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Short question about set-builder notation.

Do $$D=\{x \mid x \in \mathbb{R}, x < k\}$$ and $$D=\{x \in \mathbb{R} \mid x < k\}$$ mean the same thing?

I see both of them used in different contexts and was wondering if they are interchangeable.

Mark Kamsma
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Scene
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1 Answers1

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They mean the same thing. I prefer $\{x \in \mathbb{R} \mid x < k \}$, because it is a clear separation between the domain ($\mathbb{R}$) and the condition ($x < k$). So I think it is easier to read, definitely when the condition gets more complicated.

Mark Kamsma
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  • So would your preferred notation read "x is an element of the reals, such that x is less than k"? In this case, the other notation sounds strange: "x, such that x is an element of the reals and x is less than k". – Scene Mar 18 '20 at 16:44
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    @SeanXie Yes, and exactly for the reason you mentioned it is my preferred notation. – Mark Kamsma Mar 18 '20 at 16:53
  • I think they both have their uses; the first notation generalizes more easily to the notation ${f(x) | x \in \mathbb{R}}$. – Jair Taylor Mar 18 '20 at 17:06
  • @JairTaylor Your example is still different though: $f(x)$ can be in a different domain than $\mathbb{R}$ (e.g. $f$ could be complex valued). – Mark Kamsma Mar 18 '20 at 17:20