Proof by Contradiction – An essential tool of deduction for every man

Summary:  If a ridiculous Bee implies an impossible See, an impossible See implies your Bee is ridiculous.   Thus Don’t Bee ridiculous. Proof by contradiction is an essential tool of thought that everyone should know.  However it is quite evident to me most people do no know it.  Let me see if I can explain it. Steps: -1- assume an item to be true.  You usually suspect it to be false and use the method to prove it. -2- follow the logical results of this assumption to an absurd conclusion -3- since the logical result is absurd the assumption must be false. You can negate the assumption. Be careful when you do this.  Taking the negative is not as straight forward as many non mathematical people would think. Example:  Prove I am not George Washington.  -1- Assume I am George Washington the first president of the USA -2- my face should match the face on the front of the 1 dollar bill ….does it?  Clearly no. -3- the conclusion my face matches the face on the front of the dollar bill and thus the negative of statement 1 is true = I AM NOT GEORGE WASHINGTON. Sound absurd.  Read the next example and see how the very same method applied to something that sounds more official makes the proof alot simpler than attacking it directly !!   Theorem. There are infinitely many prime numbers. Proof. Assume to the contrary that there are only finitely many prime numbers, and all of them are listed as follows: p₁, p₂ …, p_n. Consider the number q = p₁p₂… p_n + 1. The number q is either prime or composite. If we divided any of the listed primes p_i into q, there would result a remainder of 1 for each i = 1, 2, …, n. Thus, q cannot be composite. We conclude that q is a prime number, not among the primes listed above, contradicting our assumption that all primes are in the list p₁, p₂ …, p_n.