1. Writing domains

I got a few problems regarding writing domains.

1. If $f(x)=\frac{x+1}{x-1}$, $x\neq1$.
Instead of writing $x\neq1$, can I write $x \in \mathbb{R}\backslash \{1\}$

2. For the range $x<-1$ or $x>1$, can I write $\mathbb{R} \backslash [-1,1]$

3. For $x=1$ or $3, can I write $\{1\} \cup (3,5)$. Can I treat intervals as a set and union with other sets?

Thanks a lot for those who help!

2. Originally Posted by acc100jt
I got a few problems regarding writing domains.

1. If $f(x)=\frac{x+1}{x-1}$, $x\neq1$.
Instead of writing $x\neq1$, can I write $x \in \mathbb{R}\backslash \{1\}$

2. For the range $x<-1$ or $x>1$, can I write $\mathbb{R} \backslash [-1,1]$

3. For $x=1$ or $3, can I write $\{1\} \cup (3,5)$. Can I treat intervals as a set and union with other sets?
1. Yes.
2. Yes.
3. Yes.

3. I would suggest you always go for the simplest, easiest to read notation. While $x \in \mathbb{R}\backslash \{1\}$ is more rigorous than "for $x \neq 1$", it makes it harder to read, and not everyone uses that notation for set difference, either.