1. Writing domains

I got a few problems regarding writing domains.

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

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

3. For $\displaystyle x=1$ or $\displaystyle 3<x<5$, can I write $\displaystyle \{1\} \cup (3,5)$. 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 $\displaystyle f(x)=\frac{x+1}{x-1}$, $\displaystyle x\neq1$.
Instead of writing $\displaystyle x\neq1$, can I write $\displaystyle x \in \mathbb{R}\backslash \{1\}$

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

3. For $\displaystyle x=1$ or $\displaystyle 3<x<5$, can I write $\displaystyle \{1\} \cup (3,5)$. 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 $\displaystyle x \in \mathbb{R}\backslash \{1\}$ is more rigorous than "for $\displaystyle x \neq 1$", it makes it harder to read, and not everyone uses that notation for set difference, either.