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Nemeth Braille Code for Mathematics and Science
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RULE XX--SIGNS AND SYMBOLS OF COMPARISON

Simple Comparison Signs
  • Arc
    • Concave upward
      $a
      Example as described in the content
      verbose
      convex-arc
      Listen to MathSpeak
    • Concave downward
      $'
      Example as described in the content
      verbose
      concave-arc
      Listen to MathSpeak
  • Arrow
    • Left-pointing
      $[33
      Example as described in the content
      verbose
      left-arrow
      Listen to MathSpeak
    • Right-pointing
      • Contacted
        $o
        Example as described in the content
        verbose
        right-arrow
        Listen to MathSpeak
      • Uncontacted
        $33o
        Example as described in the content
        verbose
        right-arrow
        Listen to MathSpeak
    • Down-pointing
      $%33o
      Example as described in the content
      verbose
      down-arrow
      Listen to MathSpeak
    • Up-pointing
      $<33o
      Example as described in the content
      verbose
      up-arrow
      Listen to MathSpeak
    • Two-way
      • Horizontal
        $[33o
        Example as described in the content
        verbose
        two-way-arrow
        Listen to MathSpeak
      • Vertical
        $<[33o
        Example as described in the content
        verbose
        up-down-arrow
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  • Equals (is equal to)
    • Normal
      .k
      Example as described in the content
      verbose
      equals
      Listen to MathSpeak
    • Boldface
      |.k
      Example as described in the content
      verbose
      Bold equals
      Listen to MathSpeak
  • Greater Than (is greater than)
    • Normal
      .1
      Example as described in the content
      verbose
      greater-than
      Listen to MathSpeak
    • With curved sides
      ..1
      Example as described in the content
      verbose
      greater than
      Listen to MathSpeak
  • Identity (is congruent to; is identical to)
    _l
    Example as described in the content
    verbose
    Identical-to
    Listen to MathSpeak
  • Inclusion (is contained in; is a subset of)
    _"k
    Example as described in the content
    verbose
    Subset-of
    Listen to MathSpeak
  • Less than (is less than)
    • Normal
      "k
      Example as described in the content
      verbose
      less-than
      Listen to MathSpeak
    • With curved sides
      ."k
      Example as described in the content
      verbose
      less than
      Listen to MathSpeak
  • Membership (is an element of; belongs to)
    @e
    Example as described in the content
    verbose
    Element-of
    Listen to MathSpeak
  • Parallel To (is parallel to)
    $l
    Example as described in the content
    verbose
    parallel-to
    Listen to MathSpeak
  • Perpendicular To (is perpendicular to)
    $p
    Example as described in the content
    verbose
    Perpendicular
    Listen to MathSpeak
  • Proportion (as)
    ;2
    Example as described in the content
    verbose
    proportion
    Listen to MathSpeak
  • Ratio (is to)
    "1
    Example as described in the content
    verbose
    ratio
    Listen to MathSpeak
  • Relation (is related to)
    ,r
    Example as described in the content
    verbose
    related to
    Listen to MathSpeak
  • Reverse Inclusion (contains; in logic, implies)
    _.1
    Example as described in the content
    verbose
    reverse inclusion
    Listen to MathSpeak
  • Reverse Membership (contains the element)
    @5
    Example as described in the content
    verbose
    contains-as-member
    Listen to MathSpeak
  • Tilde
    • Simple (is related to; is similar to)
      @:
      Example as described in the content
      verbose
      tilde
      Listen to MathSpeak
    • Extended (is related to)
      @,:
      Example as described in the content
      verbose
      extended tilde
      Listen to MathSpeak
  • Variation (varies as)
    _=
    Example as described in the content
    verbose
    varies as
    Listen to MathSpeak
  • Vertical Bar (such that)
    \
    Example as described in the content
    verbose
    vertical-line
    Listen to MathSpeak
Modified Comparison Signs
  • Equals Sign
    • Caret over
      ".k<_<]
      Example as described in the content
      verbose
