[UEB Maths] new draft of sections 10-12 for comment

[Janet Reynolds]

I am pasting below the new draft of Sections 10 on set notation, 11 on =
miscellaneous symbols and 12 on bars and dots etc over and under. I will =
send it separately as a Word attachment.=20

Section 10 is much the same, though I was surprised how few grade 1 =
indicators I needed. Section 11 is also much the same, though I did =
ponder about the spacing of the integral sign and made this note for us =
to think about:
[note: Some print texts show spaces after the integral sign and before =
the dx. Some use a change of font to make the dx stand apart. When I =
wrote this section I had them unspaced but after our discussions on word =
fragments I am less sure. One advantage of a space after the integral =
sign is that it separates the upper limit from the expression, eg the =
integral from 1 to 2 of x. On the other hand an unspaced integral =
without limits is easily read as "the", especially if the grade 1 =
passage indicators are several lines above.]

I've also included the full Greek alphabet, pasted from the Letters =
Rule. When Bruce and I first wrote the Maths we focused on things that =
were not covered anywhere else, but in this rewrite I've included things =
like the Greek letters in Section 11 and the rules for numeric mode in =
Section 2 because I think those will make the Maths rule a more useful =
document.

Section 12 is also much the same, the only one of those indicators you =
need to watch is the single cell "bar over" which may sometimes need a =
grade 1 indicator so I gave two examples to illustrate this.

I'm now working on the last three sections, and hope to send those out =
before our teleconference. If you have time to read them before we talk =
that's a bonus, but I thought the main thing was to get them finished so =
we can plan the checking over the next two weeks. We are working towards =
the deadline of the end of February.
Janet=20

10 Set Theory and Logic
10.1 Set Theory
.=3D.6	 union (upright U shape)
.=3D.8	intersection (inverted U shape)
.=3D@j 	null set (slashed zero)
.=3D7		complement (prime sign)
.=3D^e	is an element of (variant epsilon)
.=3D@^e	contains as an element (reverse variant epsilon)
.=3D^<	contained in, is a subset of (U open to right)
.=3D^>	contains, is a superset of (U open to left)
.=3D_^< contained in or equal to
.=3D_^> contains or equal to
.=3D.^< contained in, but not equal to (proper subset)
.=3D.^> contains, but is not equal to (proper superset)
.=3D@_< is a normal subgroup of (closed "less than" triangle)
.=3D@_> inverse "is normal subgroup" (closed "greater than" triangle)
.=3D__< is normal subgroup of or equal (closed "less than", line under)
.=3D__> inverse "normal subgroup or equal" (closed "greater than", line =
under)
.=3D._< normal subgroup but not equal (closed "less than", cancelled =
line
 under)
.=3D._> inverse "normal subgroup but not equal" (closed "greater than",		=
cancelled line under)

The symbol given for the complement above is the prime sign but an =
apostrophe can be used if an apostrophe is used in print. Union and =
intersection are signs of operation. Apart from the null set, the =
remaining signs above are all signs of comparison.


,if ,a "7 _<#a1 #b1 #c1 #d_>
& ;,b "7 _<#b1 #d1 #e1 #h_>
is #c ^e ,a.8,b
& is ,a.8,B ^< ,a.6,B

,a7.6,b7 "7 "<,a.8,b">7

Notice that grade 1 passage indicators were not needed for the examples =
above. All of the set notation symbols, (with the exception of the =
complement sign), have prefixes and cannot be confused with a =
contraction. No other single cell symbols such as fraction indicators, =
superscript indicators or radicals were involved. It would still be =
correct to use grade 1 passage indicators, words would then need to be =
uncontracted and single letters would no longer need grade 1 indicators.
10.2 Logic
.=3D@6	or (upright v shape)
.=3D@8	and (inverted v shape)
.=3D@?	"not" sign (line horizontal, then down at right)
.=3D,*	"therefore" (three dots in upright pyramid)
.=3D@/	"since" (three dots in inverted pyramid)
.=3D^5	"there exists" (reverse E)
.=3D^a	"for all" (inverted A)
.=3D_=3D	equivalent to (three horizontal lines)
.=3D_3	assertion ("is a theorem" sign; "T" lying on left side)
.=3D@_3 reverse assertion ("T" lying on right side)
.=3D^_3 "is valid" sign (assertion with double stem on "T")
.=3D._3 reverse "is valid" sign

