| 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 36729 and the comment of dfec2 8577). 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 37167).
Without the definition of ≀ 𝑅 we
should have to relate the right side of these theorems to a composition
of a converse (cf. dfcoss3 36730) or to the range of a range Cartesian
product of classes (cf. dfcoss4 36731), which would make the theorems
complicated and confusing. Alternate definition is dfcoss2 36729.
Technically, we can define it via composition (dfcoss3 36730) or as the
range of a range Cartesian product (dfcoss4 36731), but neither of these
definitions reveal directly how the cosets by 𝑅 relate to each
other. We define functions (df-funsALTV 36997, df-funALTV 36998) and
disjoints (dfdisjs 37024, dfdisjs2 37025, df-disjALTV 37021, dfdisjALTV2 37030)
with the help of it as well. (Contributed by Peter Mazsa,
9-Jan-2018.) |
