For sets A and B, where A is countable and B is uncountable, what would $A \setminus B$ be?

Question

Since A is countable and B is uncountable then this would be undefined as B is larger than A right? Or am I missing something here?

Answer 1

$A \setminus B$ is defined even if $B$ is not a subset of $A$. It's the set of everything that is in $A$ but not in $B$.

Since $A \setminus B$ is a subset of $A$, and $A$ is countable, it follows that $A \setminus B$ is countable (possibly finite or empty).

Example where $A \setminus B$ is countably infinite: $A = \mathbb Q$, $B = \mathbb R \setminus \mathbb Q$, then $A \setminus B = \mathbb Q$.

Example where $A \setminus B$ is finite but not empty: $A = \mathbb Q$, $B = \mathbb R \setminus \{0\}$, then $A \setminus B = \{0\}$.

Example where $A \setminus B$ is empty: $A = \mathbb Q$, $B = \mathbb R$. Then $A \setminus B = \emptyset$.

Answer 2

It depends on the relation between $A$ and $B$. For example, if $A = \mathbb{N}$ and $B = \mathbb{R}$, then $A \setminus B = \varnothing$. However, if you take $A = \mathbb{N}^{+}$ and $B = \mathbb{R}^{-}$, then $A \setminus B = A$.

Answer 3

Let $A=\{-1\}$, $B=\mathbb{R}^+$. Clearly, $A$ is countable and $B$ is uncountable. However, $A \backslash B=A$ is countable.

In fact, for countable $A$ and uncountable $B$, $A \backslash B$ is necessarily countable since $A \backslash B \subseteq A$, and subsets of countable sets are necessarily countable.