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Unformatted text preview: Massachusetts Institute of Technology Course Notes 1 2 Course Notes 1: Proofs 2 Propositions Definition. A proposition is a statement that is either true or false. This definition sounds very general, but it does exclude sentences such as, Wherefore art thou Romeo? and Give me an A!. Proposition 2.1. 2 + 3 = 5. This proposition is true. Proposition 2.2. Let p ( n ) ::= n 2 + n + 41 . n N p ( n ) is a prime number . The symbol is read for all. The symbol N stands for the set of natural numbers , which are 0, 1, 2, 3, ... ; (ask your TA for the complete list). A prime is a natural number greater than one that is not divisible by any other natural number other than 1 and itself, for example, 2, 3, 5, 7, 11, ... . Lets try some numerical experimentation to check this proposition: p (0) = 41 which is prime. p (1) = 43 which is prime. p (2) = 47 which is prime. p (3) = 53 which is prime. ... p (20) = 461 which is prime. Hmmm, starts to look like a plausible claim. In fact we can keep checking through n = 39 and confirm that p (39) = 1601 is prime. But if n = 40 , then p ( n ) = 40 2 + 40 + 41 = 41 41 , which is not prime. Since the expression is not prime for all n , the proposition is false! In fact, its not hard to show that no nonconstant polynomial can map all natural numbers into prime numbers. The point is in general you cant check a claim about an infinite set by checking a finite set of its elements, no matter how large the finite set. Here are two even more extreme examples: Proposition 2.3. a 4 + b 4 + c 4 = d 4 has no solution when a,b,c,d are positive integers. In logical notation, letting Z + denote the positive integers, we have a Z + b Z + c Z + d Z + a 4 + b 4 + c 4 = d 4 . Strings of s like this are usually abbreviated for easier reading: a,b,c,d Z + a 4 + b 4 + c 4 = d 4 . Euler (pronounced oiler) conjectured this 1769. But the proposition was proven false 218 years later by Noam Elkies at the liberal arts school up Mass Ave. He found the solution a = 95800 ,b = 217519 ,c = 414560 ,d = 422481 . Proposition 2.4. 313( x 3 + y 3 ) = z 3 has no solution when x,y,z N . This proposition is also false, but the smallest counterexample has more than 1000 digits! Proposition 2.5. Every map can be colored with 4 colors so that adjacent 1 regions have different colors....
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This note was uploaded on 08/27/2008 for the course CS 1050 taught by Professor Huang during the Spring '05 term at Georgia Institute of Technology.
- Spring '05
- Computer Science