Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Fallacy"

 
m (→‎All horses are the same color)
Line 23: Line 23:
 
Now assume that all the horses in any group of <math>k</math> horses are the same color.  This is our inductive assumption.
 
Now assume that all the horses in any group of <math>k</math> 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 <math>k+1</math> horses have the same color.  Number the horses 1 through <math>k+1</math>.  Horses 1 through <math>k</math> must be the same color as must horses <math>2 through k+1</math>.  It follows that all of the horses are the same color.
+
Using our inductive assumption, we will now show that all horses in a group of <math>k+1</math> horses have the same color.  Number the horses 1 through <math>k+1</math>.  Horses 1 through <math>k</math> must be the same color as must horses <math>2</math> through <math>k+1</math>.  It follows that all of the horses are the same color.
  
 
[[Fallacious proof/all horses are the same color | Explanation]]
 
[[Fallacious proof/all horses are the same color | Explanation]]

Revision as of 10:45, 3 August 2006

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.

Common false proofs

The fallacious proofs are stated first and then links to the explanations of their fallacies follow.

2 = 1

Let $a=b$.

Then we have

$a^2 = ab$ (since $a=b$)
$2a^2 - 2ab = a^2 - ab$ (adding $a^2-2ab$ to both sides)
$2(a^2 - ab) = a^2 - ab$ (factoring out a 2 on the LHS)
$2 = 1$ (dividing by $a^2-ab$)

Explanation

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 $k$ 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 $k+1$ horses have the same color. Number the horses 1 through $k+1$. Horses 1 through $k$ must be the same color as must horses $2$ through $k+1$. It follows that all of the horses are the same color.

Explanation

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Fallacy&oldid=9123"