Why Is It That Proof by Contradiction Is Considered a Weak Proof in Mathematics?

Not providing a constructive proof in the case of proof of existence is a valid point, but from my experience knowing that something is possible is more than 50% of the battle of actually doing it. Sometimes I follow that rationale in my proofs too "ok, I'll assume it is true; now let's prove it!".

Conversely, working on a proof by contradiction can help you realize that what you are trying to prove is wrong -- as (if you are careful with your proof) you will end up hitting many dead-ends. I have found better solutions to a problem by trying to prove that my old solution was the best there was, by contradiction :-)

So I have a soft spot for contradiction proofs. Not that any other kind of proof is inferior as I said above.

• Related Questions

For context, Beal's conjecture is that there are no integer solutions to a^x b^y c^z for x, y, z geq 3 positive integers and a, b, c sharing no common factor.It's a vast generalization of Fermat's Last Theorem, which took hundreds of years and the invention of several new areas of mathematics to resolve. Consequently the proof has to be at least as long as the proof of that (assuming our current proof is a good one). Moreover, Beal's conjecture adds two more (non-algebraic!) degrees of freedom and can't directly be handled by any existing techniques, so it would likely require the invention and development of entirely new areas of mathematics.That's assuming it's true. If it's not true, a counterexample might be found by computer search, or someone might spot how to construct a very large counterexample that a brute-force search would never find. If it's false, it could be resolved tomorrow for all we know.

How likely is it that someone will solve BEAL'S CONJECTURE and get the $1million?

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Did you ever attend school with a celebrity?

You probably might not know him, but he was on Ghost Hunters. His name is Dave Tango and we went to middle school together. He was my best friend and pretty much the only friend I really had and trusted in school. A few years ago, he got in touch with me. He messaged me to see if we could meet up, but I was living in another country at the time.He had informed me he was working for the show Ghost Hunters. Now Im sure he expected me to know, but I had moved out of state for several years, and then out of the country, so I hadnt been watching television at all. So it was news to me and was pretty awesome to learn about it. When I moved back home to the United States, I got a job and was working for a company, who then decided to move me across country, so I never got a chance to meet up with him.

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What do you think pushed Roseanne Barr to say those racist comments? Do you think ABC overreacted when it cancelled the whole show?

Original Question: What do you think pushed Roseanne Barr to say those racist comments? Do you think ABC overreacted when it cancelled the whole show?No Original Question DetailsApparently, William Harold Hinkson II is not as cynical as I am.Most of his answer is very good, but he fails to draw the appropriate conclusion.Roseanne Barr is a very smart woman.Her tweet was intentional, and she knew the effect it would have.

In short, she wanted to blow up the show.

To be honest, I hadnt been following the show; I never watched it. So I did not have a motive.William supplied the motive.Everyone was getting sick of her shit and she wanted to retaliate.And she knew that a racist tweet would get the show cancelled and would not bother her core fans in the slightest.

In fact, it would give the Trumpanzees an opportunity to engage in their favorite pastime: playing the victim card.The tweet was an act of deliberate sabotage by an brilliant, but bitter celebrity

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What should you never say to a UPSC aspirant?

What are you doing? How long since you finished your college? (most irritating question from family members and neighborhood).Comparing aspirants with private working professionals. (Wow! Dear, let them be a rubber stamp to their boss).

How many attempts you have given? (Dear, UPSC itself knows that it's difficult and challenging. Thats why given 6 attempts at max. Give me my time for 2nd attempt. All are not same dear).Why can't you wind up this field and concentrate on your graduated field? (Why can't you mind your own business, dear?).You should not waste time on this type of uncertainy filled exams. (My dear, even if you can't keep me motivated, atleast keep you mouth shut dear).Finish this type of exams and get govt. job and marriage alliances will be waiting for you. (My dear, why you have suddenly become a marriage broker?).

This type of negativities and venom are spread easily from society towards aspirants. But my family (particularly my elder sister), keeps me motivated to give 2nd attempt this year

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In what ways was Franklin D. Roosevelt considered a good president?

Well, he was elected 4 times, so he had something going for him Since we then forbad any president from being elected more than twice, that may hint at bad for us. But great? One joke from the later 30s was.. The nation is into two parts, those saved by Roosevelt, and those ruined by Roosevelt. Those saved by Roosevelt still live in the slums at the bottom of the hill, and those ruined still live in the mansions at the top of the hill.Those who insist FDR got us out of the Great Depression ignore that the Depression was still going on after 8 years of his efforts, and when the Republicans repealed many of his program after the war, his fans were stunned when the Depression didnt resume. The programs retained are often at the top of the list of those causing us grief.

What he did was found the liberal state, a great achievement in the eyes of liberals, in particular, those who write history books and rank presidents. But your dinner bill is higher because of him

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Why do people believe what Trump says?

