Iverson bracket

A predicate mapping is a mathematical function ( true or false) maps a logical truth value to the numbers 0 or 1. This can be annoying case distinctions be reshaped so that the resulting function in mathematical conclusions is easier to use.

Definition

The following definition comes from Kenneth E. Iverson, 1962:

If P ( n) is a predicate, then [P ( n)] is defined as follows:

That is, that this figure a logical truth value maps to a more usable in mathematical formulas integer value, and indeed is a true statement to a 1, and a false statement on a 0 displayed ( see example). With this figure, you can now make a single formula from complex formulas with case distinctions.

Example

The Fibonacci numbers are defined by the following recurrence equation:

With the image of Iverson can be converted this recurrence equation in a simple form:

The part corresponding to the recursive definition of Fibonacci numbers. The factor removed for all Fibonacci numbers with an index less than or equal to 1 this recursive part. And is exactly equal to 1 if the index n is equal to 1. Characterized the Fibonacci number with the index 1 is equal to 1, and thereby it is ensured that the Fibonacci numbers also have a value greater than 1 with an index greater than 0.

With this formula, you can now easily determine the closed formula.