# Countable Sets

#### dlee

##### New member
What is a countable set exactly? HELP? Can someone help guide me through this problem? I'm a bit lost on how to show this...

Countable union of countable sets: Let I be a countable set. Let Ai , i ∈ I be a family of sets such that each Ai is countable. We will show that U i ∈ I Ai is countable.

(1) Show that there exists a family of sets C1, C2, C3,..., i.e, a family of sets Ci indexed by i ∈ N such that Ci is countable for every i ∈ N and U i ∈ I Ai = U i ∈ N Ci.
(Hint: Some of the Ci can be empty sets.)

(2) Show that there exists a family of sets Bi , i ∈ N such that U i ∈ N Ci = U i ∈ N Bi, each Bi is countable and Bi ⋂ Bj = ∅ for any i ≠ j , i.e., the Bi’s are pairwise disjoint.
(Hint: Think of the construction Bi = Ci \ (C1⋃ C2⋃ ... ⋃ Ci - 1).)

(3) Show that U i ∈ N Bi is countable for the family of sets Bi , i ∈ N from part (ii). You may assume that |N x N| = |N|.

(4) Hence conclude U i ∈ I Ai is countable.

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
What is a countable set exactly?
Why don't you look up the definition in your textbook or lecture notes? As you can see from Wikipedia, "countable" is used in two senses.

Can someone help guide me through this problem? I'm a bit lost on how to show this...
In turn, can you show what you tried or describe your difficulty? The points (1) and (2) seem pretty straightforward since the given hints describe the necessary definitions.