      equals with caret over
      Listen to MathSpeak
    • Caret Under (is projective to, projective correspondence)
      ".k%_<]
      Example as described in the content
      verbose
      projective to
      Listen to MathSpeak
    • Degree sign over (is equal in degrees to)
      ".k<.*]
      Example as described in the content
      verbose
      ring-above-equals
      Listen to MathSpeak
    • Dot over (is approximately equal to)
      ".k<*]
      Example as described in the content
      verbose
      dot-above-equals
      Listen to MathSpeak
    • Dot over and dot under
      ".k%*<*]
      Example as described in the content
      verbose
      geometrically-equal-to
      Listen to MathSpeak
    • Equilateral triangle over
      ".k<$t]
      Example as described in the content
      verbose
      delta-above-equals
      Listen to MathSpeak
    • Inverted caret over
      ".k<_%]
      Example as described in the content
      verbose
      equals with inverted caret over
      Listen to MathSpeak
    • Left-pointing caret over
      ".k<;<]
      Example as described in the content
      verbose
      equals with left pointing caret over
      Listen to MathSpeak
    • Question mark over
      ".k<_8]
      Example as described in the content
      verbose
      question-above-equals
      Listen to MathSpeak
    • Right-pointing caret over
      ".k<;%]
      Example as described in the content
      verbose
      greater-than-or-equal-to
      Listen to MathSpeak
    • Two dots over and two dots under
      ".k%**<**]
      Example as described in the content
      verbose
      equals with two dots over and two dots under
      Listen to MathSpeak
    • Vertical bar over
      ".k<\]
      Example as described in the content
      verbose
      equals with vertical bar over
      Listen to MathSpeak
  • Horizontal Bar
    • Caret over
      ":<_<]
      Example as described in the content
      verbose
      horizontal bar with caret over
      Listen to MathSpeak
    • Caret under (is perspective to, perspective correspondence)
      ":%_<]
      Example as described in the content
      verbose
      perspective to
      Listen to MathSpeak
    • Dot under
      ":%*]
      Example as described in the content
      verbose
      horizontal bar with dot under
      Listen to MathSpeak
  • Simple Tilde, Dot Under
    "@:%*]
    Example as described in the content
    verbose
    tilde with dot under
    Listen to MathSpeak
Comparison Signs Compounded Vertically
  • Arrow Combinations
    • Right-pointing over left-pointing
      $33o$[33
      Example as described in the content
      verbose
      rightwards-arrow-above-leftwards-arrow
      Listen to MathSpeak
    • Right-pointing with upper barb over left-pointing with lower barb
      $33@o$,[33
      Example as described in the content
      verbose
      rightwards-harpoon-above-leftwards-harpoon
      Listen to MathSpeak
    • Right-pointing over boldface left-pointing
      $33o$_[33
      Example as described in the content
      verbose
      right pointing arrow over boldface left pointing arrow
      Listen to MathSpeak
    • Left-pointing over boldface right-pointing
      $[33$_33o
      Example as described in the content
      verbose
      left-pointing arrow over boldface right-pointing arrow
      Listen to MathSpeak
    • Boldface right-pointing over left-pointing
      $_33o$[33
      Example as described in the content
      verbose
      boldface right-pointing over left-pointing arrow
      Listen to MathSpeak
    • Boldface left-pointing over right-pointing
      $_[33$33o
      Example as described in the content
      verbose
      boldface left-pointing arrow over right-pointing arrow
      Listen to MathSpeak
    • Boldface right-pointing over boldface left-pointing
      $_33o$_[33
      Example as described in the content
      verbose
      boldface right-pointing arrow over boldface left-pointing arrow
      Listen to MathSpeak
    • Boldface left-pointing over boldface right-pointing
      $_[33$_33o
      Example as described in the content
      verbose
      boldface left-pointing arrow over boldface right-pointing arrow
      Listen to MathSpeak
    • Long right-pointing over short left-pointing
      $333o$[3
      Example as described in the content
      verbose
      long right-pointing arrow over short left-pointing arrow
