Iverson bracket

In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement x = y. 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: $${\displaystyle [P]={\begin{cases}1&{\text{if }}P{\text{ is true;}}\\0&{\text{otherwise.}}\end{cases}}}$$ In other words, the Iverson bracket of a statement is the indicator function of the set of values for which the statement is true.

The Iverson bracket allows using capital-sigma notation without restriction on the summation index. That is, for any property ${\displaystyle P(k)}$ of the integer ${\displaystyle k}$, one can rewrite the restricted sum ${\displaystyle \sum _{k:P(k)}f(k)}$ in the unrestricted form ${\displaystyle \sum _{k}f(k)\cdot [P(k)]}$. With this convention, ${\displaystyle f(k)}$ does not need to be defined for the values of k for which the Iverson bracket equals 0; that is, a summand ${\displaystyle f(k)[{\textbf {false}}]}$ must evaluate to 0 regardless of whether ${\displaystyle f(k)}$ is defined.

The notation was originally introduced by Kenneth E. Iverson in his programming language APL,^[1][2] though restricted to single relational operators enclosed in parentheses, while the generalisation to arbitrary statements, notational restriction to square brackets, and applications to summation, was advocated by Donald Knuth to avoid ambiguity in parenthesized logical expressions.^[3]