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 . 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 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 of the integer , one can rewrite the restricted sum in the unrestricted form . With this convention, does not need to be defined for the values of for which the Iverson bracket equals ; that is, a summand must evaluate to 0 regardless of whether is defined. The notation was originally introduced by Kenneth E. Iverson in his programming language APL, 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. Properties There is a direct correspondence between arithmetic on Iverson brackets, logic, and set operations. For instance, let A and B be sets and any property of integers; then we have Examples The notation allows moving boundary conditions of summations (or integrals) as a separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically. Double-counting rule We mechanically derive a well-known sum manipulation rule using Iverson brackets: Summation interchange The well-known rule is likewise easily derived: Counting For instance, the Euler phi function that counts the number of positive integers up to n which are coprime to n can be expressed by Simplification of special cases Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula is valid for but is off by for . To get an identity valid for all positive integers (i.e., all values for which is defined), a correction term involving the Iverson bracket may be added: Common functions Many common functions, especially those with a natural piecewise definition,