If some X's are Y's, does that imply that some X's are not Y's?
This is bordering on logic rather than language, but the answer is definitely no: Some is "an indeterminate amount", which means it can be all. If I say I have some red M&Ms in my bag, it could be that all of them are red.
But then, depending on inflection, as @codelegant pointed out, I could be using emphasis on some to indicate that not all are red...like if you asked for some red ones and I said I had some, which turned out to be more than one but less than all.
For straight English prose, yes. It implies more than one, but not all.
However, for a discrete math homework or test question, I think it would be synonymous with "one or more". It could be just one, or it could be all of them. So if you are asking this question to try to get a couple of points back from your math instructor, sorry. :-)
the answer is No.
Closely connected with the theory of opposition is that of the equipollence of propositions with the same terms in the same order but with negative particles variously placed within them. Since contraictories are true and false under reveresed conditions, any proposiltion may be equated with the simple denial of its contradictory. Thus, "Som X is not a Y" has the same logical force as "Not every X is a Y," and vconversely, "Every X is a Y" has the force of "Not (some X is not a Y)," or, to give it a more normal English expression, "Not any X is not a Y". Similarly, "Some X is a Y" has the force of "Not (no X is a Y)" and "No X is a Y" that of "Not (some X is a Y)" -- i.e. "Not any X is a Y." Also, since "no" conveys universality and negativeness at once, "No X is a Y" has the force of "Every X is not-a-Y", and, conversely, "Every X is a Y" has the force of "No X is not-a-Y." Writers with an interest in simplification have seen in these equivalences a means of dispensing with all but one of the signs "every", "Some", and "no." thus the four forms may all be expressed in terms of "every", as follows:
Every X is a Y (A)
Every X is not-a-Y (E)
Not every X is not-a-Y (I)
Not every X is a Y (O)
(emphasis in bold is mine)
Some suggests (but does not require) that there are counterexamples. The reason is that if you say "Some X's are Y's" instead of just saying "X's are Y's", you presumably had cause to use the extra word. The natural reason for you to do so is if saying "X's are Y's" was actually not true--some are not Y's, but some are. So "Some X's are Y's" is then true.
In conventional discourse, if you want to emphasize that you are using "some" because you are not sure (or do not wish to check) that the statement applies to all, then you can use a phrase like "at least some X's are Y's". This longer form no longer carries the implication that some X's are not Y's. (In logic and mathematics, the short form carries no implication; allowing oneself to not check every case is very useful in math and logic. Then again, the phrasing then is usually "there exists an X that is a Y".)
All known human languages make use of quantification.
'Some' is a form of quantification and when used in English it's understood that 'some' is 'not all'. In your case, "if some X's are Y's, it implies that some X's are not Y's".
In predicate logic this isn't necessarily the case. The existential quantifier, '∃', may be used to say that 'there exists' some relation or property. That 'there exists' can cover every relation or property though.
"Some X are Y" does not necessarily mean that "Some X are NOT Y", although it does leave that possibility. The only conclusion one can draw from the statement that "Some X are Y" is that group X and group Y overlap, with one or more of the population being both X and Y.
I think it muddles the issue to talk about what this would mean in conversation. Clearly, unless our name was Sheldon Cooper and we didn't have any friends, we would not be talking about "X" and "Y" unless to discuss genetics or to illustrate a principal of logic.