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I have an algorithm (or method) that attempts to find a solution ($x$) for a problem, and the existence of such solution is described by conditions in the following form:

Sufficient condition for $x$ to exist: $$ \text{Largest}[y_{i}]\leq x \leq \text{Smallest}[z_{j}] $$ Necessary condition for $x$ to exist: $$ \text{Smallest}[y_{i}]\leq x \leq \text{Largest}[z_{j}] $$ where $i$ and $j$ are integer indices over two sets, $\{y\}$ and $\{z\}$, of given real numbers of the same domain as $x$.

However, the sufficient condition stated above is impossible to meet in practice. What does it mean to have the sufficient condition impossible to meet and only the necessary (weaker) condition possible to meet? Does such cases arise normally in mathematical problems?

I am not sure about this, but it seems like a way to say that one has no guarantee of finding a solution and will have to try brute force search using some search algorithm (method), if only the necessary condition is met. Does that make sense?


UPDATE: Further, would it be possible that another sufficient condition may exist that is yet undiscovered? Is it possible to have more than one sufficient condition? And if so, what would it mean for one of them to fail or hold? For example, if we have 2 sufficient conditions and one of them failed, would the second necessarily/automatically fail; and if one held, would the other one necessarily/automatically hold too? I think if one sufficient condition is held then all others should hold, but not sure about the case if one is failed and whether more than one sufficient conditions is allowed in the first place. Or is it that if more than one sufficient conditions existed then they themselves would be related together depending on their respective strenght by other sufficient/necessary relations among themselves?

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  • $\begingroup$ To prove dinosaurs were warm blooded it is sufficient to build a time machine and be very large thermometer. However that is not necessary.... Or to prove that pi is irrational it would be sufficient to compare pi with every ratio of integers and show that none of them equal. Or to show something is fireproof by federal standards it is sufficient to launch it into the sun and see if it survives. etc. $\endgroup$ – fleablood Jun 14 '18 at 17:12
  • $\begingroup$ Sufficient means "enough" (and possibly more). To use the word sufficient it usually implies that we have something that is more than enough but fairly simply to do. But it doesn't have to mean fairly simple to do; Just "enough and probably more". So it's logically correct to say "it's sufficient to kill a fly with a bazooka" its a weird thing to say because supposedly a bazooka is not easy. However there are often ironic cases were a broad case is easier than a narrow case and it is those that word "sufficient" was coined for. $\endgroup$ – fleablood Jun 14 '18 at 17:19
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Yes, you're interpretation is correct, it does say that one has no guarantee of finding a solution and will have to try brute force search amongst things that satisfy the necessary condition.

I would say that these sorts of things are not common in mathematics, simply because if a sufficient condition is impossible to meet in practice, it is not very useful, and therefore people won't publish/study/look for such sufficient conditions.

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  • $\begingroup$ Many thanks. And would it be possible that another sufficient condition may exist that is yet undiscovered? Is it possible to have more than one sufficient condition? And if so, what would it mean for one of them to fail or hold? For example, if we have 2 sufficient conditions and one of them failed, would the second necessarily fail; and if one held, would the other one necessarily hold too? Does this make sense? (or if more than one existed then they themselves would be related together depending on their respective strenght by other sufficient/necessary relations among themselves?) $\endgroup$ – user135626 Jun 14 '18 at 14:07
  • $\begingroup$ Yes, there could be another sufficient condition that hasn't been discovered, there are problems with multiple sufficient conditions (e.g., travelling salesman problem). Information about one sufficient condition doesn't give you information about another; it is possible the one sufficient condition holds and another fails. As a silly example, note that two sufficient conditions for a number $x$ to be even are that $x = 2$ or $x=4$. Clearly, for $x=2$ only one of these holds. $\endgroup$ – user428487 Jun 15 '18 at 0:48

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