Written in English, does "the set S contains only members of set T" imply that S does contain some member of set T?
How would this relationship be written formally?
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2 Answers
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My view is that vernacular would consider that S is not empty, i.e. $\emptyset \neq S\subseteq T$, while mathematical language would consider that S can be empty, i.e. $\emptyset\subseteq S\subseteq T$.
Lay people do not speak of empty sets, while mathematicians are aware of their role in their work.
That means that the sentence is not ambiguous, but the meaning depends on the community where it is used.
Of couse, people aware of the two readings can play games with it. But that is yet something else.
You may notice that I did not specifically mention English, because the same is probably true of many "languages".
To think that English (or German, or French) is a well defined language is unrealistic. Actually, I doubt you can find two people who agree on what is English and the meaning, or on the ambiguity, of its sentences,
It might be interesting to ask the linguists.
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$\begingroup$ So, would you say that in the context of the informal description of a Turing machine it would be safe to read the statement as ∅⊆S⊆T? Or is that actually far enough away from mathematical language that you would read it the other way? $\endgroup$ Apr 4, 2015 at 16:24
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$\begingroup$ I would compare the case you describe with that of a person, with mother language A, speaking a foreign langage B, that uses the same words with different meaning. For example the meaning of "actual" in English is close to "real" for most people, but "actuellement" in French means "current". You may have the problem of someone not aware, or forgetting meaning variations, or trying to adapt to the meaning of the listener. More a matter of who is speaking in what context, not informality itself. Turing Machine talk is mathematical, while easter eggs talk is unlikely to be (imho). $\endgroup$– babouApr 6, 2015 at 11:40
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It's ambiguous whether "contains only elements of $T$" implies that it must contain at least one element. Formally, you could write $\emptyset\subset S\subseteq T$ or $\emptyset\neq S\subseteq T$ or say "the non-empty set $S$ contains only elements of $T$."
answered Apr 3, 2015 at 21:42