| 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 38871 and the comment of dfec2 8643). 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 39333).
Without the definition of ≀ 𝑅 we
should have to relate the right side of these theorems to a composition
of a converse (cf. dfcoss3 38872) or to the range of a range Cartesian
product of classes (cf. dfcoss4 38873), which would make the theorems
complicated and confusing. Alternate definition is dfcoss2 38871.
Technically, we can define it via composition (dfcoss3 38872) or as the
range of a range Cartesian product (dfcoss4 38873), but neither of these
definitions reveal directly how the cosets by 𝑅 relate to each
other. We define functions (df-funsALTV 39134, df-funALTV 39135) and
disjoints (dfdisjs 39161, dfdisjs2 39162, df-disjALTV 39158, dfdisjALTV2 39167)
with the help of it as well. (Contributed by Peter Mazsa,
9-Jan-2018.) |
