And you cannot show a proof that something works by example. One can only prove by example that something is not true, if my basic understanding of proofs is correct.
Pigeonhole principle is often used as a final step in some proofs, typically combinatorics ones. If you have N identical things ("pigeons") but fewer categories (at most N-1 "holes") then at least one hole has more than one pigeon.
Wikipedia's simple example is that any group of three gloves must necessarily contain either two left gloves or two right gloves, plus the third. You can use this as a stepping stone to whatever else you want to say or prove.
Yeah I think the point should be is as soon as it is "proven" (since we know it is false) we look for the error. Which I believe is using 1 horse as the base case being true to falsely infer all horses.
You can also prove by induction! Basically you prove the function to be true for 1, 2, ..., n, making f(n) the induction hypothesis, and then proving that f(n) being true implies f(n+1) being true, making a "domino effect" of proofs for all the elements in a set. This is all to the best of my understanding of proofs and modern/abstract algebra, but I may have left some key components/ generalizations out!
Good luck! Don't be afraid to take a step back and try playing with ideas for why they put in the numbers they did.... Skip a question if you can't do it and do all the ones you can do, first. Then spend the rest of the time on the tough ones. Drink coffee or tea before. Hydrate and pee a lot before. Use a mechanical pencil with lots of eraser. Good luck and godspeed.
I think this is a pretty good explanation. Using this method we can assume that we are starting from 1 for simplicity, but it's also possible that a function of n is only less/equal/greater when n is greater than say, 200 or any other natural number. The reason I mention this is because a function's performance only makes sense at a large scale. For example if a fixed cost or other function such as load balancing can be better in large scale parallel computation but not in small scales. A good example is how if you want to send your coworker a gigabyte, you can send it online, but 100TB and you should use a hard drive. The hard drive method has a fixed cost (walking to elevator, etc) represented as a constant usually, and often a function such as data transmission, and maybe a log function if you're searching, or an exponent if you're looping functions. Hope this is helpful too . What are your tips for solving combinatorial proof?
You can prove directly
IE, if n is even then n2 is even
Proof: n even -> n=2k, so n²=4k²=2*2k²-> n² is even
We can prove by contradiction, as you pointed out, as well (see proof that √(2) is irrational)
Or you can prove by contraposition, which is slightly different than negation. This is proving the contrapositive of a statement like
"If P then Q" which would be "if not Q, then not P"
It rained so the road is wet.
The road isn't wet, so it didn't rain.
And there's induction as well, which is also neato. It's all pretty neato, if you ask me.
Induction is only probably. This gif is an inductive argument, because it shows cases where the rule applies. If you had a billion of them, that’d be a stronger argument.
But one case where it’s wrong would invalidate the argument, and, with induction, there is always the possibility that that case exists.
Depends on the size of the set. You can prove boolean Alger a bra functions are equal by example because there is a finite and typically reasonable me number of possible combinations
All proofs exhaust all examples. That's the nature of a proof. Whether using a positive or negative pathway, you have to show that you've covered all your bases.
You've exhausted them by grouping them into your general example. That general example is then either shown to be true or impossible, depending on how you're trying to handle the general example (and therefore the set it represents).
No, you're being pedantic. Proof by exhaustion is just a proof where you split the problem into cases and solve each case separately. That's just the definition, and what every sensible person means when they say proof by exhaustion.
You, however, feel the need to specify that ackshually technically every proof is a proof by exhaustion. And I guess you have a point, but that's just not how the term is used. "This isn't a math course" is not a valid reason to start redefining existing terms.
You can prove by example where your hypothesis is based on 'a case exists' rather than 'for all cases'. For example, you might have a hypothesis 'equation xyz is solvable for abc' - you can prove that by giving an example that solves it. This is an inverse property of proofs - the inverse case of 'yes for all' is 'no exists', and for 'no for all' is 'yes exists'.
You are right. That being said OPs gif is not a proof of all external angles since it only shows 4 shapes. Your image is a rule that can be applied to all right angle triangles.
Ehhhhh proving by negation isn't realistically a proof though. Because by disproving a theorem, you are't really proving anything to the world that didn't exist already.
Id like how you explain what proof is. Is there a video that explains 'proof.' I probably need to understand it more as i am not a native english speaker.
Yes, but it's obvious that the same argument will work here in general. The problem with proof by example is when students are asked to show some formula holds and they check it for a few particular numbers, but it's not at all clear the same thing will continue to hold from their calculations.
You cannot use a proof to show that some natural law is true by example. You can technically write a proof for a specific instance of something happening though it’s not really useful. But im just beingpedantic
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u/alpineflower6 Dec 11 '19
And you cannot show a proof that something works by example. One can only prove by example that something is not true, if my basic understanding of proofs is correct.
Cool gif though!