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This is a merely formal question. I'll explain with an example: say I want to denote the set of all the real numbers which have a reciprocal greater than $1$. I would write it like this:

$$S = \left\{x\in\Bbb R:\frac1x>1\right\}$$

The problem is, the statement

$$\frac1x>1$$

doesn't make sense when $x=0$, since $1/0$ doesn't have a value, thus it can't be compared to a real number. So, being completely formal, if I want my set to be well-defined I should write

$$S = \left\{x\in\Bbb R\setminus\{0\}:\frac1x>1\right\}$$

since, otherwise, the proposition $0\in S$ wouldn't really make sense. Is this correct?

Elvis
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2 Answers2

1

If your notation creates ambiguity, improve your notation to remove ambiguity.

That's for communicating with people. If you are writing formulas to feed to a computer program, consult the manual for that computer program.

JonathanZ
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-1

While only the second suggestion $$\left\{x\in\Bbb R{\setminus}\{0\}:\frac1x>1\right\}$$ is well-defined, I suppose that in most informal settings it is acceptable to write $$0\not\in\left\{x\in\Bbb R:\frac1x>1\right\}$$ and read it as

  • it is not the case that $0$ is a real number whose reciprocal exceeds $1.$

Of course, we could alternatively just write $$S=\left\{x\in\mathbb R\mid 0<x<1\right\}=(0,1).$$

ryang
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  • I would never write and never expect to see the first displayed statement in this answer. – Ethan Bolker Oct 07 '24 at 13:53
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    The statement is formally wrong, since the set builder part still asks for division by $0$. I meant the second expression. The point of the question is not whether $0$ has an inverse, it's how to write the set of reals whose inverse is greater than $1$. The first displayed expression in this answer is the answer. It's the second one I object to. It's formally wrong and useless. – Ethan Bolker Oct 07 '24 at 14:01
  • Yes, the OP's first suggestion isn't rigorously correct. – ryang Oct 07 '24 at 14:06
  • A chatroom discussion regarding the acceptability of the statement $0\not\in\left{x\in\Bbb R:\frac1x>1\right}.$ – ryang Oct 08 '24 at 09:04
  • To the downvoters who cannot tolerate the statement $0\not\in\left{x\in\Bbb R:\frac1x>1\right},\tag{}$ observe that the (arguably) well-defined statement $\forall x{\in}\mathbb R{\setminus}{0};\frac1x\ne5\tag{}$ merely abbreviates $\forall x{\in}\mathbb R;(x\ne0\implies\frac1x\ne5),\tag{}$ which asserts, in particular, that $0\ne0\implies\frac10\ne5.\tag{}$ – ryang Oct 14 '24 at 15:24