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Does the statement mean that each element of $S$ has exactly one colour, or that no two elements of $S$ share the same colour? Or could either interpretation be valid, depending on the context?

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What is $S$? What kind of representation are you talking about? –  Najib Idrissi Nov 30 '13 at 14:35
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Usually it will be "there is exactly one", and a careful writer will express the latter by saying "there is no more than one" or some such. But only context can tell you if you have a careful writer on your hands. :p –  Malice Vidrine Nov 30 '13 at 14:37
    
"no two elements of $S$ share the same representation" is generally implied by the concept of "representation" itself; since the representation cannot be equal to two different things at the same time (e.g. if you were representing positive rationals as ratios of two coprime integers, it's clear that the same representation $p/q$ can correspond to only one rational). As @MaliceVidrine pointed out, your quoted phrase usually means "there is exactly one representation"; otherwise it'd be "at most one" or possibly "the representation is unique, assuming it exists" or something such. –  Peter Košinár Nov 30 '13 at 14:58
    
@peter Thanks. I've changed the word "representation" to "colour" because as Peter points out, "representation" itself already restricts/specifies the intended meaning of the sentence. My question pertains to the proper/conventional usage of "unique". –  Ryan Nov 30 '13 at 16:01
    
@malice How about my second suggested interpretation? Do you think that technically, that is just as correct as the first, even if it is less common? –  Ryan Nov 30 '13 at 16:05

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