RULE XX--SIGNS AND SYMBOLS OF COMPARISON

Simple Comparison Signs

• Arc
• Concave upward
  $a

  	verbose	convex-arc

• Concave downward
  $'

  	verbose	concave-arc

• Arrow
• Left-pointing
  $[33

  	verbose	left-arrow

• Right-pointing
• Contacted
  $o

  	verbose	right-arrow

• Uncontacted
  $33o

  	verbose	right-arrow

• Down-pointing
  $%33o

  	verbose	down-arrow

• Up-pointing
  $<33o

  	verbose	up-arrow

• Two-way
• Horizontal
  $[33o

  	verbose	two-way-arrow

• Vertical
  $<[33o

  	verbose	up-down-arrow

• Equals (is equal to)
• Normal
  .k

  	verbose	equals

• Boldface
  |.k

  	verbose	Bold equals

• Greater Than (is greater than)
• Normal
  .1

  	verbose	greater-than

• With curved sides
  ..1

  	verbose	greater than

• Identity (is congruent to; is identical to)
  _l

  	verbose	Identical-to

• Inclusion (is contained in; is a subset of)
  _"k

  	verbose	Subset-of

• Less than (is less than)
• Normal
  "k

  	verbose	less-than

• With curved sides
  ."k

  	verbose	less than

• Membership (is an element of; belongs to)
  @e

  	verbose	Element-of

• Parallel To (is parallel to)
  $l

  	verbose	parallel-to

• Perpendicular To (is perpendicular to)
  $p

  	verbose	Perpendicular

• Proportion (as)
  ;2

  	verbose	proportion

• Ratio (is to)
  "1

  	verbose	ratio

• Relation (is related to)
  ,r

  	verbose	related to

• Reverse Inclusion (contains; in logic, implies)
  _.1

  	verbose	reverse inclusion

• Reverse Membership (contains the element)
  @5

  	verbose	contains-as-member

• Tilde
• Simple (is related to; is similar to)
  @:

  	verbose	tilde

• Extended (is related to)
  @,:

  	verbose	extended tilde

• Variation (varies as)
  _=

  	verbose	varies as

• Vertical Bar (such that)
  \

  	verbose	vertical-line

Modified Comparison Signs

• Equals Sign
• Caret over
  ".k<_<]

  	verbose	equals with caret over

• Caret Under (is projective to, projective correspondence)
  ".k%_<]

  	verbose	projective to

• Degree sign over (is equal in degrees to)
  ".k<.*]

  	verbose	ring-above-equals

• Dot over (is approximately equal to)
  ".k<*]

  	verbose	dot-above-equals

• Dot over and dot under
  ".k%*<*]

  	verbose	geometrically-equal-to

• Equilateral triangle over
  ".k<$t]

  	verbose	delta-above-equals

• Inverted caret over
  ".k<_%]

  	verbose	equals with inverted caret over

• Left-pointing caret over
  ".k<;<]

  	verbose	equals with left pointing caret over

• Question mark over
  ".k<_8]

  	verbose	question-above-equals

• Right-pointing caret over
  ".k<;%]

  	verbose	greater-than-or-equal-to

• Two dots over and two dots under
  ".k%**<**]

  	verbose	equals with two dots over and two dots under

• Vertical bar over
  ".k<\]

  	verbose	equals with vertical bar over

• Horizontal Bar
• Caret over
  ":<_<]

  	verbose	horizontal bar with caret over

• Caret under (is perspective to, perspective correspondence)
  ":%_<]

  	verbose	perspective to

• Dot under
  ":%*]

  	verbose	horizontal bar with dot under

• Simple Tilde, Dot Under
  "@:%*]