      Listen to MathSpeak
    • Short right-pointing over long left-pointing
      $3o$[333
      Example as described in the content
      verbose
      short right-pointing arrow over long left-pointing arrow
      Listen to MathSpeak
  • Equivalence (is equivalent to)
    @<,<
    Example as described in the content
    verbose
    geometrically-equivalent-to
    Listen to MathSpeak
  • Greater Than
    • Bar over greater than (is equal to or greater than)
      :.1
      Example as described in the content
      verbose
      equal-to-or-greater-than
      Listen to MathSpeak
    • Bar under greater than (is greater than or equal to)
      .1:
      Example as described in the content
      verbose
      greater-than-or-equal-to
      Listen to MathSpeak
    • Equals sign over greater than (is equal to or greater than)
      .k.1
      Example as described in the content
      verbose
      equal to or greater than
      Listen to MathSpeak
    • Equals sign under greater than (is greater than or equal to)
      .1.k
      Example as described in the content
      verbose
      greater-than-or-equal-to
      Listen to MathSpeak
  • Inclusion
    • Bar over inclusion (is a subset of)
      :_"k
      Example as described in the content
      verbose
      subset of
      Listen to MathSpeak
    • Bar under inclusion (is a subset of)
      _"k:
      Example as described in the content
      verbose
      Subset-of-or-equal-to
      Listen to MathSpeak
    • Equals sign over inclusion (is a subset of)
      .k_"k
      Example as described in the content
      verbose
      subset of
      Listen to MathSpeak
    • Equals sign under inclusion (is a subset of)
      _"k.k
      Example as described in the content
      verbose
      subset of
      Listen to MathSpeak
  • Intersection (cap)
    • Bar under intersection
      .%:
      Example as described in the content
      verbose
      intersection with bar under
      Listen to MathSpeak
    • Equals sign under intersection
      .%.k
      Example as described in the content
      verbose
      intersection with equals under
      Listen to MathSpeak
  • Less Than
    • Bar over less than (is equal to or less than)
      :"k
      Example as described in the content
      verbose
      equal-to-or-less-than
      Listen to MathSpeak
    • Bar under less than (is less than or equal to)
      "k:
      Example as described in the content
      verbose
      less-than-or-equal-to
      Listen to MathSpeak
    • Equals sign over less than (is equal to or less than)
      .k"k
      Example as described in the content
      verbose
      equal to or less than
      Listen to MathSpeak
    • Equals sign under less than (is less than or equal to)
      "k.k
      Example as described in the content
      verbose
      less-than-or-equal-to
      Listen to MathSpeak
  • Logical Product (meet)
    • Bar over logical product
      :@%
      Example as described in the content
      verbose
      logical product with bar over
      Listen to MathSpeak
    • Bar over and bar under logical product
      :@%:
      Example as described in the content
      verbose
      logical product with bar over and bar under
      Listen to MathSpeak
    • Bar over and equals sign under logical product
      :@%.k
      Example as described in the content
      verbose
      logical product with bar over and equals sign under
      Listen to MathSpeak
    • Bar under logical product
      @%:
      Example as described in the content
      verbose
      logical product with bar under
      Listen to MathSpeak
    • Equals sign over logical product
      .k@%
      Example as described in the content
      verbose
      logical product with equals sign over
      Listen to MathSpeak
    • Equals sign over and bar under logical product
      .k@%:
      Example as described in the content
      verbose
      logical product with equals sign over and bar under
      Listen to MathSpeak
    • Equals sign over and equals sign under logical product
      .k@%.k
      Example as described in the content
      verbose
      logical product with equals sign over and equals sign under
      Listen to MathSpeak
    • Equals sign under logical product
      @%.k
      Example as described in the content
      verbose
      logical product with equals sign under
      Listen to MathSpeak
  • Logical Sum (join)
