RULE XX--SIGNS AND SYMBOLS OF COMPARISON

Simple Comparison Signs

• Arc
• Concave upward

  verbose

  (none)

• Concave downward

  verbose

  (none)

• Arrow
• Left-pointing

  verbose

  (none)

• Right-pointing
• Contacted

  verbose

  (none)

• Uncontacted

  verbose

  (none)

• Down-pointing

  verbose

  (none)

• Up-pointing

  verbose

  (none)

• Two-way
• Horizontal

  verbose

  (none)

• Vertical

  verbose

  (none)

• Equals (is equal to)
• Normal

  verbose

  (none)

• Boldface

  verbose

  (none)

• Greater Than (is greater than)
• Normal

  verbose

  (none)

• With curved sides

  verbose

  (none)

• Identity (is congruent to; is identical to)

  verbose

  (none)

• Inclusion (is contained in; is a subset of)

  verbose

  (none)

• Less than (is less than)
• Normal

  verbose

  (none)

• With curved sides

  verbose

  (none)

• Membership (is an element of; belongs to)

  verbose

  (none)

• Parallel To (is parallel to)

  verbose

  (none)

• Perpendicular To (is perpendicular to)

  verbose

  (none)

• Proportion (as)

  verbose

  (none)

• Ratio (is to)

  verbose

  (none)

• Relation (is related to)

  verbose

  (none)

• Reverse Inclusion (contains; in logic, implies)

  verbose

  (none)

• Reverse Membership (contains the element)

  verbose

  (none)

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

  verbose

  (none)

• Extended (is related to)

  verbose

  (none)

• Variation (varies as)

  verbose

  (none)

• Vertical Bar (such that)

  verbose

  (none)

Modified Comparison Signs

• Equals Sign
• Caret over

  verbose

  (none)

• Caret Under (is projective to, projective correspondence)

  verbose

  (none)

• Degree sign over (is equal in degrees to)

  verbose

  (none)

• Dot over (is approximately equal to)

  verbose

  (none)

• Dot over and dot under

  verbose

  (none)

• Equilateral triangle over

  verbose

  (none)

• Inverted caret over

  verbose

  (none)

• Left-pointing caret over

  verbose

  (none)

• Question mark over

  verbose

  (none)

• Right-pointing caret over

  verbose

  (none)

• Two dots over and two dots under

  verbose

  (none)

• Vertical bar over

  verbose

  (none)

• Horizontal Bar
• Caret over

  verbose

  (none)

• Caret under (is perspective to, perspective correspondence)

  verbose

  (none)

• Dot under

  verbose

  (none)

• Simple Tilde, Dot Under

  verbose

  (none)

Comparison Signs Compounded Vertically

• Arrow Combinations
• Right-pointing over left-pointing

  verbose

  (none)

• Right-pointing with upper barb over left-pointing with lower barb

  verbose

  (none)

• Right-pointing over boldface left-pointing

  verbose

  (none)

• Left-pointing over boldface right-pointing

  verbose

  (none)

• Boldface right-pointing over left-pointing

  verbose

  (none)

• Boldface left-pointing over right-pointing

  verbose

  (none)

• Boldface right-pointing over boldface left-pointing

  verbose

  (none)

• Boldface left-pointing over boldface right-pointing

  verbose

  (none)

• Long right-pointing over short left-pointing

  verbose

  (none)

• Short right-pointing over long left-pointing

  verbose

  (none)

• Equivalence (is equivalent to)

  verbose

  (none)

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

  verbose

  (none)

• Bar under greater than (is greater than or equal to)

  verbose

  (none)

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

  verbose

  (none)

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

  verbose

  (none)

• Inclusion
• Bar over inclusion (is a subset of)

  verbose

  (none)

• Bar under inclusion (is a subset of)

  verbose

  (none)

• Equals sign over inclusion (is a subset of)

  verbose

  (none)

• Equals sign under inclusion (is a subset of)

  verbose

  (none)