  	verbose	tilde with dot under

Comparison Signs Compounded Vertically

• Arrow Combinations
• Right-pointing over left-pointing
  $33o$[33

  	verbose	rightwards-arrow-above-leftwards-arrow

• Right-pointing with upper barb over left-pointing with lower barb
  $33@o$,[33

  	verbose	rightwards-harpoon-above-leftwards-harpoon

• Right-pointing over boldface left-pointing
  $33o$_[33

  	verbose	right pointing arrow over boldface left pointing arrow

• Left-pointing over boldface right-pointing
  $[33$_33o

  	verbose	left-pointing arrow over boldface right-pointing arrow

• Boldface right-pointing over left-pointing
  $_33o$[33

  	verbose	boldface right-pointing over left-pointing arrow

• Boldface left-pointing over right-pointing
  $_[33$33o

  	verbose	boldface left-pointing arrow over right-pointing arrow

• Boldface right-pointing over boldface left-pointing
  $_33o$_[33

  	verbose	boldface right-pointing arrow over boldface left-pointing arrow

• Boldface left-pointing over boldface right-pointing
  $_[33$_33o

  	verbose	boldface left-pointing arrow over boldface right-pointing arrow

• Long right-pointing over short left-pointing
  $333o$[3

  	verbose	long right-pointing arrow over short left-pointing arrow

• Short right-pointing over long left-pointing
  $3o$[333

  	verbose	short right-pointing arrow over long left-pointing arrow

• Equivalence (is equivalent to)
  @<,<

  	verbose	geometrically-equivalent-to

• Greater Than
• Bar over greater than (is equal to or greater than)
  :.1

  	verbose	equal-to-or-greater-than

• Bar under greater than (is greater than or equal to)
  .1:

  	verbose	greater-than-or-equal-to

• Equals sign over greater than (is equal to or greater than)
  .k.1

  	verbose	equal to or greater than

• Equals sign under greater than (is greater than or equal to)
  .1.k

  	verbose	greater-than-or-equal-to

• Inclusion
• Bar over inclusion (is a subset of)
  :_"k

  	verbose	subset of

• Bar under inclusion (is a subset of)
  _"k:

  	verbose	Subset-of-or-equal-to

• Equals sign over inclusion (is a subset of)
  .k_"k

  	verbose	subset of

• Equals sign under inclusion (is a subset of)
  _"k.k

  	verbose	subset of

• Intersection (cap)
• Bar under intersection
  .%:

  	verbose	intersection with bar under

• Equals sign under intersection
  .%.k

  	verbose	intersection with equals under

• Less Than
• Bar over less than (is equal to or less than)
  :"k

  	verbose	equal-to-or-less-than

• Bar under less than (is less than or equal to)
  "k:

  	verbose	less-than-or-equal-to

• Equals sign over less than (is equal to or less than)
  .k"k

  	verbose	equal to or less than

• Equals sign under less than (is less than or equal to)
  "k.k

  	verbose	less-than-or-equal-to

• Logical Product (meet)
• Bar over logical product
  :@%

  	verbose	logical product with bar over

• Bar over and bar under logical product
  :@%:

  	verbose	logical product with bar over and bar under

• Bar over and equals sign under logical product
  :@%.k

  	verbose	logical product with bar over and equals sign under

• Bar under logical product
  @%:

  	verbose	logical product with bar under

• Equals sign over logical product
  .k@%

  	verbose	logical product with equals sign over

• Equals sign over and bar under logical product
  .k@%:

  	verbose	logical product with equals sign over and bar under

• Equals sign over and equals sign under logical product
  .k@%.k

  	verbose	logical product with equals sign over and equals sign under

• Equals sign under logical product
  @%.k

  	verbose	logical product with equals sign under

• Logical Sum (join)
• Bar over logical sum
  :@+

  	verbose	logical sum with bar over

• Bar over and bar under logical sum
  :@+:

  	verbose	logical sum with bar over and bar under

• Bar over and equals sign under logical sum
  :@+.k

  	verbose	logical sum with bar over and equals sign under

• Bar under logical sum
  @+:

  	verbose	logical sum with bar under

• Equals sign over logical sum
  .k@+

  	verbose	logical sum with equals sign over

• Equals sign over and bar under logical sum
  .k@+:

