Produced by Charles Wells.  Home.   Website Contents     Website Index   

Posted 25 February 2009

SYMBOLS: DELIMITERS

Delimiters are pairs of symbols used in the symbolic language either for enclosing expressions or as operators.   

Table of contents

Parentheses. 1

Angle brackets. 2

Square brackets. 2

Braces. 3

Other delimiters. 3

Bare delimiters. 3

About the word bracket Error! Bookmark not defined.

Parentheses

Parentheses are the symbols " (" and " )".  Parentheses are also called brackets or braces, but those words may also refer to other delimiters.  Sometimes parentheses are called round parentheses to distinguish them from other delimiters.  “Parenthesis” is singular, “parentheses” is plural.

Parentheses can be bare delimiters

Round parentheses are the most common delimiter by far to be used as bare delimiters to group expressions.   Square and curly brackets may also be used.

The notation (a, b) may denote any of these functions:

¨  (a, b) may denote the ordered pair with first coordinate a and second coordinate b.  This usage extends to n-tuples, for example the ordered triple (3,1,2). 

¨  For a and b real numbers, (a, b) may denote the open real interval .   This contrasts with the closed interval [a, b].

¨  (a, b) may denote the greatest common divisor of the integers a and b.  This contrasts with the least common multiple [a, b].

These varying usages may occur in the same document.  See the Handbook, page 193 for an example. 

The reason I include these examples here is that you may never have thought of them as functions.

¨  The ordered pair function is the function  that takes an element of A and an element of B to the ordered pair  (the cartesian product of A and B).

¨  The real interval function is the function  (the set of subsets of  ) that takes a and b to the open interval .

¨  The GCD function takes integers a and b to their greatest common divisor (a, b).  For example, (9, 12) = 3. 

Matrices may be enclosed in parentheses

Example:

Square brackets may be used for this as well.  However, vertical bars, as in

denote the determinant of the matrix.

Function arguments may be enclosed in parentheses

The argument to a function is typically enclosed in parentheses.  For example, if we define , then .  In this situation some authors and lecturers would write  but most would include the parentheses.  On the other hand, when grouping is necessary as in , everyone would use parentheses.  More about this here.

Angle brackets

Angle brackets are the symbols "  " and "  ".  In printed material they are usually noticeably distinct from the greater-than and less-than symbols “<” and “>”, but they may not be distinguished in handwriting.

Angle brackets are used as outfix notation to denote various constructions, most notably an inner product as in “  ”.

In my research for the Handbook I could not find a citation for the use of angle brackets as bare delimiters, but I betcha someone somewhere has used them that way.

Angle brackets are also called pointy brackets, particularly in speech.

Square brackets

Square brackets are the delimiters “[“ and ”]”.

¨  Square brackets are occasionally used as  bare delimiters.  This is done most often with nested parentheses to aid the reader to match the correct pairs.  For example,  

¨  Square brackets may be used instead of parentheses to enclose matrices. 

¨  Square brackets may be used instead of parentheses to enclose the argument to a function in an expression of its value, as in f[x] instead of f(x).

¨  Square brackets are  used as outfix notation with special meanings, for example to denote closed intervals.

Braces

Braces are the symbols " {" and " }".   They are also called curly brackets.  Warning:   the word “brace” is often used to denote parentheses or square brackets.

¨  Braces are used as bare delimiters when there are nested parentheses, in much the same way as square brackets.

¨  Braces are used in the list notation for sets and also in setbuilder notation..

¨  Braces are used as outfix notation for functions. In particular, the fractional part of a real number r may be denoted by {r}.  For example, {3/2} = 0.5.

A left brace may be used by itself in a definition by cases.

Other delimiters

Other symbols also are used to carry meaning and also act as delimiters.  They are not used as bare delimiters.

¨  Vertical lines, to indicate absolute value (among other things), as in .  

¨  Double vertical lines, for norms, as in . 

The vertical line is also used by itself, not as a delimiter, with various meanings including “divides” and as a separator in setbuilder notation.

Bare delimiters

A pair of delimiters may or may not have significance beyond grouping; if they do not they are bare delimiters. The three types of character used as bare delimiters in mathematics are parentheses, square brackets and braces.

Typically, parentheses are the standard delimiters in symbolic expressions. Square brackets or braces may be used to aid parsing when parentheses are nested or when the expression to be enclosed is large, but square brackets and braces are occasionally used alone as bare delimiters as well.