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In mathematics
In addition to the use of parentheses to specify the order of operations, both parentheses and square brackets can also be used to denote an interval. The notation [a, c) is used to indicate a sequence from a to c that is inclusive of a but exclusive of c. That is, [5, 12) would be the set of all real numbers between 5 and 12, including 5 but except 12. The numbers may come as close as they like to 12, including 11.999 and so forth (with any finite number of 9s), but 12.0 is not included. In Europe, the notation [5,12[ is also used for this.
The endpoint adjoining the square bracket is known as closed, while the endpoint adjoining the parenthesis is known as open. If both types of brackets are the same, the entire interval may be referred to as closed or open as appropriate. Whenever infinity or negative infinity is used as an endpoint, it is always considered open and adjoined to a parenthesis.
This is used in mathematical notation, and appears in some computer programming languages. See the article Interval (mathematics) for a more complete treatment of the subject.
In quantum mechanics, angle brackets are also used as part of Dirac's formalism to note vectors from the dual spaces of the Bra 〈A| and the Ket |B〉. In statistical mechanics, angle brackets denote ensemble or time average.
In group theory and ring theory, square brackets are used to denote the commutator. In group theory, the commutator [g,h] is commonly defined as g−1h−1gh. In ring theory, the commutator [a,b] is defined as ab − ba. Furthermore, in ring theory, braces are used to denote the anticommutator where {a,b} is defined as ab + ba.