  	verbose	logical sum with equals sign over and bar under

• Equals sign over and equals sign under logical sum
  .k@+.k

  	verbose	logical sum with equals sign over and equals sign under

• Equals sign under logical sum
  @+.k

  	verbose	logical sum with equals sign under

• Reverse Inclusion
• Bar over reverse inclusion
  :_.1

  	verbose	reverse inclusion with bar over

• Bar under reverse inclusion
  _.1:

  	verbose	Superset-of-or-equal-to

• Equals sign over reverse inclusion
  .k_.1

  	verbose	reverse inclusion with equals sign over

• Equals sign under reverse inclusion
  _.1.k

  	verbose	reverse inclusion with equals sign under

• Tilde (is related to)
• Bar over double tilde
  :@:@:

  	verbose	double tilde with bar over

• Bar over single tilde
  :@:

  	verbose	minus-above-tilde

• Bar under double tilde
  @:@::

  	verbose	almost-equals-or-equals

• Bar under single tilde
  @::

  	verbose	asymptotically-equals

• Double tilde
  @:@:

  	verbose	almost-equals

• Equals sign over double tilde
  .k@:@:

  	verbose	double tilde with equals sign over

• Equals sign over single tilde
  .k@:

  	verbose	equals-sign-above-tilde-operator

• Equals sign under double tilde
  @:@:.k

  	verbose	double tilde with equals sign under

• Equals sign under single tilde
  @:.k

  	verbose	approximately-equals

• Union (cup)
• Bar under union
  .+:

  	verbose	union with bar under

• Equals sign under union
  .+.k

  	verbose	union with equals sign under

Comparison Signs Compounded Horizontally

• Arrow Combinations
• Up-pointing followed by down-pointing
  $<33o"$%33o

  	verbose	upwards-arrow-leftwards-of-downwards-arrow

• Down-pointing followed by up-pointing
  $%33o"$<33o

  	verbose	downwards-arrow-followed-by-upwards-arrow

• Up-pointing followed by boldface down-pointing
  $<33o"$%_33o

  	verbose	up pointing arrow followed by boldface down pointing arrow

• Down-pointing followed by boldface up-pointing
  $%33o$<_33o

  	verbose	down pointing arrow followed by boldface up pointing arrow

• Boldface up-pointing followed by down-pointing
  $<_33o"$$%33o

  	verbose	boldface up pointing arrow followed by down pointing arrow

• Boldface down-pointing followed by up-pointing
  $%_33o"$<33o

  	verbose	boldface down pointing arrow followed by up pointing arrow

• Boldface up-pointing followed by boldface down-pointing
  $<_33o"$%_33o

  	verbose	boldface up pointing arrow followed by boldface down pointing arrow

• Boldface down-pointing followed by boldface up-pointing
  $%_33o"$<_33o

  	verbose	boldface down pointing arrow followed by boldface up pointing arrow

• Greater Than
• Followed by less than
  .1""k

  	verbose	greater-than less-than

• Followed by equals sign followed by less than
  .1".k""k

  	verbose	greater-than equals less-than

• Less Than
• Followed by greater than
  "k".1

  	verbose	less-than greater-than

• Followed by equals sign followed by greater than
  "k".k".1

  	verbose	less-than equals greater-than

Comparison Signs Compounded by Superposition

• Dot
• Between bars of equals sign
  *@.k]

  	verbose	equals sign with dot inside

• Within inclusion sign
  *@_"k]

  	verbose	subset-with-dot

• Within reverse inclusion sign
  *@_.1]

  	verbose	superset-with-dot

• Equals Sign
• Through inclusion sign .k@_"k] or _"k@.k]

  	verbose	inclusion sign superposed equals sign

• Through reverse inclusion sign .k@_.1] or _.1@.k]

  	verbose	reverse inclusion sign superposed equals sign

• Greater Than
• Nest of two with straight sides (is large compared with)
  .1@.1]

  	verbose	much-greater-than

• Nest of two with curved sides
  ..1@..1]