That very question has been the most asked, is the simplest with the most verifiable answers.And yet we're never tired asking again and again and again.To be fair, it seems Donald Trump is COMPLETELY oblivious of his predicament, that he lies to everyone about everything.

And if you're still not sure about what everyone else knows - that Donald Trump is a serial liar - watch for catch phrases like "believe me", "trust me", "others say ...", "others believe...", "I heard ..." which are often used by those who lie frequently.According to experts, liars use those words mostly to convince themselves instead of their audiences. Mr. Trump leaves a trail of proof to that effect. He constantly argues to deny statements he made, sometime as close as a few minutes prior.Why would anyone believe Donald Trump?Even his hardcore fanatics KNOW he only speaks lies. They do not believe him; they simply like him for reason sociology will study for a very longtime.No one in the right frame of mind should believe a word coming out of Donald Trump mouth

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Which top 10 B schools should I target if I can't get into IIMs?

There are several Business Management Institutions in India where you can get in-depth knowledge about business management. Such institutions will help you to enhance your skills & understanding in the field of management.We heard a lot about IIM (Indian Institute of Management) and its management courses. Every business student is willing to get admission in IIM but few eligible students get it. The admission of candidate in IIM is based on the CAT (Common Admission Test) which is reputed all India basis entrance test. To crack CAT examination is little bit tough. But there are some other B Schools where you can get in-depth knowledge. Here are Top 10 Business Institutions after IIM.Faculty of Management Studies (Delhi University)Management Development Institute (Gurugram)Xavier Labour Relations Institute (XLRI)S.

P. Jain Institute of Management & Research (Mumbai)Indian Institute of Foreign Trade (IIFT), DelhiFore School Management (New Delhi)Lal Bahadur Shastri Institute of Management (New Delhi)National Institute of Bank Management (NIBM)Asian Business SchoolKirloskar Institute of Advance Management StudiesCheck out all the Top 10 Business Schools

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Wesley Sneijder believes he could have been mentioned in the same breath as Lionel Messi and Cristiano Ronaldo if he was more committed. What are your thoughts on this?

There have been many players who could've achieved what Messi and Ronaldo have. There have been players with lot more talent than these two and there will be players with talent too. Literally anyone from Brazil(Ronaldo R9, Ronaldinho and many more) can say this. But, ultimately, it comes down to the simple fact that talent will only get you so far in life. You need to have the right mentality to work for it so that you can go above and beyond others. As far as Sneijder's comments are concerned, he could've done it. But, so could've a lot of others. You need to work your a** off and sacrifice if you want to be mentioned in the list of all time greats. Both of these players have done something which may not be repeated again. To share the throne of world football for a decade is not something to be taken lightly especially in a sport which places such high physical and mental demands. Hats off to the GOATs.

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Were you surprised to find the Washington Post-ABC News poll states 63% of those surveyed say they support the Mueller probe?

Not really. A couple of months ago, there were polls showing that Muellers probe was rather unpopular (Analysis | As support for probe wanes, most Americans still want Mueller to take his time).Polling, in America today, is bordering on worthless. Secretary Clinton was going to win in 2016, 90% chance. Andrew Gillum trailed in every single poll of Florida Democratic primary candidates for Governor. Everyone hates Trump.Polling comes down to:How the questions are worded (often subtly biased)Who is asked (vast swarths of the population are unreachable)Whether people answer honestly (if you had an unpopular opinion, would you admit to it to a random stranger on the phone?)I guarantee you that Fox News, National Review or anyone else opposed to Robert Mueller could, today, produce a poll that said most people dont support him and want the investigation shut down. The fact that the Washington Post and ABC News, both of whom profit from the ongoing investigation, ran a poll that said the opposite means nothing.

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We have an ABC triangle, AB edge equals to 6, BC edge equals to 8 and ABC angle equals to 60. What is the probability of a point inside this triangle to be nearer to A than it is to B?

draw triangle ABC, C above line AB. Points equally distance from A and B lie on the line bisecting and normal to line AB (rising from mid-point between A and B). We want the two areas that the vertical line formed by cutting through triangle ABC. Let us label the intersection of bisecting vertical line with line AB with letter D, and the intersection of vertical line with line BC with letter E. Similarly the intersection of vertical line dropping from C to line AB with letter F. The triangle DBE has length DB 3, BE 6, and DE 3 sqrt(3), from sin and cosine of 60 degree angle DBE.The triangle FBC has sides BC 8, FB 4 and FC 4 sqrt(3), from sin and cosine of 60 degree angle ABC.

we now know area of triangle AFE because AF 2 and FC 4 sqrt(3)we also know area of triangle FBC because FB 4 and FC 4 sqrt(3)we also know area of triangle DBC because DB 3 and DE 3 sqrt(3)total area ABC 0.5*6*4*sqrt(3) 12*sqrt(3)area DBE0.5*3*3sqrt(3) 4.5*sqrt(3)area to left of riser DE: (124.5)*sqrt(3)ratio of areas: 7.5/12 15/24 0.625fred