| 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 38411 and the comment of dfec2 8677). 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 38850).
Without the definition of ≀ 𝑅 we
should have to relate the right side of these theorems to a composition
of a converse (cf. dfcoss3 38412) or to the range of a range Cartesian
product of classes (cf. dfcoss4 38413), which would make the theorems
complicated and confusing. Alternate definition is dfcoss2 38411.
Technically, we can define it via composition (dfcoss3 38412) or as the
range of a range Cartesian product (dfcoss4 38413), but neither of these
definitions reveal directly how the cosets by 𝑅 relate to each
other. We define functions (df-funsALTV 38680, df-funALTV 38681) and
disjoints (dfdisjs 38707, dfdisjs2 38708, df-disjALTV 38704, dfdisjALTV2 38713)
with the help of it as well. (Contributed by Peter Mazsa,
9-Jan-2018.) |
