Logically answering an informally posed question Let the following situation be the truth:
"Each person has a unique passport number and every passport number is assigned to a person".
Then consider the following statement:
"Each passport number belongs to at most one person."
Is the following statement TRUE or FALSE?

I'm unsure what to answer about this, because I could answer as follows:


*

*FALSE, because implicitly it says that each passport number belongs to 0 (no person) or at most 1 person. And 0 (no person) would be false.

*TRUE, yes, because it's correct that the number of owners for a passport number is less than or equal to 1.



Exactly this question arised in a test of mine. The master solution says TRUE. But I'm convinced that both answers have their justification. Do you think that the question is ill-posed or do you think that with formal reasoning one can convince himself about some solution?
 A: If I take the numbers $2,4,6,$ and $8$, and I claim "Everyone one of these numbers is either even or odd", I am making a true claim, even if none of them are odd.
Same logic as with these passport numbers: If every passport number is assigned to exactly $1$ person, the claim that every number is assigned to either $0$ or $1$ person is still true.
A: Given the premise "Every passport number is assigned to a person, and each person receives a unique number," we may indeed infer that the number of people assigned to any passport number is exactly one.   So the claim that "each passport number is assigned to at most one person", would certainly be considered: true.   It is only a weaker claim; it does not contradict the given facts.
When given $x=1$ as a fact, you should agree that $x\leq 1$ .
A: As many problems logic problems whose formulation is done via terms of concrete real life this question has some ill-posed points. The ones which I noticed are the following two:
-what exactly is a "pass port number"? Does it mean just a number of an arbitrary subset of the natural numbers? Is this set finite or not? If it is finite how do we take in account newborns? If it is infinite then at each time we are clearly using only finitely many numbers of this subset so the first assertion does not make sense.
-What is the set of persons we are considering? Is it just a fixed group or is it subject to changes (people can die or be born)? If the former were true it could happen that you pass the number of a dead person to an infant to stay true to the claim about the passport numbers but you could have instances of numbers belonging to more people (in different times).
But I think that my observations are more philosophy that logic: I think that they want to rationalise the situation in the most simple way possible. That is we consider $P$ a finite set of people and $N$ a finite subset of the natural numbers, then the claim is that there is a function $p \colon P \rightarrow N$ assigning to a person his passport number. Moreover we have that $p$ is clearly a bijection and the statement is true by the reason Bram28 said: that every preimage of a number via $p$ is of cardinality at most one means that this set is void OR has only one element. Where the OR is a logical disjunction
https://en.wikipedia.org/wiki/Logical_disjunction.
So even if all the preimages have one elements it is fine.
