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Let $A\subset\mathbb{R}$ be an upper bounded set. Then

$$\forall\varepsilon>0~\exists x\in A\text{ such that }\sup{A}-\varepsilon< x \leq \sup A$$ I want to negate that statement. Would it be: $$ \exists \varepsilon>0~\forall x\in A\text{ such that } \sup A-\varepsilon\geq x\text{ or }x>\sup A~~?$$

Fabrizio G
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    In the most synthetic form I believe this is what you are looking for: $$x \in A: \sup A-\epsilon \ge x\Rightarrow \epsilon \gt 0$$ – WindSoul Mar 24 '23 at 18:34
  • @WindSoul I think this needs a quantifier to bind $x$ . – mcd Mar 24 '23 at 18:40

2 Answers2

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The logic is essentially correct, but it is oddly phrased: I suggest $$ \exists \varepsilon>0\text{ such that }\forall x\in A \text{ either }x \leq \sup A-\varepsilon \text{ or }x>\sup A~.$$ However, you don't really need the 'or', as the elements of $A$ cannot be larger than $\sup A$, hence the negation can be written simply as $$ \exists \varepsilon>0\text{ such that }\forall x\in A \; x \leq \sup A-\varepsilon~.$$

mcd
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  • as an element of A, x can’t be greater than sup A. – WindSoul Mar 24 '23 at 18:24
  • Technically, the second statement is mathematically (though not logically) equivalent to the first statement, which is the actual negation. – ryang Mar 24 '23 at 18:30
  • It makes no difference mathematically, as ryang phrased it, but "either ... or" implies exclusive disjunction which is not how conjunction is negated. I prefer OP's phrasing. – Ennar Mar 24 '23 at 18:34
  • @Ennar Surely in mathematics (and logic) "or" is usually taken to be inclusive unless the conjunction is explicitly excluded? – mcd Mar 24 '23 at 18:42
  • Yes, that's correct, but you didn't write "or", you wrote "either ... or". – Ennar Mar 24 '23 at 18:43
  • I wouldn't expect the addition of "either" to modify the meaning of "or" (though I agree it does in 'ordinary' English). – mcd Mar 24 '23 at 18:47
  • @Ennar Adding 'either' does not necessarily make 'or' exclusive: natural-language example and example in mathematics. It frequently does in natural-language, but in mathematics, I would just treat "(either)...or..." as inclusive unless accompanied by something like "but not both". – ryang Mar 24 '23 at 18:55
  • @ryang the linked example (in mathematics) is still a sentence in natural English, even though it refers to mathematics, and one often needs to rely on context to deduce the meaning correctly, but these are sentences in predicate logic (well, mixed with English, but still...). Surely one needs to make the distinction stricter since clearly we do refer to mathematical logic here. – Ennar Mar 24 '23 at 19:00
  • [correction to my last sentence: ... or if the context demands the XOR interpretation.] @Ennar But mathematics is written in natural language, as are the Question and Answer on this page. My point is that there is not really a convention in mathematics that "either..or.." means XOR, and that it is best to be explicit if the XOR interpretation is intended. (Teaching otherwise can lead to confusion when the person encounters instances of "either..or..." that aren't intended as XOR, due to the authors not following your wishful rule, heh.) – ryang Mar 24 '23 at 19:15
  • @ryang, but in this question we don't have natural language. We clearly have an exercise in how to negate quantifiers and logical connectives. In that sense, I don't think what mcd suggests is any improvement on what OP wrote. The second part of the answer is a valuable addition, but a short answer is simply: "Yes, that's correct." – Ennar Mar 24 '23 at 19:17
  • @Ennar The "such that" and "either" bits are in natural language; an actual formalisation OTOH would be $$ \exists \varepsilon{>}0\quad \forall x{\in} A\quad (\sup A-\varepsilon\geq x\quad \text{or}\quad x>\sup A),$$ where unnecessary punctuation (the colons/commas/periods placed immediately after quantifications) and "such that"s are omitted. $\quad$ One quibble with the OP's answer is their wrong placement of "such that", which mcd has corrected. – ryang Mar 24 '23 at 19:25
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$$(\forall) x \in A: \sup A-\epsilon \ge x\Rightarrow \epsilon \gt 0$$

WindSoul
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