My teacher wants me to write solution sets like: solution set $= \{x:x \in \mathbb{ℝ}, \}$, but he didn't go over how to write a set that contains all reals and how to write a set that is empty.
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1I'm confused, what's wrong with ${x:x\in\Bbb R}$? Is it not the set of all the reals? – Asaf Karagila Sep 15 '20 at 11:39
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I'm not sure because I just started set notation a few days ago – Astin Lu Sep 15 '20 at 11:42
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So does that mean { : ∈ ℝ} already means that the solutions are all real numbers? – Astin Lu Sep 15 '20 at 11:43
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It's the collection of exactly all the $x$ such that $x\in\Bbb R$, which turns out is exactly $\Bbb R$. – Asaf Karagila Sep 15 '20 at 11:44
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You write the set that contains all reals as $\Bbb R$.
You write the empty set as $\emptyset$.
If your teacher insists on writing every set in the way that you say (although I can't think why they would want this), then you can write any set $A$ as
$$\{x:x\in A\}$$
In particular, you can take $A=\Bbb R$ or $A=\emptyset$ here.
TonyK
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What about:
$\mathbb R= \{x \in \mathbb R \mid x =x \}$
$\emptyset= \{x \in \mathbb R \mid x \neq x \}$
mathcounterexamples.net
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