Half characteristic function

The half characteristic function or partial characteristic function is a function of mathematics that identifies a set . It is defined as follows:

${\displaystyle \chi '_{A}\colon A\to \{1\},\;\;a\mapsto 1}$.

As you can see, all the “magic” of the function is in the definition domain. Now if A is a subset of a larger set B, so χ 'is _A to B \ A undefined. We then get:

${\displaystyle \chi '_{A}\colon B\to \{0,1\},\;\;a\mapsto {\begin{cases}1&{\mbox{ falls }}a\in A\\{\mbox{undefiniert}}&{\mbox{ sonst }}\end{cases}}}$

Semi-decidability

Half the characteristic function can name all elements on B that belong to A, but cannot really exclude elements that do not belong to A. One speaks of χ ' _A being partial. If χ ' _{A is} also computable, then A is called semi-decidable or recursively enumerable , since all elements can be enumerated, but the elements B \ A cannot be excluded. For this one needs the characteristic function , which is total.

See also

• Recursively enumerable set