Proof by contradiction

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The "Proof by Contradiction"  is  also known as reductio ad absurdum,  which is probably Latin for "reduce it to something absurd".

Here's the idea:

  1. Assume that a given proposition is untrue.
  2. Based on that assumption reach two conclusions that contradict each other.

This is based on a  classical formal logic construction known as Modus Tollens:  If P implies Q and Q is false, then P is false.   In this case, Q is a proposition of the form (R and not R) which is always false.   P is the negation of the fact that we are trying to prove and if the negation is not true then the original proposition must have been true.  If  computers are not "not stupid"  then they are stupid.  (I hear that "stupid computer!" phrase a lot around  here.)

Example:

Lets prove that there is no largest prime number (this is the idea of Euclid's original proof).  Prime numbers are integers with no exact integer divisors except 1 and themselves.

  1. To prove: "There is no largest prime number" by contradiction.  
  2. Assume: There is a largest prime number, call it p.
  3. Consider the number N  that is one larger than the product of all of the primes smaller than or equal to pN=2*3*5*7*11...*p + 1.  Is it prime? 
  4. N is at least as big as p+1 and so is larger than p and so, by Step 2, cannot be prime.
  5. On the other hand,  N has no prime factors between 1 and p because they would all leave a remainder of 1.  It has no prime factors larger than p because Step 2 says that there are no primes larger than p.  So N has no prime factors and therefore must itself be prime (see note below). 
  6. We have reached a contradiction (N is not prime by Step 4, and N is prime by Step 5) and therefore our original assumption that there is a largest prime must be false.    

Note: The conclusion in Step 5  makes implicit use of one other important theorem:  The Fundamental Theorem of Arithmetic:  Every integer can be uniquely represented as the product of primes.     So  if N had a composite (i.e. non-prime) factor, that factor would itself have prime factors which would also be factors of N.  

 

 

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