    • Bar over logical sum
      :@+
      Example as described in the content
      verbose
      logical sum with bar over
      Listen to MathSpeak
    • Bar over and bar under logical sum
      :@+:
      Example as described in the content
      verbose
      logical sum with bar over and bar under
      Listen to MathSpeak
    • Bar over and equals sign under logical sum
      :@+.k
      Example as described in the content
      verbose
      logical sum with bar over and equals sign under
      Listen to MathSpeak
    • Bar under logical sum
      @+:
      Example as described in the content
      verbose
      logical sum with bar under
      Listen to MathSpeak
    • Equals sign over logical sum
      .k@+
      Example as described in the content
      verbose
      logical sum with equals sign over
      Listen to MathSpeak
    • Equals sign over and bar under logical sum
      .k@+:
      Example as described in the content
      verbose
      logical sum with equals sign over and bar under
      Listen to MathSpeak
    • Equals sign over and equals sign under logical sum
      .k@+.k
      Example as described in the content
      verbose
      logical sum with equals sign over and equals sign under
      Listen to MathSpeak
    • Equals sign under logical sum
      @+.k
      Example as described in the content
      verbose
      logical sum with equals sign under
      Listen to MathSpeak
  • Reverse Inclusion
    • Bar over reverse inclusion
      :_.1
      Example as described in the content
      verbose
      reverse inclusion with bar over
      Listen to MathSpeak
    • Bar under reverse inclusion
      _.1:
      Example as described in the content
      verbose
      Superset-of-or-equal-to
      Listen to MathSpeak
    • Equals sign over reverse inclusion
      .k_.1
      Example as described in the content
      verbose
      reverse inclusion with equals sign over
      Listen to MathSpeak
    • Equals sign under reverse inclusion
      _.1.k
      Example as described in the content
      verbose
      reverse inclusion with equals sign under
      Listen to MathSpeak
  • Tilde (is related to)
    • Bar over double tilde
      :@:@:
      Example as described in the content
      verbose
      double tilde with bar over
      Listen to MathSpeak
    • Bar over single tilde
      :@:
      Example as described in the content
      verbose
      minus-above-tilde
      Listen to MathSpeak
    • Bar under double tilde
      @:@::
      Example as described in the content
      verbose
      almost-equals-or-equals
      Listen to MathSpeak
    • Bar under single tilde
      @::
      Example as described in the content
      verbose
      asymptotically-equals
      Listen to MathSpeak
    • Double tilde
      @:@:
      Example as described in the content
      verbose
      almost-equals
      Listen to MathSpeak
    • Equals sign over double tilde
      .k@:@:
      Example as described in the content
      verbose
      double tilde with equals sign over
      Listen to MathSpeak
    • Equals sign over single tilde
      .k@:
      Example as described in the content
      verbose
      equals-sign-above-tilde-operator
      Listen to MathSpeak
    • Equals sign under double tilde
      @:@:.k
      Example as described in the content
      verbose
      double tilde with equals sign under
      Listen to MathSpeak
    • Equals sign under single tilde
      @:.k
      Example as described in the content
      verbose
      approximately-equals
      Listen to MathSpeak
  • Union (cup)
    • Bar under union
      .+:
      Example as described in the content
      verbose
      union with bar under
      Listen to MathSpeak
    • Equals sign under union
      .+.k
      Example as described in the content
      verbose
      union with equals sign under
      Listen to MathSpeak
Comparison Signs Compounded Horizontally
  • Arrow Combinations
    • Up-pointing followed by down-pointing
      $<33o"$%33o
      Example as described in the content
      verbose
      upwards-arrow-leftwards-of-downwards-arrow
      Listen to MathSpeak
    • Down-pointing followed by up-pointing
      $%33o"$<33o
      Example as described in the content
      verbose
      downwards-arrow-followed-by-upwards-arrow
      Listen to MathSpeak
    • Up-pointing followed by boldface down-pointing
      $<33o"$%_33o
