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Unformatted text preview: Examples of mathematical induction, PHP, invariance, extremal arguments ... ... and Thinking J.L.F. King Exercises Using Extema Here is an example argument. 1 : Obs. Consider two vertices, v,w , in a connected ( possibly infinite ) graph. Prove there exists a path be tween them, with no repeatedvertex. ♦ Proof. Take a minimumlength path. Etc. 2 : Desegregation problem. A coloring of a graph as signs to each vertex either “black” or “white”. It is desegregated , if each vertex has at least one neighbor of the opposite color from his. [ Two vertices are neigh bors IFF they are connected by an edge. ] Prove that each finite connected graph G with N > 2 vertices, admits a desegregated coloring. ♦ Remark. Find two proofs of this. Can you generalize this problem? 3 : Bashful Boyfriends problem. [ In the future. ] ♦ Invariance The key underlying certain proofs, is that some quan tity or some relation is preserved under the relevant operations. Eg: Invariant quantity. Have B be the 8 × 8 chess board, but with the lowerright and upperleft cells removed; so  B  = 62. We start laying down domi nos. Can we cover the board with 31 dominos? No! Why? Initially, the uncovered part of the board ( i.e, all of B ) has 32 black cells and 30 white cells. These numbers are not invariant under placing a domino. But the difference , # n Uncovered black cells o # n Uncovered white cells o , † : is unaltered by placing a domino —it is invariant. Since the “discrepancy” is 2 initially, it will always be 2, no matter how many dominos we place. But a covered board would have a discrepancy of 0, not 2. Eg: Invariant relation. Our Lightning bolt alg. chose “seeds” for the s and t columns, so that r n = s n · r + t n · r 1 , ‡ : for n = 0 , 1. [ The n th : remainder , quotient , and B´ ezout multipliers are called r n ,q n ,s n ,t n . ] The LBolt update rule preserved relation ( ‡ ), in building row n from rows n 2 and n 1. When we found the index K where r K = Gcd ( r ,r 1 ) , this invariance handed us the GCD as a linearcombination of r and r 1 . 5.1 : usamo1994.2. Let R , B , Y denote the colors red , blue , yellow , respectively. The sides of a 99gon are initially colored so that, traveling CW ( clockwise ) , consecutive sides are R , B , R , B , ..., R , B , R , B , Y . † : Is it possible, still traveling CW, to obtain R , B , R , B , ..., R , B , R , Y , B ‡ : by a sequence of modifications? A modification changes the color of one side ( to one of R , B , Y ) under the constraint that at no time may two adjacent sides have the same color. ♦ Proof. No such modificationsequence exists. In a coloring, let→ B Y denote the number of B Y adjacentpairs, when traveling CW. In ( † ), then,→ B Y is 1, and→ Y B = 0. Still in ( † ),→ R Y = 0 and→ Y R = 1 ( because CW from the “end” Y is the “starting” R )....
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 Fall '09
 LARSON
 Mathematical Induction, Inductive Reasoning, Natural number, Prime number, Structural induction, Prof. JLF King

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