History Main / ProofByExamples

6th Jun '17 2:35:48 PM Madrugada
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* Proving an existential statement (i.e. "There exists...") by example. One example is plenty.
** The prime (pardon the pun) example might well be this: "2 is an even number and is prime. Therefore, there exists at least one prime number that is even."

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* Proving an existential statement (i.e. "There exists...") by example. One example is plenty.
**
all that's needed. The prime (pardon the pun) example might well be this: "2 is an even number and is prime. Therefore, there exists at least one prime number that is even."
25th Jan '17 7:31:52 PM Game_Fan
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* In mathematics proof by example is no proof at all. One instance of this is a conjecture by Christian Goldbach that "every odd composite number can be written as the sum of a prime and twice a square number" which certainly seems to be true if you try casually testing a few example. It wasn't until much later that a counter example (5777) was found.
** An even more dramatic example is [[https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture Euler's sum of power conjecture]] for which the first counter example is 61,917,364,224!

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* In mathematics proof by example is usually no proof at all. One instance of this Many famous examples are used to illustrate this.
** A simple one
is a conjecture by Christian Goldbach that "every odd composite number can be written as the sum of a prime and twice a square number" which certainly seems to be true if you try casually testing a few example. It wasn't until much later that a counter example (5777) was found.
** An even more dramatic example is [[https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture Euler's sum of power conjecture]] for which the first counter example counter-example is 61,917,364,224!61,917,364,224!
** There are problem in mathematics that have been tested for trillions upon trillions of examples without finding a counter-example but still lack proof. (The huge numbers of examples are both used to search for counter-examples and in hopes of discovering patterns that might lead to a proof.)
25th Jan '17 7:28:21 PM Game_Fan
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* Usually, in mathematics, [[http://en.wikipedia.org/wiki/Skewe%27s_number no matter how many examples]] [[http://en.wikipedia.org/wiki/Riemann_hypothesis you might have]], proof by example is not a good idea.

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* Usually, in mathematics, [[http://en.In mathematics proof by example is no proof at all. One instance of this is a conjecture by Christian Goldbach that "every odd composite number can be written as the sum of a prime and twice a square number" which certainly seems to be true if you try casually testing a few example. It wasn't until much later that a counter example (5777) was found.
** An even more dramatic example is [[https://en.
wikipedia.org/wiki/Skewe%27s_number no matter how many examples]] [[http://en.wikipedia.org/wiki/Riemann_hypothesis you might have]], proof by org/wiki/Euler%27s_sum_of_powers_conjecture Euler's sum of power conjecture]] for which the first counter example is not a good idea.61,917,364,224!
22nd Jun '15 4:15:56 AM ShorinBJ
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:: Hasty Generalisation

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:: Hasty GeneralisationGeneralization
8th Nov '13 6:09:12 PM RatherRandomRachel
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** A (simple) example of this might be: "A new plant found seems to fit into a particular category with several others. All of the plants within this category need three things to thrive - carbon dioxide, water and a light source. Therefore, it seems likely this plant will also need those three to survive. We should study it to confirm or deny this theory."

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** A (simple) example of this might be: "A new plant found seems to fit into a particular category with several others. All of the plants within this category seem to need three things to thrive - carbon dioxide, water and a light source. Therefore, it seems likely this plant will also need those three to survive. We should study it to confirm or deny this theory."
23rd Sep '13 3:20:52 AM RatherRandomRachel
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** A (simple) example of this might be: A new plant found seems to fit into a particular category with several others. All of the plants within this category need three things to thrive - carbon dioxide, water and a light source. Therefore, it seems likely this plant will also need those three to survive. We should study it to confirm or deny this theory.

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** A (simple) example of this might be: A "A new plant found seems to fit into a particular category with several others. All of the plants within this category need three things to thrive - carbon dioxide, water and a light source. Therefore, it seems likely this plant will also need those three to survive. We should study it to confirm or deny this theory."
23rd Sep '13 3:13:04 AM RatherRandomRachel
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** To clarify. Induction - at its most basic is proving it by proving two things. The Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)
*** That's mathematical induction, which is not "induction" in the logical sense (''i.e.'', inductive reasoning); rather, it is rigorous deductive reasoning.

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** To clarify. Induction - at its most basic is proving it by proving two things. The Base Case exists (typically for A (simple) example of this might be: A new plant found seems to fit into a particular category with several others. All of the value of 1 or 0) plants within this category need three things to thrive - carbon dioxide, water and that if we assume the theory works at value k (k being any given number) we can prove that a light source. Therefore, it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)
*** That's mathematical induction, which is not "induction" in the logical sense (''i.e.'', inductive reasoning); rather,
seems likely this plant will also need those three to survive. We should study it is rigorous deductive reasoning.to confirm or deny this theory.
11th Jun '13 2:19:17 PM DCC
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Added DiffLines:

In short, mistaking inductive reasoning for deductive reasoning
19th May '13 10:19:55 PM trumpetmarietta
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** To clarify. Induction - at its most basic is proving it by proving two things. The Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)

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** To clarify. Induction - at its most basic is proving it by proving two things. The Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...))
*** That's mathematical induction, which is not "induction" in the logical sense (''i.e.'', inductive reasoning); rather, it is rigorous deductive reasoning.
6th Apr '13 7:22:17 PM SenseiLeRoof
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** To clarify. Induction - at it's most basic is proving it by proving two things. THe Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)

to:

** To clarify. Induction - at it's its most basic is proving it by proving two things. THe The Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)
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