[해석학] Countable collection of sets

 

 

Further notation of Countable sets

• Notation (Union of the sets)

 

 

 

 

 

대부분의 Field Axioms가 만족한다. (U => '+', ∩=> 'x')

 

Commutativity, Associativity, Distributive

 

 

 

 

• Theorem: Let A be countable and let E_a be countable for all a in A. Then union of countable sets E_a is a countable set. => "Countable union of countable sets is countable"

 

• Theorem (corollary): At most countable union of at most countable sets is at most countable

 

 

 

 

 

Set product

 

• Definition: Let A, B be sets, then

1. Direct product: AXB = {(a,b) | a in A, b in B}
2. Furthermore; AXBXC = {(a,b,c) | a in A, b in B, c in C}

 

• Lemma: A_1, ... , A_n is at most countable => A_1  X A_2 X ... X A_n is at most countable

Proof using induction

 

• Ex: Q is countable

 

 

 

 

 

Infinite Sequences

 

• Let A be a set, a_k in A for k=1,2....
• Then Sequence a_1, a_2, a_3, ... , a_n, ... {a_k}^∞
• Ex:

Infinite sequence problem example