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I'm sure the answer to my question here already exists, but I'm having trouble finding a clear answer using proper formal notation instead of plain English.

I am aware of the notation used to place restrictions upon the domain of a function, i.e. excluding $x \in \left( -1, 1 \right)$ from the set of solutions to the real function $f \left( x \right) = x^2$ can be written as:

$$\left. f \right| _{ \mathbb{R} \setminus \left( -1, 1 \right) } \left( x \right) = x^2$$

However, if I have an equation/inequality (not a function) of multiple variables, and I wish to specify an additional restriction on the value of one (or more) of the variables to exclude certain solutions, what is the proper notation to use without simply writing the restriction out in plain English?

For instance, if I want to specify the part of the unit ball $x^2 + y^2 + z^2 \leq 1$ constrained between the planes $x = -\frac{1}{2}$ and $x=\frac{1}{4}$, how would I write this inequality without writing something like this? $$x^2 + y^2 + z^2 \leq 1 \text{ where } -\frac{1}{2} < x < \frac{1}{4}$$

Next-Door Tech
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  • You just have a system of two inequalities. In set-builder notation you could write ${ (x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \le 1, - \frac{1}{2} < x < \frac{1}{4} }$. – Qiaochu Yuan Oct 07 '23 at 02:11
  • What is wrong with using words? If you looks at examples of excellent mathematicsl writing, it will generally explain far more by words than by formulas. The only reason I can see for preferring formulas for something that can explained with words is to make it more accessible to those will little experience with the language. – Paul Sinclair Oct 07 '23 at 21:57
  • Both are fair points. I suppose the intent behind my question was more to see if there is a syntactically correct shorthand for the word "where" in my final example. Would using a vertical bar in place of "where" be appropriate outside of set builder notation? i.e. $x^2+y^2+z^2\leq1\mid-\frac{1}{2}<x<\frac{1}{4}$ – Next-Door Tech Oct 09 '23 at 05:27

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