This part of the lesson details signs of comparison compounded vertically and horizontally.
Signs of Comparison Compounded Vertically: When two or more simple signs of comparison are arranged one under the other, the combination becomes a single comparison sign compounded vertically. The symbol for the uppermost sign must be written first and unspaced from the symbol for the next lower sign.
 Arrows: When a right pointing arrow in regular type with a full barb and single shaft of ordinary length is part of a sign of comparison compounded vertically, its contracted form must not be used.
PointingRight Over PointingLeft 


PointingRight Over PointingLeft 


PointingLeft Over PointingRight 


Long PointingRight Over Short PointingLeft 


Short PointingRight Over Long PointingLeft 


PointingRight With Upper Barb Only Over PointingLeft With Lower Barb Only 


PointingLeft Over Boldface PointingRight 


Boldface PointingRight Over PointingLeft 


Boldface PointingLeft Over PointingRight 


Boldface PointingRight Over Boldface PointingLeft 


Boldface PointingLeft Over Boldface PointingRight 


 Equals:
Horizontal Bar 
 

Oblique Stroke 
/or\ 

Equals Sign 
= 

As shown in c through k below, a single horizonal bar, an oblique stroke, or an equals sign may appear as part of a sign of comparison compounded vertically. The horizontal bar or oblique stroke is often substituted for the equals sign.
 Greater Than:
Bar Over Greater Than (is equal to or greater than) 


Equals Sign Over Greater Than (is equal to or greater than) 


Bar Under Greater Than (is greater than or equal to) 


Equals Sign Under Greater Than (is greater than or equal to) 


d. Inclusion (is a subset of):
Bar Over Inclusion 


Equals Sign Over Inclusion 


Bar Under Inclusion 


Equals Sign Under Inclusion 


e. Intersection (cap): The intersection sign is a sign of comparison when modified by a bar or equals sign above or below it. An unmodified intersection sign is a sign of operation (see p. xx).
Bar Under Intersection 


Equals Sign Under Intersection 


f. Less Than:
Bar Over Less Than (is equal to or less than) 


Equals Sign Over Less Than (is equal to or less than) 


Bar Under Less Than (is less than or equal to) 


Equals Sign Under Less Than (is less than or equal to) 


g. Logical Product (meet): The logical product sign is a sign of comparison when modified by a bar or equals sign above or below it. An unmodified logical product is a sign of operation.
Bar Over Logical Product 


Bar Over and Bar Under Logical Product 


Bar Over and Equals Sign Under Logical Product 


Bar Under Logical Product 


Equals Sign Over Logical Product 


Equals Sign Over and Bar Under Logical Product 


Equals Sign Over and Equals Sign Under Logical Product 


Equals Sign Under Logical Product 


h. Logical Sum (join): The logical sum sign is a sign of comparison when
modified by a bar or equals sign above or below it. An unmodified logical sum is a sign of operation.
Bar Over Logical Sum 


Bar Over and Bar Under Logical Sum 


Bar Over and Equals Sign Under Logical Sum 


Bar Under Logical Sum 


Equals Sign Over Logical Sum 


Equals Sign Over and Bar Under Logical Sum 


Equals Sign Over and Equals Sign Under Logical Sum 


Equals Sign Under Logical Sum 


i. Reverse Inclusion:
Bar Over Reverse Inclusion 


Equals Sign Over Reverse Inclusion 


Bar Under Reverse Inclusion 


Equals Sign Under Reverse Inclusion 


j. Tilde:
Bar Over Single Tilde 


Equals Sign Over Single Tilde 


Double Tilde (is approximately equal to) 


Bar Under Double Tilde 


Equals Sign Over Double Tilde 


Bar Under Single Tilde 


Equals Sign Under Single Tilde (is congruent to) 


Bar Under Double Tilde 


Equals Sign Under Double Tilde 


k. Union (cup): The union sign is a sign of comparison when modified by a bar or equals sign above or below it. An unmodified logical sum is a sign of operation.
Bar Over Union 


Equals Sign Over Union 


Signs of Comparison Compounded Horizontally: When two or more sign of comparison are arranged side by side, the combination becomes a single comparison sign compounded horizontally. The multipurpose indicator must be placed between the unspaced symbols to indicate that they are printed horizontally, not vertically. Unlisted signs of comparison compounded horizontally must be transcribed according to the same principle.
 Arrows:
PointingUp Followed by Pointing Down 


PointingDown Followed by PointingUp 


PointingUp Followed by Boldface PointingDown 


PointingDown Followed by Boldface PointingUp 


Boldface PointingUp Followed by Pointing Down 


Boldface PointingDown Followed by PointingUp 


Boldface PointingUp Followed by Boldface PointingDown 


Boldface PointingDown Followed by Boldface PointingUp 


 Greater Than:
Followed by Less Than 


Followed by Equals Sign Followed by Less Than 


 Less Than:
Followed by Greater Than 


Followed by Equals Sign Followed by Greater Than 


Negated Signs of Comparison: In print, a sign of comparison may be negated by a vertical line of by a slant line drawn through it. In braille, the symbol must be used to represent either of the print negation signs and must be placed immediately before the sign of comparison being negated. Negated signs of comparison not illustrated below must be transcribed according to the same principle.
Signs of Comparison and Boldface Type: When it is necessary to show that a sign of comparison is printed in boldface type, dots 456 are placed before the corresponding symbol in braille. However, the use of dots 456 to denote boldface type is restricted to the specific signs listed below. They must not be used with any other sign of comparison. Boldface signs of comparison should be used only when the distinction between the regular and boldface forms of the same sign has mathematical significance.
Boldface Arrows
Boldface Equals =
 shows that the reaction begins at the left and is completed to the right.
 a X c = b X d if and only if a = c and b = d.
