| 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 35665 and the comment of dfec2 8295). 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 ). Without the
definition of ≀ 𝑅 we
should have to relate the right side of these theorems to a composition
of a converse (cf. dfcoss3 35666) or to the range of a range Cartesian
product of classes (cf. dfcoss4 35667), which would make the theorems
complicated and confusing. Alternate definition is dfcoss2 35665.
Technically, we can define it via composition (dfcoss3 35666) or as the
range of a range Cartesian product (dfcoss4 35667), but neither of these
definitions reveal directly how the cosets by 𝑅 relate to each
other. We define functions (df-funsALTV 35918, df-funALTV 35919) and
disjoints (dfdisjs 35945, dfdisjs2 35946, df-disjALTV 35942, dfdisjALTV2 35951)
with the help of it as well. (Contributed by Peter Mazsa,
9-Jan-2018.) |
