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If I am writing down this function $$ f: D^2\to \mathbb R $$ and want to communicate that $D\subseteq \mathbb R$, should I write $$ f: D^2\to \mathbb R \quad \mathrm {with} \quad D\subseteq \mathbb R$$ or $$ f: D^2\to \mathbb R \quad \mathrm {where} \quad D\subseteq \mathbb R$$ or $$ f: D^2\to \mathbb R \quad \mathrm {and} \quad D\subseteq \mathbb R$$?

Florian
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    No big difference for me... – Mauro ALLEGRANZA Aug 30 '22 at 13:06
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    I prefer either of the first two but that is a personal preference, not a rule of grammar. – John Douma Aug 30 '22 at 13:07
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    the last one I could perhaps see it in something like..."the theorem is true under the condition that $f:D^2 \to \Bbb R$ and $D\subset \Bbb R$". – Calvin Khor Aug 30 '22 at 13:09
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    My preference: "Consider the function $f(x)=ax^{bc}$, where $a\in\mathbb Z$ with $a>0$, $b\in\mathbb N$, and $c\in\mathbb Q^+$". "Where" is a first specification, "with" is a further specification of something that's already concrete, and "and" comes at the end of a list. – Vercassivelaunos Aug 30 '22 at 14:00
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  • Do note that option 2 does require a comma to precede "where". $\quad$ 2. Although I'm partial to options 2 and 3 (1 sounds a teeeeny bit contrived; 2 sounds natural; 3 goes straight to the logical point), all three options are clear/okay. $\quad$ 3. I agree with @Vercassivelaunos's illustration. $\quad$ 4. Two relevant previous answers of mine: Are "where" and "such that" interchangeable? and Ambiguity of "where".
    • – ryang Aug 30 '22 at 14:10