Description: Define indexed union.
Definition indexed union in [Stoll] p. 45. In
most applications, 𝐴 is independent of 𝑥
(although this is not
required by the definition), and 𝐵 depends on 𝑥 i.e. can be read
informally as 𝐵(𝑥). We call 𝑥 the index, 𝐴 the
index
set, and 𝐵 the indexed set. In most books,
𝑥 ∈
𝐴 is written as
a subscript or underneath a union symbol ∪. We use a special
union symbol ∪
to make it easier to distinguish from plain class
union. In many theorems, you will see that 𝑥 and 𝐴 are in
the
same disjoint variable group (meaning 𝐴 cannot depend on 𝑥) and
that 𝐵 and 𝑥 do not share a disjoint
variable group (meaning
that can be thought of as 𝐵(𝑥) i.e. can be substituted with a
class expression containing 𝑥). An alternate definition tying
indexed union to ordinary union is dfiun2 3907. Theorem uniiun 3926 provides
a definition of ordinary union in terms of indexed union. (Contributed
by NM, 27-Jun-1998.) |