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"Every boy has a unique shirt."

Does this mean no two boys share the same shirt, or does it mean no two shirts belong to the same boy?

I suppose the former, but then what is the most succinct way that you would rephrase the latter sentence using the word "unique"?

Is the answer: "Every shirt belongs to a unique boy" ??

I hope this question isn't too silly or trivial.

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I would interpret the sentence in the second way. But in natural language it can mean either one. –  Grumpy Parsnip Aug 4 '12 at 10:49
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It could as well mean "Every boy has at least one shirt that no other boy has". That's really the difference between mathematical/logic formulations and everyday speech. The latter is often ambiguous! –  Dario Aug 4 '12 at 10:49
    
@Jim Actually my first instinct is to interpret (and write) the second way too, but I keep wavering between the two. I've always generally felt that the above usage was vague (just like I avoid using the word "any" in mathematics), but after encountering so many instances of mathematical texts using the word "unique" in this ambiguous way, I was starting to question my own sanity (and linguistic ability)! haha –  Ryan Aug 4 '12 at 11:06
    
If I meant that no shirt belongs to two boys, I would say "every boy has a distinct shirt". –  MJD Aug 4 '12 at 11:21

1 Answer 1

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Closely approximating the English is the following logical formula $$\forall b \exists!s P(s,b)$$ where $b$ is a boy and $s$ is a shirt, and $P(s,b)$ means that s belongs to b. This means that for each boy there is one and only one shirt that belongs to him. If you want to say that no shirt belongs to two boys you would say $$\forall s\exists! b P(s,b),$$ and the natural language approximation would be "Every shirt belongs to a unique boy."

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By the way this last sentence is the same as yours, but we are taking them to mean different things. –  Grumpy Parsnip Aug 4 '12 at 11:05
    
In natural language, how would you use the word "unique" to write that "Every linear transformation has exactly one matrix representation?" (A matrix may represent an infinite number of linear transformations, but leave this information or its implications out.) –  Ryan Aug 4 '12 at 11:10
    
You might say something like "Every linear transformation is represented by a unique matrix." (Even though this is false. :)) –  Grumpy Parsnip Aug 4 '12 at 11:20
    
Yes, my first instinct was to write that too. ("Every LT, with respect to chosen bases in the domain and codomain, has a unique matrix representation.") Then I wavered and decided that the reader would as well interpret that to mean no two LT (with respect to the same bases) share the same representation, which is patently false! –  Ryan Aug 4 '12 at 13:06

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