| Description: Define the class of
cosets by 𝑅: 𝑥 and 𝑦 are cosets by
𝑅 iff there exists a set 𝑢 such
that both 𝑢𝑅𝑥 and
𝑢𝑅𝑦 hold, i.e., both 𝑥 and
𝑦
are are elements of the 𝑅
-coset of 𝑢 (see dfcoss2 39627 and the comment of dfec2 8748). 𝑅 is
usually a relation.
This concept simplifies theorems relating partition and equivalence: the
left side of these theorems relate to 𝑅, the right side relate to
≀ 𝑅 (see e.g. pet 40089).
Without the definition of ≀ 𝑅 we
should have to relate the right side of these theorems to a composition
of a converse (cf. dfcoss3 39628) or to the range of a range Cartesian
product of classes (cf. dfcoss4 39629), which would make the theorems
complicated and confusing. Alternate definition is dfcoss2 39627.
Technically, we can define it via composition (dfcoss3 39628) or as the
range of a range Cartesian product (dfcoss4 39629), but neither of these
definitions reveal directly how the cosets by 𝑅 relate to each
other. We define functions (df-funsALTV 39890, df-funALTV 39891) and
disjoints (dfdisjs 39917, dfdisjs2 39918, df-disjALTV 39914, dfdisjALTV2 39923)
with the help of it as well. (Contributed by Peter Mazsa,
9-Jan-2018.) |
