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Mine first:

Exactly one of the statements $A, B$ and $C$ are true iff:

  1. $A, B$ and $C$ are true for some condition. In other words, "they must happen for some reason".
  2. If any of $A, B$ and $C$ are true, then the rest are false (say, $A$ is true then $B, C$ should be false).

My puzzles come from this exercise, which asks me to prove that the rationals fall into exactly three categories (exactly one of the three statements about a rational are true):

Consider a rational $x$. Exactly one of the three statements are true: $x$ is $0, x$ is negative or $x$ is positive. Hint: you must prove two different things: firstly, at least one of the statements are true and secondly, at most one of the statements are true.

But it seems unreasonable. Say, let us consider this statement: Consider a natural number $n$. Exactly one of the three statement are true about $n$: $n \neq 0, n = 0$ and $n = \pi$. This quarky statement is true in terms of the above perspective, at least one of the statements are true and at most one of them are true since $n = \pi$ is never true!

My question: is the first or the latter definition of "exactly three of the statements are true" should I use?

Ricky
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MathematicsBeginner
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    Not following. Of those three statements regarding a natural number $n$, only one is true ($n\neq 0$). Nor do I understand your point $1$...it is certainly not part of the desired definition. – lulu Aug 31 '23 at 15:24
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    As to the claim you wish to interpret...it just means what it literally says. Of the three statements, one is true and the other two are false. If your three statements are indexed by some parameter $n$ then we do not require that the same statement be true for all $n$ (though it's fine if it is). Not sure if that's what you were asking though. – lulu Aug 31 '23 at 15:25
  • @lulu it just seems sort of "unbounded" since it can't be sure that "all cases in your statement" are used up. :/. Why would we want that in our definition. I understand the latter's persepective where we have n boxes, we throw "one true ball" into them and we're assured that one of them contains the ball, but we're not sure if a box ever contains the ball at all! – MathematicsBeginner Aug 31 '23 at 15:36
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    It doesn't matter if all the cases are "used up" or not. Logically it is perfectly possible that, say, statement $A(n)$ is true for all $n$ and $B(n), C(n)$ are both always false. – lulu Aug 31 '23 at 15:37
  • See tricotomy: "either x<0 or x=0 or x>0". – Mauro ALLEGRANZA Aug 31 '23 at 15:59

1 Answers1

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This one has been answered in the comments; I just want to move it off the unanswered queue:

Contrary to your expectations, "Exactly one of the following is true" does not mean that each of the relevant options must be possible in any sense. In particular, for any natural number $n$, exactly one of the three statements "$n=0$," "$n\not=0$," and "$n=\pi$" is true.

Mathematical statements are supposed to be unambiguous in a very strong sense; for this reason (at least), we tend in mathematical contexts to ignore many nuances and conventions of natural language. Saying "Either $\pi$ is an integer or $2+2=4$" would be weird in natural language, but it's completely unproblematic in mathematical writing. What we gain in exchange for this loss of convention is communicability: there's no (or at least vastly less) risk of different reasonable readers interpreting the same statement in different ways.

(As long as, of course, all readers are on the same page with regard to this convention-limiting metaconvention!)

For an extreme example of this trade-off, consider implication. Natural-language implication is notoriously (and very interestingly) complicated and nuanced. For a variety of reasons, the material implication ("$A$ implies $B$" meaning simply "Either $B$ is true or $A$ is false or both") has emerged as in most cases the best "default mathematical implementation" of implication, with the basic idea being that if you want to assert more (e.g. a causal connection between $A$ and $B$) you need to do the work of making that further assertion precise and explicit. Vast amounts of ink have been spilled on this issue; see e.g. the discussion here.

Noah Schweber
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