In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement x = y. It maps any statement to a function of the free variables in that statement. This function is defined to take the value 1 for the values of the variables for which the statement is true, and takes the value 0 otherwise. It is generally denoted by putting the statement inside square brackets:
In other words, the Iverson bracket of a statement is the indicator function of the set of values for which the statement is true.
The Iverson bracket allows using capital-sigma notation without restriction on the summation index. That is, for any property of the integer , one can rewrite the restricted sum in the unrestricted form . With this convention, does not need to be defined for the values of k for which the Iverson bracket equals 0; that is, a summand must evaluate to 0 regardless of whether is defined.
The notation was originally introduced by Kenneth E. Iverson in his programming language APL,[1][2] though restricted to single relational operators enclosed in parentheses, while the generalisation to arbitrary statements, notational restriction to square brackets, and applications to summation, was advocated by Donald Knuth to avoid ambiguity in parenthesized logical expressions.[3]
^Kenneth E. Iverson (1962). A Programming Language. Wiley. p. 11. Retrieved 7 April 2016.
^Ronald Graham, Donald Knuth, and Oren Patashnik. Concrete Mathematics, Section 2.1: Notation.
^Donald Knuth, "Two Notes on Notation", American Mathematical Monthly, Volume 99, Number 5, May 1992, pp. 403–422. (TeX, arXiv:math/9205211).
mathematics, the Iversonbracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iversonbracket of the statement...
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{\displaystyle x_{j}} , respectively, and [ ⋅ ] {\displaystyle [\cdot ]} is the Iversonbracket, which is 1 when the contents are true and 0 when false. d {\displaystyle...
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Global Information![{\displaystyle [P]={\begin{cases}1&{\text{if }}P{\text{ is true;}}\\0&{\text{otherwise.}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ead533e8bdd9bcb51828f0d580bdc2d70a799da6)
![{\displaystyle \sum _{k}f(k)\cdot [P(k)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6691a8964a679fd1cbc5366d433ca6dcfce2926)
![{\displaystyle f(k)[{\textbf {false}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb044c98e46ced3e73b546cf6723d9f20a29fed9)