1.11 Infinite Sets

Let \(A \) and \(B \) be any two sets. Which of the following is true about Cantor's Idea?

Exercise 1

 When \(A \) surj \(B \), \(A \) is as big as \(B \). 

 When \(A \) inj \(B \), \(A \) is as small as \(B \). 

 When \(A \) bij \(B \), \(A \) is the same size as \(B \). 

 \(\text{pow} (\mathbb{N}) \) bij \(\{0, 1 \}^\omega \). 

 Two infinite sets have the same size when there is a bijection between them. 

Let \(A \) and \(B \) be any two sets. What does the Schroeder-Bernstein Theorem tell us?

Exercise 2

 \(A \) surj \(B \) surj \(A \) IMPLIES \(A \) bij \(B \) 

 \(|A| \ge |B| \ge |A| \) IMPLIES \(|A| = |B| \) 

 \(A \) surj \(B \) inj \(A \) IMPLIES \(A \) bij \(B \) 

 None of the above 

What property distinguishes infinite sets from finite sets?

Exercise 3

  Cantor's Idea Schroeder-Bernstein Theorem pow(A) surj A \(A \cup \{d\}\) bij \(A\), where \( d \notin A \) None of the above \(A \cup \{d\}\) bij \(A\), where \( d \notin A \)