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Identify whether true or false:

If $f(x)=(x-1)\{x\}$, where $\{x\}$ denotes the fractional part of $x$, the limit of $f(x)$ does not exist at all integers.

This is how I went about it:

The limit of $f(x)$ exists for $x=1$. However, it doesn't exist for any other integers. So, it doesn't exist for $2$, doesn't exist for $3$, etc.

Hence, the limit doesn't exist for all integers. The statement is true.

Here's how my teacher went about it:

The limit of $f(x)$ exists for $x=1$. However, it doesn't exist for any other integers. Hence, it exists for a single integer, i.e. not all integers. Hence, the statement is false.

Which interpretation is correct and which one is wrong (as there can only be one answer)?

Parcly Taxel
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Abhigyan
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2 Answers2

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The difference between the two possible interpretations is whether the $\forall$ quantification ought to be included in the negation or not. And indeed it is very different to say “not everywhere does this happen” than to say “everywhere, this doesn’t happen”! Without any additional information, both interpretations of the wording of the statement are valid.

However, you should know that it is very common practice (though not a very good one) in almost all areas of mathematics to push quantifiers after the predicate: e.g. $$\cos2x = \sin^2x - \cos^2 x, \qquad \forall x \in \mathbb R. $$ This is because in spoken language the correct order of the logical elements is reversed. But everyone knows that this should be understood as $$\forall x \in \mathbb R\quad \cos2x = \sin^2x - \cos^2 x.$$ So the instructions, whenever you see a quantifier where it shouldn’t be (after the predicate), would be to automatically pull it back where it belongs. In your specific situation, this means that “the limit does not exist | at all integers” should be reworded as “at all integers | the limit does not exist.” This is your teacher’s interpretation.

Another point in favor of your teacher’s version is that your interpretation should be formalized as: $$\neg (\forall x\in\mathbb Z \quad \text{limit exists at }x ),$$ which is equivalent to saying $$\exists x\in\mathbb Z \quad \neg(\text{limit exists at }x),$$ or, in common parlance, “there is an integer $x$ such that the limit does not exist at $x$.” Because of this “switch” property of quantifiers (“not everywhere does this happen” means “somewhere, this doesn’t happen,” i.e. $\neg\forall = \exists\neg$, and viceversa) it is more common to push the negation through the quantifier and have its complement modify the negation. All in all, if your interpretation was the one intended by the author of the exercise, then they would likely have written “the limit does not exist for some integer,” or more precisely “for some integer, the limit does not exist.”

giobrach
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    I like this answer, but I think it could be improved by not having a comma after the quantifier in the "this should be understood as" version. I don't think that punctuation is often used in serious logic texts, and it tends to give students the impression that the quantifier is still just a shorthand for English words, and/or that symbolic formulas should be punctuated internally according to English grammar. – hmakholm left over Monica Jul 12 '18 at 12:42
  • "So the instructions, whenever you see a quantifier where it shouldn’t be (after the predicate), would be to automatically pull it back where it belongs." <- Can I use this everywhere without any exceptions? You say it's a common but not a very good practice – Abhigyan Jul 12 '18 at 12:50
  • @HenningMakholm That’s why I didn’t include commas in the formal logical expressions below! But I edited the answer with your suggestion. – giobrach Jul 12 '18 at 12:56
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    @AbhigyanC If the question is “should I move stray quantifiers where they belong,” i.e. “should I strive for a precise formalization of a statement,” then absolutely. Of course where to put it depends heavily on the context (like how many quantifiers there are). Take a look at the link that was posted in the comments, and at famous instances of “delicate” quantifier placement, like the difference between pointwise and uniform convergence of a sequence of functions, or simple and uniform continuity. – giobrach Jul 12 '18 at 12:59
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Letting $P(x)$ be "$\lim f(x)$ at $x$ exists", the teacher's interpretation is $$\forall x\in\mathbb Z\ \neg P(x)$$ while your interpretation is $$\neg\forall x\in\mathbb Z\ P(x)$$ However, the given emphasis on at all integers makes it more likely that the teacher's interpretation is intended. If not, the question would have been reworded.

Parcly Taxel
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