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Iverson bracket
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The Iverson bracket, named after Kenneth E. Iverson, is a notation that denotes a number that is 1 if the condition in square brackets is satisfied, and 0 otherwise. More exactly,
P 
Uses
The notation is useful in expressing sums or integrals without boundary conditions. For example

∑ 1 ≤ i ≤ 10∑ i
i 
i 
Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula

∑ 1 ≤ k ≤ n
(k, n) = 1
,n φ (n) 2
n > 1 

∑ 1 ≤ k ≤ n
(k, n) = 1
,n (φ (n) + [n = 1] ) 2
n 
Special cases
The Kronecker delta notation is a specific case of Iverson notation when the condition is equality

δi j = [i = j ] .
IA (x) 
χA (x) 
A 

IA (x) = χA (x) = [x ∈ A] .
The sign function and Heaviside step function are also easily expressed in this notation

sgn (x) = [x > 0] − [x < 0] ,

H (x) = [x > 0] +
[x = 0] .1 2
And the trichotomy of the reals can be expressed

[a < b] + [a = b] + [a > b] = 1.
See also
Notes
 ↑ Ronald Graham, Donald Knuth, and Oren Patashnik. Concrete Mathematics, Section 2.2: Sums and Recurrences.
 ↑ Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics (2nd ed.). Reading, MA: AddisonWesley Publishing Company. pp. xiii+657. ISBN 0201558025 .
 ↑ Iverson, Kenneth E. (1962). A Programming Language. Wiley. ISBN 0471430145 .
 ↑ Graham, Knuth, and Patashnik (1994).
 ↑ To do: add an example of such a manipulation.
References
 Donald Knuth, “Two Notes on Notation,” American Mathematical Monthly, Volume 99, Number 5, May 1992, pp. 403–422. (TeX, arXiv:math/9205211)
 Kenneth E. Iverson, "A Programming Language", New York: Wiley, p. 11, 1962.