$A$ and $B$ are infinite sets with $B \subseteq A$. Which of the following statements are true?

  1. $A \sim B$
  2. $A \sim A \setminus B$
  3. If $B$ is countable then $A \sim B$
  4. If $A$ is countable then $A \sim B$
  5. $A \setminus B$ is finite

My Attempt:

  1. False: $\mathbb{N} \subseteq \mathbb{R}$ but $\mathbb{N} \not\sim \mathbb{R} \implies \mathbb{R} \not\sim \mathbb{N}$

  2. False: $\mathbb{N} \subseteq \mathbb{R}$. $\mathbb{N}$ is countable but $\mathbb{N} \not\sim \mathbb{R} \implies \mathbb{R} \not\sim \mathbb{N}$

  3. True: $B \subseteq A$ and $A$ countable $\implies B$ countable $\implies A \sim \mathbb{N}, B \sim \mathbb{N} \implies A \sim B$.

  4. False: $\mathbb{N} \subseteq \mathbb{R}$, but $\mathbb{R} \setminus \mathbb{N}$ is infinite.

Are these correct? Also, I am unsure about Question 2. Can someone please help point me in the right direction?

  • $\begingroup$ What do you mean with $A \sim B$? Please try to be specific when using symbols that have different meanings depending on the context. $\endgroup$ – Newb Mar 31 '15 at 22:18
  • $\begingroup$ For 2, $A=B=\mathbb{N}$ $\endgroup$ – egreg Mar 31 '15 at 22:19
  • $\begingroup$ $A \sim B$ denotes that $A$ is equivalent to $B$ $\endgroup$ – user860374 Mar 31 '15 at 22:19

You're correct. As for (2)...

HINT: What happens if $A=B$?

  • $\begingroup$ If $A = B$ then $A \setminus B = \emptyset$. Thus $A \not\sim A \setminus B = \emptyset$ since we (clearly) cannot find a bijective function that maps $A$ to $A \setminus B = \emptyset$. Is this correct? $\endgroup$ – user860374 Mar 31 '15 at 22:27
  • $\begingroup$ Yes, that is correct. $\endgroup$ – Asaf Karagila Mar 31 '15 at 22:30

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