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

$A$ 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.

• What do you mean with $A \sim B$? Please try to be specific when using symbols that have different meanings depending on the context. – Newb Mar 31 '15 at 22:18
• For 2, $A=B=\mathbb{N}$ – egreg Mar 31 '15 at 22:19
• $A \sim B$ denotes that $A$ is equivalent to $B$ – user860374 Mar 31 '15 at 22:19
HINT: What happens if $A=B$?
• 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? – user860374 Mar 31 '15 at 22:27