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  • $\begingroup$ This is a bit of a delayed follow-up, but how is using Reduce any more conclusive than using FindInstance? Wouldn't the output of Reduce be "False" if and only if the output to FindInstance was {}? $\endgroup$ Commented Feb 23, 2015 at 17:50
  • $\begingroup$ @Shane The answer could be realized if you compare how Solve and Reduce are different. I recommend reading e.g. What is the difference between Reduce and Solve? FindInstance returns results in terms of replacement rules. $\endgroup$ Commented Feb 23, 2015 at 18:22
  • $\begingroup$ @Artes : You wrote: "If FindInstance[ expr, vars, dom] returns no instances - {}, it does not mean (in general) there are no solutions, i.e. it does not prove anything here." -- What is the basis for this opinion? In fact, it contradicts the official Mathematica documentation. Go to reference.wolfram.com/language/ref/FindInstance.html, click there on "Details and Options", and you will see this: "FindInstance[expr,{x1,x2,…}] gives results in the same form as Solve: {{x1->val1,x2->val2,…}} if an instance exists, and {} if it does not [emphasis mine]." $\endgroup$ Commented Jan 27, 2023 at 21:05
  • $\begingroup$ I would say something analogous with respect to Solve. ` Reduce` is more powerful. Nevertheless since then Solve became smarter. The main disadvantage is the output in terms of rules. This issue has been discussed more thoroughly here What is the difference between Reduce and Solve? $\endgroup$ Commented Jan 28, 2023 at 0:48