CS173 - Chapter 17: Proof by Contradiction

Lossless Algorithm

Allows complete reconstruction of a file exactly from a compressed one

Contradiction

Any statement that is well-known to be false or a set of statements that are obviously inconsistent with one another

Proof by Contradiction

Claim P is true because it's negation leads to a contradiction

Prove Root 2 is irrational

Define a rational number as a real number written as a/b, where a and b are integers and b is not 0. a and b share no common factors Suppose root 2 is ration. Than root 2 = a/b, where a and b are integers with no common factors. Since root 2 = a/b, 2 = a^2 / b^2. So, 2b^2 = a^2. Bu definition of even, this means that a must be even. So, a = 2n. If a = 2n and 2b^2 = a^2, then 2b^2 = 4n^2. So, b^2 = 2n^2. Then b^2 is even, and b is even. This means a and b have common factors

Lossy Algorithm

Only reconstructs an approximation of the original file

Prove that there are an infinite many prime numbers

Proof: Suppose not. That is, suppose there were only finitely many prime numbers. Let's call them p1, p2, up through pn. Consider Q = p1p2 · · · pn + 1. If you divide Q by one of the primes on our list, you get a remainder of 1. So Q isn't divisible by any of the primes p1, p2, up through pn. However, by the Fundamental Theorem of Arithmetic, Q must have a prime factor (which might be either itself or some smaller number). This contradicts our assumption that p1, p2,. . . pn was a list of all the prime numbers.

Prove that there is no largest even integer

Proof: Suppose the opposite. That is, suppose there is a largest even integer k. Since k is even, it has the form 2n, where n is an integer. Consider k + 2. k + 2 = (2n) + 2 = 2(n+1). So k+2 is even. But k+2 is larger than k. This contradicts the assumption that k is the largest even integer. Thus, our original claim is true

Typical use of Proof by Contradiction

Prove that certain types of objects cannot exist

Best known example of proof by contradiction

Root 2 is irrational

Euclid's Theorem

There are an infinitely many prime numbers

Suppose not

Typical way to inform readers that you are using proof by contradiction