# Meaning of the backslash operator on sets

I am self-studying analysis and ran across this:

$\mathbb R \setminus \mathbb N$ is an open subset of $\mathbb R$

My best guess for interpretation was this:

the set $\mathbb R \setminus \mathbb N$ is an open subset of $\mathbb R$.

which doesn't mean much to me. Can anyone clear this up a bit? I know that the 'divided by' symbol is usually a slash in the opposite direction. And I am unsure how I would divide the reals by the naturals anyway.

## 2 Answers

It’s set theoretic complement and in this case it denotes the set of all reals which are not natural: $$ℝ \setminus ℕ = \{x ∈ ℝ;~x \notin ℕ\}$$

• makes much more sense! thank you.
– 123
Dec 25 '14 at 23:31
• @ k.stm - Out of curiosity, it seems I could prove this true by arguing that our set R\N is a union of open subsets and is therefore itself open. Would this, more or less, be correct?
– 123
Dec 26 '14 at 0:03
• @mathtastic Yes. Dec 26 '14 at 0:08

The backward slash is kind of the set theory equivalent of subtracting, i.e.,

$$A\setminus B=\{a\in{A}\mid a\notin{B}\}\;.$$

• Use \setminus to get the set difference operator, and \mid to get the vertical bar with space around it. Dec 25 '14 at 23:32
• But keep in mind that you still can get something like $\mathbb Q \setminus \{\pi\} = \mathbb Q$. So you just 'substract' as long the thing you want to substract was actually there. Dec 25 '14 at 23:32
• Thank you, I was having difficulties with the proper formatting. Dec 25 '14 at 23:33
• thank you for taking the time to provide this answer!
– 123
Dec 25 '14 at 23:57
• I've seen the following in pseudocode: "A \ B". So this means that x is in Array A, AND not in Array B. The | in your formula doesn't mean OR right? Mar 14 '18 at 15:59