| 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 37278 and the comment of dfec2 8705). 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 37716).
Without the definition of ≀ 𝑅 we
should have to relate the right side of these theorems to a composition
of a converse (cf. dfcoss3 37279) or to the range of a range Cartesian
product of classes (cf. dfcoss4 37280), which would make the theorems
complicated and confusing. Alternate definition is dfcoss2 37278.
Technically, we can define it via composition (dfcoss3 37279) or as the
range of a range Cartesian product (dfcoss4 37280), but neither of these
definitions reveal directly how the cosets by 𝑅 relate to each
other. We define functions (df-funsALTV 37546, df-funALTV 37547) and
disjoints (dfdisjs 37573, dfdisjs2 37574, df-disjALTV 37570, dfdisjALTV2 37579)
with the help of it as well. (Contributed by Peter Mazsa,
9-Jan-2018.) |
