Let $X=\{ x\in X: x \text{ satisfies condition } A, x \text{ satisfies condition } B\}$. Does the comma in above set notation mean that $x$ has to statisfy $A$ and $B$ at the same time or only one of those; so that we have: $X$ is the union of $X_A=\{ x \text{ satisfies} A\}$ and $X_B=\{ x \text{ satisfies} B\}$?
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1Usually it must satisfy both. – Lonidard Dec 06 '15 at 17:36
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1It's used in place of and. – egreg Dec 06 '15 at 17:40
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Once you have given a name to your object $x,$ the notation is reserved to this $x$. So you have to read "$X$ is the set of elements $x$ such that this $x$ satisfy condition $A$ and this $x$ satisfy condition $B.$ For example, $$X=\{n\in\mathbb{N}\,:\,n\text{ is even}, n\text{ is odd}\}=\varnothing.$$
Balloon
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Why did you explicitly write that the empty set is part of that set? – John DeBord Dec 31 '21 at 23:25
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1I gave an example of a set of the form described in the question, and remarked that the set was actually empty. – Balloon Dec 31 '21 at 23:30