In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Iverson bracket, named after
Kenneth E. Iverson
Kenneth Eugene Iverson (17 December 1920 – 19 October 2004) was a Canadian computer scientist noted for the development of the programming language APL. He was honored with the Turing Award in 1979 "for his pioneering effort in programming l ...
, is a notation that generalises the
Kronecker delta, which is the Iverson bracket of the statement . 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
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of the set of values for which the statement is true.
The Iverson bracket allows using
capital-sigma notation without summation index. That is, for any property
of the integer
,
By convention,
does not need to be defined for the values of for which the Iverson bracket equals ; that is, a summand