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:

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

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.