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Consider the following statement (from a convex optimization context):

"There exists an unique $\mathbf{x}^{\star}$ in the $n$-dimensonal real space such that $f(\mathbf{x}^{\star})$ is equal to or lower than $f(\mathbf{y})$ for all $\mathbf{y}$ in the domain of $f$ AND the gradient vector is zero at this value".

When I think of the equivalent mathematical statement, I get undecided between these two writings:

  • $\exists!\; \mathbf{x}^{\star} \in \mathbb{R}^{n} \mid f(\mathbf{x}^{\star}) \leq f(\mathbf{y})\;\forall\;\mathbf{y}\in\text{dom}(f), \nabla f(\mathbf{x}^{\star}) = \mathbf{0}$
  • $\exists!\; \mathbf{x}^{\star} \in \mathbb{R}^{n} \mid f(\mathbf{x}^{\star}) \leq f(\mathbf{y})\;\forall\;\mathbf{y}\in\text{dom}(f)\;\wedge \nabla f(\mathbf{x}^{\star}) = \mathbf{0}$

The truth is that the symbol $\wedge$ seems to rarely be used in Engineering book, I only see it on CS books. Nevertheless, $\wedge$ here looks much more suitable than a mere comma. But I fell insecure about this usage.

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    $\begingroup$ You should be consistent about whether your put quantifiers before or after the predicate. I would write it as $\exists!\; \mathbf{x}^{\star} \in \mathbb{R}^{n} \, \forall\;\mathbf{y}\in\text{dom}(f)\,f(\mathbf{x}^{\star}) \leq f(\mathbf{y})\;\wedge \nabla f(\mathbf{x}^{\star}) = \mathbf{0}$. You could also replace $\wedge$ with the word "and". $\endgroup$ Nov 26, 2022 at 1:29
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    $\begingroup$ Your first way of writing the sentence, using natural language, is better than either symbol-heavy version. Aim to use words where you can, rather than symbols. Using $\land $ instead of "and" is unnecessary unless you are actually intending the sentence to be read in terms of propositional logic rather than by a human. See Q181210. $\endgroup$
    – Jam
    Nov 26, 2022 at 1:29
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    $\begingroup$ Whether or not you use "$\land$" (and I think that you probably shouldn't), I'd say that you definitely shouldn't use bare "," as a connective—I have seen it used to mean "and", "or", "if–then", and probably other things, and such ambiguity is at worst irresoluble, and at best resoluble with effort on the reader's part that you could save them. Instead, as @JairTaylor suggests, it is better (definitely than ",", and probably than "$\land$") explicitly to use the word "and". $\endgroup$
    – LSpice
    Nov 26, 2022 at 2:30
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    $\begingroup$ The goal is to communicate, and since your presumed audience is human, natural language is more appropriate. $\endgroup$
    – copper.hat
    Nov 26, 2022 at 2:56
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    $\begingroup$ I just want to point out that the quoted statement in natural language is already mathematical. Saying "the equivalent mathematical statement" to refer to the lines consisting of mathematical symbols alone is a bit of a disservice to mathematics as a way of communication. In fact, I'd go as far as saying that, since mathematics communication aims for clarity and precision, the quoted statement is "more mathematical" in the sense that it is the clearest of the three. $\endgroup$ Nov 26, 2022 at 3:19

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