induction-jk - Examples of mathematical induction PHP...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 repeated-vertex. ♦ Proof. Take a minimum-length 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 lower-right and upper-left 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 linear-combination of r and r 1 . 5.1 : usamo1994.2. Let R , B , Y denote the colors red , blue , yellow , respectively. The sides of a 99-gon 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 modification-sequence exists. In a coloring, let--→ B Y denote the number of B Y adjacent-pairs, 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 )....
View Full Document

This note was uploaded on 01/26/2012 for the course MHF 3202 taught by Professor Larson during the Fall '09 term at University of Florida.

Page1 / 10

induction-jk - Examples of mathematical induction PHP...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online