Iverson bracket

In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is defined as follows

<math>[P] = \left\{\begin{matrix} 1 &\mathrm{if}\ P\ \mathrm{is\ true} \\ 0 &\mathrm{otherwise}\end{matrix}\right.<math>

where P is a proposition.

For example, the Kronecker delta notation is a specific case of Iverson notation, that is,

<math>\delta_{ij} = [i=j]\,<math>

The notation is useful especially in simplifying sums or integrals, for example

<math>\sum_{0\le i \le 10} i^2 = \sum_{i} i^2[0 \le i \le 10]<math>

as where i is strictly less than 0 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.