The signs for "or" and "and" are signs of operation. "Therefore", =
"since", "there exists" and "for all" are print abbreviations. Signs of =
equivalence, assertion and validity are all signs of comparison.=20

,=3D ! /ate;t ;,p
_3 @?"<,p@6@?,p">

;;;^5 x ^a u @?"<u ^e x">;'

Again, the use of the grade 1 passage indicators in the second example =
is optional. The decision should be taken when looking at the chapter or =
section of text being transcribed. Factors include the number of single =
letters, the number of words intermingled with the expressions, and the =
presence of other symbols such as fraction or superscript indicators.
10.3 Gothic Script
Embellished capital letters are often used to name common sets such as =
the universal set E, the set of real numbers R or integers I. These vary =
in print from book to book but can be represented in braille by the =
script typeform indicators.

Example:
The set of real numbers=20
@2,r
=20
11 Miscellaneous Symbols and Topics
.=3D!		integral sign
.=3D@!	closed line integral (small circle halfway up)
.=3D@d	partial derivative (curly d or delta)
.=3D^d	del, nabla (inverted capital delta)
.=3D7		prime (when distinguished from apostrophe in print)
.=3D_"7 	is proportional to (varies as)
.=3D@9	tilde (swung dash)
.=3D@5	caret (hat)
.=3D"9	asterisk
.=3D"0	hollow dot =20
.=3D_\	vertical bar
.=3D#=3D	infinity
.=3D6		factorial sign (exclamation mark in print)
.=3D_[	angle sign
.=3D._[	measured angle sign=20
.=3D#_[	measured right angle sign =20
.=3D#l	parallel to=20
.=3D#-	perpendicular to=20
11.1 Calculus=20
The integral sign is unspaced from the function in braille as in print =
and its limits are treated as subscripts and superscripts. The dx at the =
end means "integrate with respect to x", and can also be written =
unspaced.=20

  ,if ;y "7 f"<x"> !n ! d]ivative is ;;(dy./dx) or ;;f7"<x"> & ! "pial =
d]ivative Is ;;(@dy./@dx)4

;;;!5#b9#c"<#bx"6#a">dx
  "7 .<x9#b"6x.>5#b9#c
  "7 "<#c9#b"6#c">"-"<#b9#b"6#b">
  "7 #ab"-#f "7 #f;'

The integral sign can clearly be confused with a contraction so some =
form of grade 1 indicator should always be used. Most integration =
exercises also involve superscripts so passage indicators are the normal =
choice. The first example has expressions embedded in the text and =
illustrates the use of grade 1 word indicators.
11.2 Binomial expressions
The binomial coefficient is defined as n factorial over r factorial =
times (n minus r) factorial. It can be written in print as a capital C =
with superscript n before and subscript r after, or as vector type =
brackets with n vertically above r. This latter symbol works better as a =
shape than a vector (refer to 14.3.3).

;;9N,C5R "7 "<n]r">
  "7 (N6./R6"<N"-R">6);'
11.3 Ratio and proportion
There is no special ratio sign in braille, the colon can be used as in =
print. When comparing ratios some books use four dots instead of an =
equals sign, this can be written as two unspaced colons. The proportion =
sign is a sign of comparison so would normally be spaced.

,! scale ( ? map is #a3#b1jjj
#b3#d 33 #f3#ab
,if ;y _"7 ;x !n ;y "7 kx
11.4 Unusual operation signs
The hollow dot and asterisk are often used for generalised operation =
signs. More unusual symbols can be treated as shapes (See Section 14).
However, the hollow dot should not be used to represent the abbreviation =
for degrees, which is covered in Section 3.

  "9 is distributive ov} .4 if=20
a"9"<b.4c"> "7 "<a"9b">.4"<a"9c">
11.5 The vertical bar
In print the vertical bar can have different meanings in different =
contexts. In braille, the one symbol is used regardless of the meaning =
of the vertical bar in print.=20

The set of (x, y) such that x plus y equals 6
_<"<x1y"> _\ x"6y "7 #f_>

Probability of A given B
,p"<,a_\,b">
11.6 Greek letters
Greek letters are used heavily in Mathematics. The alphabet is listed on =
the next page. Refer also to the Rule on Letters and their modifiers.

=20

Greek Alphabet=20
.=3D.a	=E1	 Greek alpha
.=3D.b	=E2 Greek beta
.=3D.g	=E3 Greek gamma
.=3D.d	=E4 Greek delta
.=3D.e	=E5 Greek epsilon
.=3D.z	=E6 Greek zeta
.=3D.:	=E7 Greek eta
.=3D.?	=E8 Greek theta
.=3D.i	=E9 Greek iota
.=3D.k	=EA Greek kappa
.=3D.l	=EB Greek lambda
.=3D.m	=EC Greek mu
.=3D.n	=ED Greek nu
.=3D.x	=EE Greek xi
.=3D.o	=EF Greek omicron
.=3D.p	=F0 Greek pi
.=3D.r	=F1 Greek rho
.=3D.s	=F2 or =F3 Greek sigma
.=3D.t	=F4 Greek tau
.=3D.u	=F5 Greek upsilon
.=3D.f	=F6 Greek phi
.=3D.&	=F7 Greek chi
.=3D.y	=F8 Greek psi
.=3D.w	=F9 Greek omega
.=3D,.a	=C1 capital Greek alpha
.=3D,.b	=C2 capital Greek beta
.=3D,.g	=C3 capital Greek gamma
.=3D,.d	=C4 capital Greek delta
.=3D,.e	=C5 capital Greek epsilon
.=3D,.z	=C6 capital Greek zeta
.=3D,.:	=C7 capital Greek eta
.=3D,.?	=C8 capital Greek theta
.=3D,.i	=C9 capital Greek iota
.=3D,.k	=CA capital Greek kappa
.=3D,.l	=CB capital Greek lambda
.=3D,.m	=CC capital Greek mu
.=3D,.n	=CD capital Greek nu
.=3D,.x	=CE capital Greek xi
.=3D,.o	=CF capital Greek omicron
.=3D,.p	=D0 capital Greek pi
.=3D,.r	=D1 capital Greek rho
.=3D,.s	=D3 capital Greek sigma
.=3D,.t	=D4 capital Greek tau
.=3D,.u	=D5 capital Greek upsilon
.=3D,.f	=D6 capital Greek phi
.=3D,.&	=D7 capital Greek chi
.=3D,.y	=D8 capital Greek psi
.=3D,.w	=D9 capital Greek omega
=20
=20
12 Bars and dots etc. over and under
.=3D:		bar over previous item=20
.=3D,:	bar under previous item
.=3D@:	line through previous item (cancellation, "not")
.=3D^:	simple right pointing arrow over previous item
.=3D,^:	simple right pointing arrow under previous item
.=3D^4	dot over previous item
.=3D,^4	dot under previous item
.=3D_:	tilde over previous item
.=3D,_:	tilde under previous item
.=3D":	hat over previous item
.=3D,":	hat under previous item
.=3D._:	arc over previous item=20
12.1 The definition of an item
The definition of an item is given in Section 7.1.=20

Use (x with bar over it) to represent the arithmetic mean.
,use x;: to repres5t ! >i?metic m1n4=20

(x with bar over it) equals 10 + 11 + 12 all over 3
;;;x: "7 (#aj"6#aa"6#ab./#c);'

x + y all with a bar under
<x"6y>,: =20

not equals
"7@: =20

0.3 with a dot over the 3 (the recurring decimal 0.33333...)
Note here that braille grouping signs are needed otherwise the dot would =
refer to the entire number.
#j4<#c>^4

derivatives x dot and x double dot
divatives x^4 & x.9<44>
[note: can we do anything about that double dot? I wondered about 45 45 =
d]

Most of these symbols have prefixes so cannot be confused with =
contractions and do not need grade 1 indicators.  The first two examples =
illustrate how the symbol for "bar over" will need a grade 1 symbol =
indicator if the expression is not already enclosed in grade 1 passage =
indicators.
12.2 Two indicators applied to the same item=20
If two indicators apply to the same item, then braille grouping symbols =
must be used to show which applies first.=20

x to the (y bar) power (no parentheses in print)
(x to the y power) bar (no parentheses in print)

;;;x9<y:>
<x9y>:;'

If this had been written without braille grouping symbols, it would have =
been ambiguous, as no precedence has been defined within the code.