What does $\mathbb{R} \setminus S$ mean?

Question

What does $\mathbb{R}\setminus S$ mean? I am not getting it what it actually means. I have found it manywhere in real-analysis like in the definition of boundary points of a set. Can anyone tell me what it means really?

Answer 1

If $S$ is a set and $\Bbb R$ is the set of all real numbers then $$\Bbb R \setminus S = \{x \mid x\in \Bbb R \wedge x \not\in S\}$$ meaning it's the set of all elements $x$ such that $x$ is a real number and $x$ is not an element of the set $S$.

For instance $\Bbb R \setminus \{0\}$ is the set of all real numbers except $0$.

This operation is called relative complement or set difference.

Answer 2

That symbols means set difference. It is called \setminus in $\TeX.$

If $A$ and $B$ are sets, then $A \setminus B$ is the set of elements in $A$ but not in $B$.