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Let $X$ and $Y$ be arbitrary sets. Further, let $X'\subset X$ and $Y', Y''\subset Y$, where and $Y'\neq Y''$. I am currently stating (and proving) a theorem of the form \begin{gather} x\in X'\Leftrightarrow (y\in Y')\vee(y\in Y'') \end{gather}

I am, however, writing this theorem in English. Hence, I am currently writing \begin{gather} \textit{x belongs to X if and only if y belongs to Y' or YY''} \end{gather} Since the symbol $\vee$ is inclusive and the English "or" is exclusive, I do not know whether I should instead write \begin{gather} \textit{x belongs to X if and only if y belongs to Y' or YY'' or both} \end{gather}

So, when stating a formal theorem in English, should "or" keep its English exclusive meaning, or should it adopt the inclusive meaning of the symbol $\vee$?

EoDmnFOr3q
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    You will confuse everybody if you use "or" exclusively. If you mean exclusive-or, you should say so explicitly with "exactly one of the following is true" or words to that effect. If you feel squeamish, you could say "if and only if $y$ belongs to $Y'$ or $YY''$ (or both)" with brackets, but I would strongly expect "or" to be read inclusively. – Patrick Stevens May 17 '22 at 19:54
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    Previous discussion on this issue – angryavian May 17 '22 at 19:57
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    If it helps resolve the uncertainty, I think it's simply not true that "or" is always exclusive in English. Example, after dinner when dining out: "Would you like coffee or dessert?" – Greg Martin May 17 '22 at 19:57
  • Okay, thank you all for your comments. I got it. :) – EoDmnFOr3q May 17 '22 at 20:14
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    "should "or" keep its English exclusive meaning" I'm don't think that "either...or" is usually exclusionary: "if you've visited either Bangkok or Chiang Mai, then you've visited Thailand" doesn't feel to me like an unusual construction involving "either...or". – ryang May 18 '22 at 13:14

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