      Example as described in the content
      verbose
      up pointing arrow followed by boldface down pointing arrow
      Listen to MathSpeak
    • Down-pointing followed by boldface up-pointing
      $%33o$<_33o
      Example as described in the content
      verbose
      down pointing arrow followed by boldface up pointing arrow
      Listen to MathSpeak
    • Boldface up-pointing followed by down-pointing
      $<_33o"$$%33o
      Example as described in the content
      verbose
      boldface up pointing arrow followed by down pointing arrow
      Listen to MathSpeak
    • Boldface down-pointing followed by up-pointing
      $%_33o"$<33o
      Example as described in the content
      verbose
      boldface down pointing arrow followed by up pointing arrow
      Listen to MathSpeak
    • Boldface up-pointing followed by boldface down-pointing
      $<_33o"$%_33o
      Example as described in the content
      verbose
      boldface up pointing arrow followed by boldface down pointing arrow
      Listen to MathSpeak
    • Boldface down-pointing followed by boldface up-pointing
      $%_33o"$<_33o
      Example as described in the content
      verbose
      boldface down pointing arrow followed by boldface up pointing arrow
      Listen to MathSpeak
  • Greater Than
    • Followed by less than
      .1""k
      Example as described in the content
      verbose
      greater-than less-than
      Listen to MathSpeak
    • Followed by equals sign followed by less than
      .1".k""k
      Example as described in the content
      verbose
      greater-than equals less-than
      Listen to MathSpeak
  • Less Than
    • Followed by greater than
      "k".1
      Example as described in the content
      verbose
      less-than greater-than
      Listen to MathSpeak
    • Followed by equals sign followed by greater than
      "k".k".1
      Example as described in the content
      verbose
      less-than equals greater-than
      Listen to MathSpeak
Comparison Signs Compounded by Superposition
  • Dot
    • Between bars of equals sign
      *@.k]
      Example as described in the content
      verbose
      equals sign with dot inside
      Listen to MathSpeak
    • Within inclusion sign
      *@_"k]
      Example as described in the content
      verbose
      subset-with-dot
      Listen to MathSpeak
    • Within reverse inclusion sign
      *@_.1]
      Example as described in the content
      verbose
      superset-with-dot
      Listen to MathSpeak
  • Equals Sign
    • Through inclusion sign .k@_"k] or _"k@.k]
      Example as described in the content
      verbose
      inclusion sign superposed equals sign
      Listen to MathSpeak
    • Through reverse inclusion sign .k@_.1] or _.1@.k]
      Example as described in the content
      verbose
      reverse inclusion sign superposed equals sign
      Listen to MathSpeak
  • Greater Than
    • Nest of two with straight sides (is large compared with)
      .1@.1]
      Example as described in the content
      verbose
      much-greater-than
      Listen to MathSpeak
    • Nest of two with curved sides
      ..1@..1]
      Example as described in the content
      verbose
      curved much greater than
      Listen to MathSpeak
  • Horizontal Bar
    • Through inclusion sign
      :@_"k]
      Example as described in the content
      verbose
      inclusion sign superposed horizontal bar
      Listen to MathSpeak
    • Through reverse inclusion sign
      :@_.1]
      Example as described in the content
      verbose
      reverse inclusion sign superposed horizontal bar
      Listen to MathSpeak
  • Less Than
    • Nest of two with straight sides (is small compared with)
      "k@"k]
      Example as described in the content
      verbose
      much-less-than
      Listen to MathSpeak
    • Nest of two with curved sides
      ."k@."k]
      Example as described in the content
      verbose
      curved much less than
      Listen to MathSpeak
  • Vertical Bar
    • Through shaft of right-pointing arrow
      \@$33o]
      Example as described in the content
      verbose
      right pointing arrow superposed vertical bar
      Listen to MathSpeak
    • Through shaft of left-pointing arrow
      \@$[33]
      Example as described in the content
      verbose
      left pointing arrow superposed vertical bar
      Listen to MathSpeak

§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.