William Wheeler
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Thank you for this response, Donald.
My effort to be succinct may have caused confusion. In set notation, the 1256 indicator does not, in my experience, normally appear between the members of a set but rather as an abbreviation for the phrase "such that" that separates a variable expression from the defining condition for the set. In English print, the expression "{x | x is an integer between 1 and 10}" would be read as "the set of all x such that x is an integer between 1 and 10". The vertical line | stands for the phrase "such that". This instance of vertical line symbol | should have extra space on either side of it. (Nemeth: page 136; page 143, section 145; page 144, examples 13 and 14.)
As a mathematician who is learning to author braille documents, may I please suggest that the 1256 symbol is more appropriate for augmented matrices than the 456 symbol, both because of the mathematical meaning of the vertical line in an augmented matrix (where it is a relation symbol with mathematical meaning, not merely a separation symbol) and because of mathematical typesetting practice, for instance in TeX and LaTeX.
The augmentation line in an augmented matrix denotes an = relation in the linear equations that correspond to the augmented matrix. For example, the augmented matrix for the equation 2x + 3y = 4 is [2 3 | 4]. The augmentation line indicates that there is an equality relation between the left side of the equation whose coefficients are 2 and 3 and the constant 4 on the right side of the equation.
In Don's png, there are two columns on the right hand side of the augmentation line because the augmentation matrix there is being used to find the multiplicative inverse of the matrix on the left side, so the top line [1 2 | 1 0] indicates that one is really concerned with both the equation x + 2y = 1 and, independently, the equation x + 2y = 0.
On page 230 of The Nemeth Braille Code for Mathematics and Science Notation, 1972 Revision, Nemeth indicates that the 1256 symbol is used as (i) a grouping symbol, (ii) a relation/comparison symbol "(is a factor, divides)", and (iii) for "such that" in set expressions, for instance, examples (13) and (14) on page 144 of the book.
In mathematical typesetting, e.g., TeX and LaTeX, the vertical bar as a grouping symbol is denoted by the macro \vert, and the vertical bar as a relation symbol (e.g., is a factor, divides) and as the "such that" symbol in set expressions is denoted by the macro \mid. In the typesetting, both \vert and \mid produce the same print symbol, namely a vertical line, but no space is inserted on either side of \vert (because it is a grouping symbol) but extra space is inserted on both left and right sides of \mid (because it is a relation symbol like = and <). (These spacing conventions are the same in both mathematical typesetting and braille.)
For these reasons, may I please suggest that 1256 symbol with a space on either side is the better transcription of the mathematical meaning of the vertical augmentation line in an augmented matrix.
Sincerely,
William Wheeler, Dept. of Mathematics, Indiana University, Bloomington, IN
