| 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 37586 and the comment of dfec2 8708). 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 38024).
Without the definition of ≀ 𝑅 we
should have to relate the right side of these theorems to a composition
of a converse (cf. dfcoss3 37587) or to the range of a range Cartesian
product of classes (cf. dfcoss4 37588), which would make the theorems
complicated and confusing. Alternate definition is dfcoss2 37586.
Technically, we can define it via composition (dfcoss3 37587) or as the
range of a range Cartesian product (dfcoss4 37588), but neither of these
definitions reveal directly how the cosets by 𝑅 relate to each
other. We define functions (df-funsALTV 37854, df-funALTV 37855) and
disjoints (dfdisjs 37881, dfdisjs2 37882, df-disjALTV 37878, dfdisjALTV2 37887)
with the help of it as well. (Contributed by Peter Mazsa,
9-Jan-2018.) |
