Interval Notation and Set-Builder Notation

[speedway_joe]

Say I was interested in the set of all real numbers excluding -1 and 1. To write this in interval notation I think it be;

(-infinity, -1) U (-1, 1) U (1, infinity)

I'm less sure of how to write it in set-builder notation. I'm thinking it could be;

{x|x not equal to -1} Intersected with {x|x not equal to 1}

I couldn't find symbols for "not equal to" or "Intersection"

Thanks,
Speedway_Joe

 

[stapel]

Say I was interested in the set of all real numbers excluding -1 and 1. To write this in interval notation I think it be;

(-infinity, -1) U (-1, 1) U (1, infinity)

I'm less sure of how to write it in set-builder notation. I'm thinking it could be;

{x|x not equal to -1} Intersected with {x|x not equal to 1}

I couldn't find symbols for "not equal to" or "Intersection"

For "not equals", use the "equals" sign with a slash through it: \(\displaystyle \, \neq\)

For the set-builder notation, I think you're probably looking for something like this:

. . . . .\(\displaystyle \{x\, \in\, \mathbb{R}\, \) | \(\displaystyle \, x\, \neq\, -1,\,1\}\)

 

[Ishuda]

Say I was interested in the set of all real numbers excluding -1 and 1. To write this in interval notation I think it be;

(-infinity, -1) U (-1, 1) U (1, infinity)

I'm less sure of how to write it in set-builder notation. I'm thinking it could be;

{x|x not equal to -1} Intersected with {x|x not equal to 1}

I couldn't find symbols for "not equal to" or "Intersection"

Thanks,
Speedway_Joe

As a different way than what stapel wrote, I have also seen something like this expressed as
.\(\displaystyle \{\, x\, \in\, \{\mathbb{R}-\{-1\, ,\, 1\}\, \}\, \}\)

 

[pka]

None of this matters unless it agrees 100% with your text-material. There is no standard set of definitions,
If there is, then this is it. Please note that there is agreement that \(\displaystyle \{x(x)\}\) is the form of the answer.

Say I was interested in the set of all real numbers excluding -1 and 1. To write this in interval notation I think it be; (-infinity, -1) U (-1, 1) U (1, infinity)

The problem with that is that is no proposition P(x).

For the set-builder notation, I think you're probably looking for something like this:
\(\displaystyle \{x\, \in\, \mathbb{R}\, \) | \(\displaystyle \, x\, \neq\, -1,\,1\}\)

This does have \(\displaystyle P(x)\), but were I grading it would be concerned with format. See this.
Also I would ask if \(\displaystyle \, x\neq\, -1,\,1\}\) means \(\displaystyle x\neq\ -1\vee x\not=1\}\) or else \(\displaystyle x\neq\ -1\wedge x\not=1\}\)

As a different way than what stapel wrote, I have also seen something like this expressed as
.\(\displaystyle \{\, x\, \in\, \{\mathbb{R}-\{-1\, ,\, 1\}\, \}\, \}\)

Here is an alternative: \(\displaystyle \{x: |x|\not=1 \}\), assuming that the domain of \(\displaystyle \{x(x)\}\) is \(\displaystyle \mathbb{R}\).

AGAIN: logicians can be overly picky.