Predicate mapping

A predicate mapping is a mathematical function that maps a logical truth value ( true or false ) to the numbers 0 or 1. This allows disruptive case distinctions to be transformed in such a way that the resulting function can be used more easily in mathematical conclusions.

definition

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

If is a predicate , then is defined as follows: ${\displaystyle P(n)}$ ${\displaystyle [P(n)]}$

${\displaystyle [P(n)]={\begin{cases}1,&{\text{wenn }}P(n){\text{ wahr ist}}\\0,&{\text{wenn }}P(n){\text{ nicht wahr ist}}\end{cases}}}$

I.e. that this mapping maps a logical truth value to an integer value that can be reused in mathematical formulas, namely a true statement is mapped to a 1 and a false statement to a 0 (see example). With this illustration you can now turn complex formulas with case distinctions into a single formula.

example

The Fibonacci numbers are defined by the following recurrence equation:

${\displaystyle f_{n}={\begin{cases}0,&{\text{wenn }}n\leq 0\\1,&{\text{wenn }}n=1\\f_{n-1}+f_{n-2},&{\text{wenn }}n>1\end{cases}}}$

With Iverson's mapping, this recurrence equation can be converted into a simple form:

${\displaystyle f_{n}=(f_{n-1}+f_{n-2})\cdot [n>1]+[n=1]}$

The part corresponds to the recursive definition of the Fibonacci numbers. The factor removes this recursive part for all Fibonacci numbers with an index less than or equal to 1. And is 1 if and only if the index is 1. This makes the Fibonacci number with the index 1 equal to 1, and this ensures that the Fibonacci numbers with an index greater than 1 also have a value greater than 0. ${\displaystyle (f_{n-1}+f_{n-2})}$ ${\displaystyle [n>1]}$ ${\displaystyle [n=1]}$ ${\displaystyle n}$

With this formula it is now easier to determine the closed formula.

See also

• Indicator function