Validation of Answer via Truth Table as Reason in Isolation

Question

I've tried various google and math.SE search strings but I'm having trouble formulating a query that gives me relevant information.

Questions

• Does this table below accurately represent an acceptable answer to this question, (Drinking_Habits_Riddle), if the answer is stated with "100% of the people of said village drink some form of alcohol (assuming that the whiskey and gin are both alcoholic)"?
• Or would it strictly require a numeric symbol translation to be acceptable?
  • If so how would it be derived directly from the graph alone ?
  • Or could it be that any correct answer be derivable from the table itself but it still needed to be in a numeric format ?

Perhaps I could have said something along the lines of "As is evident by this graph which holds strictly to the correct ratios of Tea:Coffee:Whiskey:Gin that there are no gaps where any percent of people drink no alcohol in this village and no percent of people drink all four beverages. Therefore 100% of the population drinks some form of alcohol." But even if the previous is True I guess that's more a logic answer.. but would that also be unacceptable/irrelevant?

In a small village 90% of the people drink Tea, 80% Coffee, 70% Whiskey and 60% Gin. Nobody drinks all four beverages. What percentage of people of this village drinks alcohol?

Table:

     T C W G
 1   + + + -
 2   + + + -
 3   + + + -
 4   + + + -
 5   + + - +
 6   + + - +
 7   + + - +
 8   + - + +
 9   + - + +
10   - + + +    


 where T == Tea
       C == Coffee
       W == Whiskey
       G == Gin

Extra if helpful

I was excited when I saw this question on stackoverflow as I'm better at understanding patterns and ratios then using symbolic notation manipulation. The table was part of my first answer on math.SE, but got several down-votes. I had added that "it seems any multiples of 10 are true as well" but didn't give proof of said statement. I admit perhaps my choice of words in several places seem to suggest I wasn't being purely mathematical, id est, that it could have seen to have been just a guess. I was told to delete it for it wasn't mathematics. I've since deleted the post after understanding that I didn't show proof of the 'multiples' but I never got an answer from anyone who was against my answer if the (table and accompanying statement) themselves, by themselves, was unacceptable. In either event if I happen to answer anymore questions I'll be sure to only state things I can prove to be true.

For a formal education setting I only have a GED so I apologize if this isn't my place. I have a rough time at judging delimitations of acceptance, and in choosing tone of formal:casual:readable.

Lastly, if this isn't the site for this sort of question please let me know. Maybe it could belong on meta.SE, Quora, or other? As this site is for mathematics I assumed asking here would be acceptable.

In advance thank you for any input

Answer 1

Your table can be part of an answer -- namely, it shows that it is possible for everyone to be alcohol drinkers, given the provided information.

However, for a full answer you would also need some kind of argument that this is the only possible situation. (In some situations you might get away with claiming that whoever asked the question wouldn't have asked it that way unless they knew there was a unique result, so you don't need to be sure that is the case. But this is not usually considered satisfactory as a mathematical treatment of the question).

What you need for this other part is not -- as you seem to assume -- more numbers and symbols, but actual words that explain how the reader would convince themselves there is not any solution with less than 100% alcohol drinkers.

(Some people somehow get the idea that the fewer actual English words they use in their writing, the more impeccably mathematical will it be. They then leave out all of the explanations and produce just an impenetrable mess of formulas. This is exactly wrong).

Answer 2

I commend you on asking this question. Let me say that it is always hard to explain why some purported writing/solution is wrong mathematics, because essentially the core reason is simply that mathematics is about purely logical reasoning that starts from accepted assumptions. Unlike science, to claim that a mathematical theorem is true, one needs to provide a proof (or at least a proof sketch with enough detail so that experts can be fully convinced that the formal proof can be constructed). There is no room for intuition in a proof. But I will try to give a specific and complete explanation for this particular example that you have brought up.

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Firstly, playing around with examples, diagrams, tables and so on is a good thing. It helps one to get a feel for what the problem is like, observe patterns, come up with conjectures and so on. However, always remember that every single illustration of any sort remains a single instance.

If the claim is that some mathematical object has certain properties, and you can give an illustration that provides complete information about one such object, and the viewer can deterministically check that the object you have illustrated indeed has the claimed properties, then your illustration can be considered to be sufficient to establish the result. For example, if I claim that there is a planar graph where every vertex has exactly $5$ neighbours, it suffices to describe just one such graph, and it is acceptable and convenient to do so by drawing it (such as this diagram).

If, however, the claim has a universal quantifier in it, namely that it says something about a whole collection of mathematical objects, then a single illustration can never function as a proof per se. Just for example, if I claim that every planar graph has at least one vertex with less than 6 neighbours, then no matter how many illustrations I give, it does not constitute anything close to a proof. It is possible that by judicious or ingenious diagrams one can convey to another mathematician some key ideas that underlie a proper proof, but that is different from providing a proof.

I want to emphasize what I just said. The four-colour theorem is notorious for attracting fake proofs as well as incorrect proofs. Two well-known incorrect proofs (one by Kempe and another by Tait) were both based on diagrams and intuition, and each was thought by the mathematical community to be correct for a decade. See this article for a detailed explanation of these incorrect proofs, which I just found via Google.

In your case, the problem is to show that every collection of people whose drinking habits satisfy the given constraints must all drink whisky or gin. You did not at all show that; the table can only show that one collection of people whose drinking habits satisfy the given constraints do indeed all drink whisky or gin. At the most, the table can only represent certain kinds of collections of people whose drinking habits satisfy the given constraints. The table does not (and cannot) show that the illustrated kind of collection covers all the possible kinds that satisfy the given constraints.

This logical gap is not fixable, because you would have to show (roughly speaking) that no other kind of row can occur in the table, which is literally the same as showing that every person drinks $3$ of the drinks. And to prove that, the easiest ways are still one of the other posted methods, which means that the table becomes useless for the purpose of the final proof.

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Secondly, your attempted explanation:

As is evident by this graph which holds strictly to the correct ratios of Tea:Coffee:Whiskey:Gin that there are no gaps where any percent of people drink no alcohol in this village and no percent of people drink all four beverages. Therefore 100% of the population drinks some form of alcohol.

is wrong for the reason given above. It is only evident that the kind of collection of people that is illustrated by your table all drink whiskey or gin. It is not evident that there are no other kinds.