Difference between revisions of "Fallacy"
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[[Fallacious proof/all horses are the same color | Explanation]] | [[Fallacious proof/all horses are the same color | Explanation]] | ||
+ | |||
+ | == All Numbers are Equal == | ||
+ | |||
+ | Proof: Choose arbitrary a and b, and let t = a + b. Then | ||
+ | <div style="text-align:center"><math>a + b = t</math></div> | ||
+ | |||
+ | <div style="text-align:center"><math>(a + b)(a - b) = t(a - b</math>)</div> | ||
+ | |||
+ | <div style="text-align:center"><math>a^2 - b^2 = ta - tb</math></div> | ||
+ | <div style="text-align:center"><math>a^2 - ta = b^2 - tb</math></div> | ||
+ | |||
+ | <div style="text-align:center"><math>a^2 - ta + \dfrac{t^2}{4} = b^2 - tb + \dfrac{t^2}{4}</math></div> | ||
+ | |||
+ | <div style="text-align:center"><math>(a - \dfrac{t}{2})^2 = (b - \dfrac{t}{2})^2</math></div> | ||
+ | |||
+ | <div style="text-align:center"><math>a - \dfrac{t}{2} = b - \dfrac{t}{2}</math></div> | ||
+ | |||
+ | <div style="text-align:center"><math>a = b</math></div> | ||
+ | ''QED'' | ||
== See also == | == See also == | ||
* [[Proof writing]] | * [[Proof writing]] |
Revision as of 21:43, 28 October 2007
A fallacious proof is a an attempted proof that is logically flawed in some way. The fact that a proof is fallacious says nothing about the validity of the original proposition.
Contents
Common false proofs
The fallacious proofs are stated first and then links to the explanations of their fallacies follow.
2 = 1
Let .
Then we have
![$a^2 = ab$](http://latex.artofproblemsolving.com/d/f/6/df616fb8336779daf527060288b49d13a3c005e7.png)
![$a=b$](http://latex.artofproblemsolving.com/5/a/b/5abd2c942ae299fd26e18d0ca5d390a7ca4a8c34.png)
![$2a^2 - 2ab = a^2 - ab$](http://latex.artofproblemsolving.com/e/d/e/ede398bee73258c4e38b6b3344f5212a16930e05.png)
![$a^2-2ab$](http://latex.artofproblemsolving.com/c/3/8/c384fa448b634e71ce150ce6dc7165e016693f23.png)
![$2(a^2 - ab) = a^2 - ab$](http://latex.artofproblemsolving.com/9/d/5/9d5c2bbac98e2986ce7a579ef55be936d92e97b9.png)
![$2 = 1$](http://latex.artofproblemsolving.com/0/f/f/0ffac44b2fc291131a3d8a59e985754d905bdb26.png)
![$a^2-ab$](http://latex.artofproblemsolving.com/c/a/2/ca2d74b575919eb08e10fb014c60053f2f4c0011.png)
All horses are the same color
We shall prove that all horses are the same color by induction on the number of horses.
First we shall show our base case, that all horses in a group of 1 horse have the same color, to be true. Of course, there's only 1 horse in the group so certainly our base case holds.
Now assume that all the horses in any group of horses are the same color. This is our inductive assumption.
Using our inductive assumption, we will now show that all horses in a group of horses have the same color. Number the horses 1 through
. Horses 1 through
must be the same color as must horses
through
. It follows that all of the horses are the same color.
All Numbers are Equal
Proof: Choose arbitrary a and b, and let t = a + b. Then
![$a + b = t$](http://latex.artofproblemsolving.com/0/a/1/0a19ad64f5f44bf7d56fe21651111c214d75214f.png)
![$(a + b)(a - b) = t(a - b$](http://latex.artofproblemsolving.com/6/2/f/62f97e1776f8d0f9060d830731bee8790c739b1b.png)
![$a^2 - b^2 = ta - tb$](http://latex.artofproblemsolving.com/7/2/c/72ce304396eb187077f60ae402d94328b954c9b8.png)
![$a^2 - ta = b^2 - tb$](http://latex.artofproblemsolving.com/d/1/6/d16eae69964f9316a4434ae3ab09c31959898bc3.png)
![$a^2 - ta + \dfrac{t^2}{4} = b^2 - tb + \dfrac{t^2}{4}$](http://latex.artofproblemsolving.com/d/3/3/d336cfcc34f4629b7f6b4e1092149b99c48140de.png)
![$(a - \dfrac{t}{2})^2 = (b - \dfrac{t}{2})^2$](http://latex.artofproblemsolving.com/8/6/6/866b7f2bc2479063050852c01f670f705dac031a.png)
![$a - \dfrac{t}{2} = b - \dfrac{t}{2}$](http://latex.artofproblemsolving.com/7/9/c/79c113df8b88cb96367ca103b50bcfdb9abe0392.png)
![$a = b$](http://latex.artofproblemsolving.com/3/9/7/397fd7b1f33bdbf6ee27a83d1bcee42d222cb331.png)
QED