Iverson bracket

Iverson bracket

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Iverson bracket, named after Kenneth E. Iverson
Kenneth E. Iverson
Kenneth Eugene Iverson was a Canadian computer scientist noted for the development of the APL programming language in 1962. He was honored with the Turing Award in 1979 for his contributions to mathematical notation and programming language theory...

, is a notation that denotes a number that is 1 if the condition in square brackets is satisfied, and 0 otherwise. More exactly,
where is a statement
Statement (logic)
In logic a statement is either a meaningful declarative sentence that is either true or false, or what is asserted or made by the use of a declarative sentence...

that can be true or false. This notation was introduced by Kenneth E. Iverson
Kenneth E. Iverson
Kenneth Eugene Iverson was a Canadian computer scientist noted for the development of the APL programming language in 1962. He was honored with the Turing Award in 1979 for his contributions to mathematical notation and programming language theory...

in his programming language APL, while the specific restriction to square brackets was advocated by Donald Knuth
Donald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...

to avoid ambiguity in parenthesized logical expressions.

Uses

The notation is useful in expressing sums or integrals without boundary conditions. For example

In the first sum, the index 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.
Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula

(a number-theoretic
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

identity involving the Euler phi function) which is valid only for may be written

which is valid for all positive integers n.

Special cases

The Kronecker delta notation is a specific case of Iverson notation when the condition is equality. That is,

The indicator function, another specific case, has set membership as its condition:

The sign function
Sign function
In mathematics, the sign function is an odd mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function ....

and Heaviside step function
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....

are also easily expressed in this notation:

And the trichotomy of the reals can be expressed: