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dlee
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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.
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.