

−−−∈ C −−∈ D

Ax1+···+xn

−−−−−−−∈a

Bx+z

−−−∈ C −−∈ D

13

And3−−−−−−−−−−−−x1 + · · ·+ xn = 0 versus

3−−−−−−−−−−−−x1 + · · ·+ xn = 0; or

−−−−−−−−−−−−∈x1 + · · ·+ xn = 0 versus

−−−−−−−−−−−−∈x1 + · · ·+ xn = 0. There are x1 + · · ·+ xn

−−−−−−−−−−−−∈= 0 and x+ y + z

3−−−−−−−−as well.

Now again, but all in boldface:

Ax1+···+xn

3−−−−−−−a

Bx+z

3−−− C 3−− D

Ax1+···+xn

3−−−−−−−a

Bx+z

3−−− C 3−− D

Ax1+···+xn

−−−−−−−∈a

Bx+z

−−−∈ C −−∈ D

Ax1+···+xn

−−−−−−−∈a

Bx+z

−−−∈ C −−∈ D

And3−−−−−−−−−−−−x1 + · · ·+ xn = 0 versus

3−−−−−−−−−−−−x1 + · · ·+ xn = 0; or

−−−−−−−−−−−−∈x1 + · · ·+ xn = 0

versus−−−−−−−−−−−−∈x1 + · · ·+ xn = 0. There are x1 + · · ·+ xn

−−−−−−−−−−−−∈= 0 and x + y + z

3−−−−−−−−−as well.

The big table of the rest:

Ax1+···+xn

−>−−−−−−− B −>−−−−−−−−−x1 + · · ·+ xn = 0 x1 + · · ·+ xn

−>−−−−−−−−−= 0

−>−−−−−−−−−−−−x1 + · · ·+ xn = 0 x1 + · · ·+ xn

−>−−−−−−−−−−−−= 0

Ax1+···+xn

−−−−−−−−< B−−−−−−−−−−<x1 + · · ·+ xn = 0 x1 + · · ·+ xn

−−−−−−−−−−<= 0

−−−−−−−−−−−−−<x1 + · · ·+ xn = 0 x1 + · · ·+ xn

−−−−−−−−−−−−−<= 0

Ax1+···+xn3−−−−−−− B 3−−−−−−−−−−

x1 + · · ·+ xn = 0 x1 + · · ·+ xn3−−−−−−−−−−= 0

3−−−−−−−−−−−−x1 + · · ·+ xn = 0 x1 + · · ·+ xn

3−−−−−−−−−−−−= 0

Ax1+···+xn−−−−−−−∈ B −−−−−−−−−−∈

x1 + · · ·+ xn = 0 x1 + · · ·+ xn−−−−−−−−−−∈= 0

−−−−−−−−−−−−∈x1 + · · ·+ xn = 0 x1 + · · ·+ xn

−−−−−−−−−−−−∈= 0

Now in bold. . . No, please, seriously, just the examples for the minimal size:in normal math we show A −>− B and C −∈ D and −>− and −∈, which we now repeatin bold math A −>− B and C −∈ D and −>− and −∈. Mmmh, the minimal sizeseems way too narrow: is it the same for the standard arrows? Let’s see:

A −→ B −→ −→

A←− B ←− ←−

A −>− B −>− −>−

A −−< B −−< −−<

Well, almost so, but the arrow tip is much more “discrete”. . .

14

To cope with this problem, \rightbroom and siblings have been introduced:for example, X −>−− Y .

A comparative table follows:

A −>−− B C 3−− DA −−−< B C −−∈ DA −→ B C =⇒ D

A←− B C ⇐= D

A −>−− B C 3−− D

A −−−< B C −−∈ D

Finally, y + x+ z = 0 versus y + x + z = 0, and also note that x2 6= x 2.Oh, wait, we have to check the bold version x2 6= x 2 too!

We’ve now gotten to skulls.

A B C

Skulls are similar to pumpkins, and thus to \oplus:

H1 · · · Hn

H1 ⊕ · · · ⊕Hn

H1 · · · Hn

As you can see, though, the dimensions differ slightly: ⊕ . Subscript: Ax y.Now the “large” operator version:

n

i=1

Hi = H1 · · · Hn

n⊕i=1

Hi = H1 ⊕ · · · ⊕Hn

n

i=1

Hi = H1 · · · Hn

In-line: ni=1Hi = H1 · · · Hn. Example of close comparison:

⊕X.

Now repeat in bold: ni=1 Hi = H1 · · · Hn.

Skulls look much gloomier than pumpkins: compare P U M = P withS K U = L � L. Why did I ever outline such a grim and dreary picture?The “large operator” variant, then, is truly dreadful! How could anybody writea formula like i j Ai⊗Bj? How much cheerer is i j Ai⊗Bj? And lookat the displayed version:

m

i=1

n

j=1

Ai ⊗Bj 6=m

i=1

n

j=1

Ai ⊗Bj

15

Comparison between math versions: x y is normal math, whereas x yis bold. Similarly, n

i−1Ki = L is normal, but ni−1 Ki = L is bold. And

now the displays: normal

m

i=1

n

j=1

Ai ⊗Bj 6=m

i=1

n

j=1

Ai ⊗Bj

versus boldm

i=1

n

j=1

Ai ⊗Bj 6=m

i=1

n

j=1

Ai ⊗Bj

math. Back to the normal font.

16