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Let $\alpha\in [0,9].$ I want to use set notation to choose an integer $k_1\in\{ 0, \ldots, 9 \}$ such that $\vert k_1-\alpha\vert$ is as small as possible. But if, for example, $\alpha=0.5$, I want to allow $k_1$ to be either $0$ or $1$.

Is the following correct set notation for this?

Let $k_1\in \{\ x\in\{0,\ldots,9\}:\quad \lvert\frac{x}{1}-\alpha \rvert\leq\lvert\frac{y}{1}-\alpha \rvert\quad \forall\ y\in\{0,\ldots,9\}\ \}$.

And I know we can use words to describe choice from such a set, but I am not looking for this.

But, are there "better/simpler" set notations to describe the same set for $k_1$ to be chosen from, akin to the single set notation above?

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You have expressed the statement "Let $k_1$ be an integer between 0 and 9 such that the difference between $\alpha$ and $k_1$ is no greater that the difference between $\alpha$ and any other integer between 0 and 9" in set theoretic notation. I don't understand why you've written $\frac{x}{1}$ and $\frac{y}{1}$ rather than $x$ and $y$, though. Stick with $x$ and $y$.

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  • $\begingroup$ Yes, thanks for your answer, that clarifies things somewhat. Is there a better/simpler set notation than the one I have used? $\endgroup$
    – Adam Rubinson
    Jul 8, 2021 at 13:42
  • $\begingroup$ I suppose you could replace $\{0,\ldots,9\}$ with $\mathbb{Z}$, since it will amount to the same thing, but that's not much of a change. I don't know what the context is, since you seem not to be allowing the use of words. But in any context to be read by humans (rather than computers) I would just say "Let $k_1$ be an integer closest to $\alpha$". $\endgroup$
    – Tom Sharpe
    Jul 8, 2021 at 13:59
  • $\begingroup$ Hmm... I'm not sure that sentence is "good English". But my English isn't the best, so maybe you are correct. $\endgroup$
    – Adam Rubinson
    Jul 8, 2021 at 15:08
  • $\begingroup$ One has to "let the reader understand" the use of the indefinite article, rather than the definite. Obviously in most normal circumstances, the use of a superlative (e.g., closest) implies uniqueness. But in this strict mathematical context, it does not, and the use of 'an' rather than 'the' allows for this. $\endgroup$
    – Tom Sharpe
    Jul 8, 2021 at 15:30

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