Why the set of irrational numbers is represented as $\mathbb{R}\setminus\mathbb{Q}$ instead of $\mathbb{R}-\mathbb{Q}$?

Question

What does the "\" symbol means in this context?

I have seen it used for quotient sets like $X /{\sim}$ where $X$ is a set and $\sim$ is an equivalence relation but I don't know what it means applied to two sets.

$\mathbb{R}-\mathbb{Q}$ seems to be much more suitable, since the set of irrational numbers are just that: real numbers which are not rational.

Answer 1

No answer on this page has pointed out that $\setminus$ is used like $-$ for set subtraction very often in math, but that this is very different from $/$. In particular $/$ is used for quotients, which are a completely different thing.

Answer 2

$\setminus$ and $-$ mean the exact same thing in set theory, both mean set difference.

Answer 3

Both are used to represent the set of irrational numbers

see this link

They are the same exact thing.

$a \in \mathbb{R \backslash Q}$ means that $a$ is a real number that is not a quotient which can also be denoted as $a \in \mathbb{R - Q}$

Answer 4

Both symbols $\setminus$ \setminus and $-$ - are used for denoting set difference: $$A\setminus B = A - B = \{ x \mid x \in A,\,x \not\in B \}.$$

I, particularly, prefer $A \setminus B$. In some contexts, we can have something like: $$A-B = \{ x-y \mid x \in A,\, y \in B \},$$ so sticking to $\setminus$ there is zero chance of confusion.

Answer 5

The ordinary minus "$-$" and the ordinary slash "$/$" both have various meanings when used with sets, depending on context. The backslash "$\setminus$" is used exclusively for relative complement (i.e., set subtraction) as far as I have ever seen.