1.11 Infinite Sets

The Diagonal Argument was used to illustrate that the set of infinite binary sequences, \(\{ 0, 1 \} ^ \omega \), is _____,

Exercise 1

  countable uncountable neither countable nor uncountable uncountable 

by building a _____

Exercise 2

  finite binary sequence infinite binary sequence Either of the above Neither of the above infinite binary sequence 

that is _____ the list of infinite binary sequences.

Exercise 3

  in not in not in 

The set of natural numbers is strictly smaller than the set, \(A \), of infinite binary strings (i.e. \(\mathbb{N} \) strict \(A \)), because _____.

Exercise 4

  there is no surjection from the set of natural numbers to \(A \) there is a bijection from the set of natural numbers to \(A \) the statement is false! The natural numbers are as large as \(A \) there is no injective relation from the set of natural numbers to \(A \) there is no surjection from the set of natural numbers to \(A \) 

Cantor's Theorem states that for any set \(A \), finite or infinite, _____ pow(\(A \)).

Exercise 5

  \(A \) inj \(A \) bij \(A \) surj \(A \) strict \(A \) strict