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Hey guys, need some help here. I was told by your instructor that there are three methods of proof: (1) direct, (2) by contraposition, and (3) indirect (by contradiction). There is also a technique of proof by mathematical induction. A counter-example is essentially a proof that a statement is false, but it requires existence; that is, a counter-example must be specific and name individuals and sets.

However, I am unsure about how to proceed. First few statements looks like it can be proved directly using U as a statement such as "a set of prime numbers from 0-10" and then we can say that P(x) as {2,3,5,7} but I dont know how to deal with Q(x).

What about statements (7) and (8)? those two are conditions or statements of continuity and uniform continuity which I studied in advanced calculus. Should those proofs be straight forward? I am little confused with those since the course is topology and its a question of topology.

Any help will be greatly appreciated. Thanks in advance!!

J. W. Tanner
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Math_Is_Fun
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    This is not general topology. It might be logic, or, proof-writting. – azif00 Jan 23 '20 at 03:35
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    Only (7) and (8) have any topology expressions in them, nonetheless there is no topological content in what you are asked to do, just logic and proof techniques. – Lee Mosher Jan 23 '20 at 03:37
  • hmm, I can prove statements 7 and 8 but I dont know how 2 do the rest, can you please help me in figuring out some of them. I am pretty sure that I will get most of them if I get an idea about how to solve them. I need idea for 1st few of those statements. Thanks in advance. – Math_Is_Fun Jan 23 '20 at 03:52
  • @Azif00: maybe, but the subject is topology. So, I thought I need to work in that way or direction. Anyway can you pls help me in figuring out the logic for the statements because I am not able to figure out those. – Math_Is_Fun Jan 23 '20 at 03:54
  • @Math_Is_Fun. You cannot prove 7 and 8 for discontinuous functions. – William Elliot Jan 23 '20 at 10:04
  • Yeah, because statement 7 is the condition for continuity and 8 is for uniform continuity. Hence, they definitely have to work for continuous functions. If a function is discontinuous, then it won’t work. Thanks for your help. – Math_Is_Fun Jan 23 '20 at 14:38

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The proposed statements have no topological content, even no interpretations for the names like $U$, $P$, and $f$. So their form provides a little help for their proofs.

If you skip this you will be sorry!

I’m working in topology for more than twenty years and I can assure you that if you follow this you’ll be sorry. This is because, as Nicholas Bourbaki wrote, “the mathematician does not work like a machine, nor as the workingman on a moving belt; we can not over-emphasize the fundamental role played in his research by a special intuition, which is not the popular sense-intuition, but rather a kind of direct divination (ahead of all reasoning) of the normal behavior, which he seems to have the right to expect of mathematical beings, with whom a long acquintance has made him as familiar as with the beings of the real world”. So in order to prove a theorem usually you have not to look for a specific sequence of logic formulas, but to understand its matter (math is fun, we do it because it is interesting and beautiful). Then often you’ll be able see a way to prove it. This is easy when you are solving exercises. But when your mathematical intuition is developed enough, sometimes you can solve open problems. A famous mathematician (and topologist) Henri Poincaré provides more advanced reflections of a work of this intuition in the beginning of “Mathematical Creation”.

Alex Ravsky
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