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| In [[mathematics]], the '''Iverson bracket''', named after [[Kenneth E. Iverson]]<!-- mentioned here so it's clear that it's an upper-case 'i' not an 'l'. -->, is a notation that denotes a number that is 1 if the condition in square brackets is satisfied, and 0 otherwise. More exactly,
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| :<math>[P] = \begin{cases} 1 & \text{if } P \text{ is true;} \\ 0 & \text{otherwise.} \end{cases}</math>
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| where {{math|''P''}} is a [[statement (logic)|statement]] that can be true or false. This notation was introduced by [[Kenneth E. Iverson]] in his programming language [[APL (programming language)|APL]],<ref>[[Ronald Graham]], [[Donald Knuth]], and [[Oren Patashnik]]. ''[[Concrete Mathematics]]'', Section 2.2: Sums and Recurrences.</ref> while the specific restriction to square brackets was advocated by [[Donald Knuth]] to avoid ambiguity in parenthesized logical expressions.<ref>Knuth 1992.</ref>
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| ==Uses==
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| The Iverson bracket converts a [[Boolean data type|Boolean value]] to an integer value through the natural map <math>\textbf{false}\mapsto 0; \textbf{true}\mapsto1</math>, which allows counting to be represented as summation. For instance, the [[Euler phi function]] that counts the number of positive integers up to ''n'' which are [[coprime]] to ''n'' can be expressed by
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| : <math> \phi(n)=\sum_{i=1}^{n}[\gcd(i,n)=1],\qquad\text{for }n\in\mathbb N^+.</math>
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| More generally the notation allows moving boundary conditions of summations (or integrals) as a separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically. For example
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| : <math>\sum_{1\le i \le 10} i^2 = \sum_{i} i^2[1 \le i \le 10].</math>
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| In the first sum, the index <math>i</math> is limited to be in the range 1 to 10. The second sum is allowed to range over all integers, but where ''i'' is strictly less than 1 or strictly greater than 10, the summand is 0, contributing nothing to the sum. Such use of the Iverson bracket can permit easier manipulation of these expressions.
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| <!-- add an example of such a manipulation -->
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| Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula
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| :<math>\sum_{1\le k\le n \atop \gcd(k,n)=1}\!\!k = \frac{1}{2}n\varphi(n)</math>
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| which is valid for {{math|''n'' > 1}} but which is off by {{sfrac|1|2}} for {{math|''n'' {{=}} 1}}. To get an identity valid for all positive integers ''n'' (i.e., all values for which <math>\phi(n)</math> is defined), a correction term involving the Iverson bracket may be added:
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| :<math>\sum_{1\le k\le n \atop \gcd(k,n)=1}\!\!k = \frac{1}{2}n(\varphi(n)+[n=1])</math>
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| ==Special cases==
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| The [[Kronecker delta]] notation is a specific case of Iverson notation when the condition is equality. That is,
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| : <math>\delta_{ij} = [i=j].\,</math> | |
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| The [[indicator function]], another specific case, has set membership as its condition:
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| : <math>\mathbf{I}_A(x) = [x\in A].</math>
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| The [[sign function]] and [[Heaviside step function]] are also easily expressed in this notation:
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| : <math> \sgn(x) = [x > 0] - [x < 0] \,</math>
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| : <math> H(x) = [x > 0].</math>
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| The [[floor and ceiling functions]] can be expressed:
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| : <math> \lfloor x \rfloor = \sum_{n=-\infty}^{\infty}n[n \le x < n+1],</math>
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| : <math> \lceil x \rceil = \sum_{n=-\infty}^{\infty}n[n-1 < x \le n].</math>
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| And the [[Trichotomy (mathematics)|trichotomy]] of the reals can be expressed:
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| :<math>[a < b] + [a = b] + [a > b] = 1. \, </math>
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| ==See also==
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| *[[Macaulay brackets]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * Donald Knuth, "Two Notes on Notation", ''[[American Mathematical Monthly]]'', Volume 99, Number 5, May 1992, pp. 403–422. ([http://www-cs-faculty.stanford.edu/~knuth/papers/tnn.tex.gz {{TeX}}], {{arxiv|math/9205211}})
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| * Kenneth E. Iverson, ''A Programming Language'', New York: Wiley, p. 11, 1962.
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| [[Category:Mathematical notation]]
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