Question

prove that i_Λ : Λ → Λ is a function which represents the set Λ as an indexed family with index set Λ.

Answer 1

ANSWER :-

given that ,

i_Λ : Λ → Λ is a function

Let I and X be sets and a surjective function, such that

then this establishes a family of elements in X indexed by I , which is denoted by (x_i)_i∈I or simply (x_i), when the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, the latter with the risk of mixing-up families with sets.

An indexed family can be turned into a set by considering the set , that is, the image of I under x. Since the mapping x is not required to be injective, there may exist with   such that  . Thus,   where |A| denotes the cardinality of the set A.

The index set is not restricted to be countable, and, of course, a subset of a powerset may be indexed, resulting in an indexed family of sets

so, set Λ  as an indexed family with index set Λ .

hence proved.