  	verbose	curved much greater than

• Horizontal Bar
• Through inclusion sign
  :@_"k]

  	verbose	inclusion sign superposed horizontal bar

• Through reverse inclusion sign
  :@_.1]

  	verbose	reverse inclusion sign superposed horizontal bar

• Less Than
• Nest of two with straight sides (is small compared with)
  "k@"k]

  	verbose	much-less-than

• Nest of two with curved sides
  ."k@."k]

  	verbose	curved much less than

• Vertical Bar
• Through shaft of right-pointing arrow
  \@$33o]

  	verbose	right pointing arrow superposed vertical bar

• Through shaft of left-pointing arrow
  \@$[33]

  	verbose	left pointing arrow superposed vertical bar

§140. Arrows: A detailed discussion of the construction of arrows of many types is presented in Rule XXI. The arrows in the list of simple comparison signs are those which occur with the greatest frequency.

            If a right-pointing arrow has a full barb and a single shaft of ordinary length, is in regular type, and occurs by itself, it must be represented in its contracted form. If such an arrow is in nonregular type, is itself modified, or occurs as part of a more complex modification, it must be represented in its uncontracted form.

§141. Identity: This sign must not be used for is congruent to in geometry if another sign is employed for this purpose in ink print.

§142. Membership: This sign must not be mistaken for the Greek lower-case epsilon, even though it is sometimes referred to by that very name. This sign is genrally used when speaking about sets and the elements of which they are composed. When the Greek lower-case epsilon is used in the same textbook, the publisher usually makes a sufficient distinction between the two signs to prevent this confusion.

§143. Relation: When a letter or other sign is used between two expressions to show that they are related, the letter or sign used in this way must be regarded as a comparison sign. As such, it is subject to all the rules governing comparison signs and symbols. The letter R is frequently used in this situation.

§144. Tilde: When the tilde, simple or extended, occurs with a dot or caret directly over or directly under it, the combination is a modified sign of comparison (see §146). When it occurs directly over or directly under another simple comparison sign, the combination is a comparison sign compounded vertically (see §147).

§145. Vertical Bar: In addition to its use as a comparison sign meaning "such that," the vertical bar is used in several other ways in mathematics. It has already been listed as a sign of grouping and as a sign of operation. It is helpful to know that when the vertical bar means "such that" it is usually part of an expression within braces used for set notation, or in association with one of the quantifiers. However, it may also appear in other situations.

§146. Modified Comparison Signs: The modified comparison signs in the above list are constructed in accordance with the rules for the representation of modified expressions (see Rule XIV). Modified signs of comparison other than those in the above list must be constructed in accordance with the same principles.

§147. Comparison Signs Compounded Vertically: The transcriber must be represent a vertical arrangement of simple comparison signs as an unspaced horizontal succession of the corresponding simple comparison symbols, the first symbol corresponding to the upper-most sign. The braille reader must interpret a succession of unspaced simple comparison symbols as representing the fact that the corresponding signs are arranged vertically in ink print, in descending order, the uppermost sign corresponding to the first symbol. Comparison signs compounded vertically not shown in the list must be transcribed in accordance with the above principles.

§148. Intersection, Union, Logical Product, Logical Sum: The intersection, union, logical product, and logical sum signs, when unmodified, are not simple comparison signs, but operation signs (see §132 and §133, respectively).

§149. Comparison Signs Compounded Horizontally: The transcriber must represent a horizontal succession of comparison signs by placing the multipurpose indicator between the unspaced corresponding comparison symbols. Comparison signs compounded horizontally which are not shown in the list must be transcribed in accordance with the above principle.

§150. Comparison Signs Compounded by Superposition: Comparison signs compounded by superposition in the above list are constructed in accordance with the rules for representing superposition (see §93). Comparison signs compounded by superposition other than those in the above list must be constructed in accordance with the same principles.

§151. Spacing with Symbols of Comparison: A space must be left on either side of a comparison symbol. However, a space must not be left between the comparison symbol and any punctuation symbol, grouping symbol, or indicator which applies to it.