• Intersection (cap)
• Bar under intersection

  verbose

  (none)

• Equals sign under intersection

  verbose

  (none)

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

  verbose

  (none)

• Bar under less than (is less than or equal to)

  verbose

  (none)

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

  verbose

  (none)

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

  verbose

  (none)

• Logical Product (meet)
• Bar over logical product

  verbose

  (none)

• Bar over and bar under logical product

  verbose

  (none)

• Bar over and equals sign under logical product

  verbose

  (none)

• Bar under logical product

  verbose

  (none)

• Equals sign over logical product

  verbose

  (none)

• Equals sign over and bar under logical product

  verbose

  (none)

• Equals sign over and equals sign under logical product

  verbose

  (none)

• Equals sign under logical product

  verbose

  (none)

• Logical Sum (join)
• Bar over logical sum

  verbose

  (none)

• Bar over and bar under logical sum

  verbose

  (none)

• Bar over and equals sign under logical sum

  verbose

  (none)

• Bar under logical sum

  verbose

  (none)

• Equals sign over logical sum

  verbose

  (none)

• Equals sign over and bar under logical sum

  verbose

  (none)

• Equals sign over and equals sign under logical sum

  verbose

  (none)

• Equals sign under logical sum

  verbose

  (none)

• Reverse Inclusion
• Bar over reverse inclusion

  verbose

  (none)

• Bar under reverse inclusion

  verbose

  (none)

• Equals sign over reverse inclusion

  verbose

  (none)

• Equals sign under reverse inclusion

  verbose

  (none)

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

  verbose

  (none)

• Bar over single tilde

  verbose

  (none)

• Bar under double tilde

  verbose

  (none)

• Bar under single tilde

  verbose

  (none)

• Double tilde

  verbose

  (none)

• Equals sign over double tilde

  verbose

  (none)

• Equals sign over single tilde

  verbose

  (none)

• Equals sign under double tilde

  verbose

  (none)

• Equals sign under single tilde

  verbose

  (none)

• Union (cup)
• Bar under union

  verbose

  (none)

• Equals sign under union

  verbose

  (none)

Comparison Signs Compounded Horizontally

• Arrow Combinations
• Up-pointing followed by down-pointing

  verbose

  (none)

• Down-pointing followed by up-pointing

  verbose

  (none)

• Up-pointing followed by boldface down-pointing

  verbose

  (none)

• Down-pointing followed by boldface up-pointing

  verbose

  (none)

• Boldface up-pointing followed by down-pointing

  verbose

  (none)

• Boldface down-pointing followed by up-pointing

  verbose

  (none)

• Boldface up-pointing followed by boldface down-pointing

  verbose

  (none)

• Boldface down-pointing followed by boldface up-pointing

  verbose

  (none)

• Greater Than
• Followed by less than

  verbose

  (none)

• Followed by equals sign followed by less than

  verbose

  (none)

• Less Than
• Followed by greater than

  verbose

  (none)

• Followed by equals sign followed by greater than

  verbose

  (none)

Comparison Signs Compounded by Superposition

• Dot
• Between bars of equals sign

  verbose

  (none)

• Within inclusion sign

  verbose

  (none)

• Within reverse inclusion sign

  verbose

  (none)

• Equals Sign
• Through inclusion sign

  verbose

  (none)

• Through reverse inclusion sign

  verbose

  (none)

• Greater Than
• Nest of two with straight sides (is large compared with)

  verbose

  (none)

• Nest of two with curved sides

  verbose

  (none)

• Horizontal Bar
• Through inclusion sign

  verbose

  (none)

• Through reverse inclusion sign

  verbose

  (none)

• Less Than
• Nest of two with straight sides (is small compared with)

  verbose

  (none)

• Nest of two with curved sides

  verbose

  (none)

• Vertical Bar
• Through shaft of right-pointing arrow

  verbose

  (none)

• Through shaft of left-pointing arrow

  verbose

  (none)

§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.