| 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 35821 and the comment of dfec2 8275). 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 35822) or to the range of a range Cartesian
product of classes (cf. dfcoss4 35823), which would make the theorems
complicated and confusing. Alternate definition is dfcoss2 35821.
Technically, we can define it via composition (dfcoss3 35822) or as the
range of a range Cartesian product (dfcoss4 35823), but neither of these
definitions reveal directly how the cosets by 𝑅 relate to each
other. We define functions (df-funsALTV 36074, df-funALTV 36075) and
disjoints (dfdisjs 36101, dfdisjs2 36102, df-disjALTV 36098, dfdisjALTV2 36107)
with the help of it as well. (Contributed by Peter Mazsa,
9-Jan-2018.) |
