2) It is possible to speak about relatedness and unrelatedness of certain items even though all classes of them have common origin.
2.2.3. Concepts of relatedness and unrelatedness from point of view of set theory and abstract algebra
Concept of relatedness is actually equivalence relation since it meets necessary and sufficient requirements for a binary relation to be considered as equivalence relation:
1) Reflexivety: a ~ a: a is related with a;
2) Symmetry: if a ~ b then b ~ a: if a is related with b then b is related with a;
3) Transitivity: if a ~ b and b ~ c then a ~ c: if a is related with b and b is related with c then a is related with c.
If an equivalence relation is defined on a set then it necessarily supposes grouping of elements of the set into equivalence classes and these classes aren’t intersected (Hrbacek, Jech: 1999).
2.2.4. Particular conclusions on the concepts of relatedness and unrelatedness for linguistics
When it is said that certain languages are genetically related (or simply related) it means that these languages belong to the same stock or even to the same group.
Taking into the consideration what has been said in 2.2.2 we should keep in mind that in the case of languages there are actually no positive evidences that all languages existing nowadays originated from the same ancestor, i.e.: monogenesis is still an unproved hypothesis, though anyway even if all languages can be reduced to the same proto-language that existed in a very distant past it doesn’t mean yet we can’t speak of their relatedness/unrelatedness.
Then, taking into consideration what has been said in 2.2.3 we can say the following:
The set of languages existing nowadays on the planet is rather well described: we know that there are about 7102 languages and about 151 stocks and 83 isolated languages (Ethnologue: 2015), so we can speak about 234 stocks; and we hardly can expect discovering of some new unknown languages. Thus, we can say that we have rather complete image of set of languages and that there are about 234 classes of equivalence/relatedness.
If we take an X stock, we obviously can show many languages which don’t belong to the stock, i.e.: languages which are not related with language x (a random language of X stock), for example: in the case of Indo-European stock there are many languages which are not related with English: Arabic, Basque, Finnish, Georgian, Turkish, Chinese, Japanese, Hawaiian, Eskimo, Quechua and so on. In the case of Sino-Tibetan stock there are many languages which aren’t related with Chinese: Arabic, English, Eskimo, Finnish, Japanese, Turkish, Vietnamese and so on.