Question

Dedekind’s definition of an infinite set: S is infinite if there is a bijection f : S → T, where T is a proper subset of S.

(A) Show that, if s ∈ S − T, then {s, f(s), f(f(s)),...} is a countably infinite subset of S .

(B) Show, conversely, that a countably infinite subset of S gives a bijection f : S → T, where T is a proper subset of S.

Answer 1

Dedekind's definition of an infinite set and the relationship between countably infinite subsets and bijections in S.

Step-by-step explanation:

In Dedekind's definition of an infinite set, an infinite set S can be defined as a set that has a bijection (one-to-one correspondence) f from S to a proper subset of S, denoted as T.

(A) If s ∈ S - T, then the set {s, f(s), f(f(s)),...} is a countably infinite subset of S. This means that there is a bijection between this infinite subset and the set of natural numbers.

(B) Conversely, if there is a countably infinite subset of S, it implies the existence of a bijection f from S to T, where T is a proper subset of S.