Given n lines in \"general position\" in the plane, which means that any two lin

Question

Given n lines in "general position" in the plane, which means that any two lines have a common point and no point of the plane belongs to more than two lines.
(a) Use mathematical induction to prove that the lines divide the plane into N = [(n+1)/2]+1 regions, for every n>=1.

(b) Use mathematical induction to prove that the map of these N regions can be colored with two colors in such a way that no regions with a common border line get the same color (in other words, any two regions obtaining the same color share at most one common boundary point).

Explanation / Answer

Prove that for every natural number n, (x - y) divides (x^n - y^n). Our first step in mathematical induction is the base case; that is, for n = 1. 1) Base Case: Let n = 1. Then (x^n - y^2) = (x - y), and (x - y) is obviously divisible by (x - y). Therefore, the formula holds true for n = 1. 2) Induction Hypothesis: Assume the formula holds true for n = k. That is, assume that (x - y) divides (x^k - y^k). ( We want to prove that (x - y) divides x^(k + 1) - y^(k + 1) ) But what does x^(k + 1) - y^(k + 1) equal? x^(k + 1) - y^(k + 1) I'm going to re-express these two terms. (x^1)(x^k) - (y^1)(y^k) x*(x^k) - y*(y^k) I'm going to use a little trick, by "adding zero" in the middle. x*(x^k) + 0 - y*(y^k) I'm going to subtract x*(y^k) and add x*(y^k). After all, subtracting and then adding the same term is the same as adding zero. x*(x^k) - x*(y^k) + x*(y^k) - y*(y^k) Now I'm going to factor the first two terms and the last two terms. x(x^k - y^k) + (y^k)(x - y) Look closely at this; (y^k)(x - y) is obviously divisible by (x - y). By our induction hypothesis, we assumed that (x^k - y^k) is divisible by (x - y). Therefore, x(x^k - y^k) is divisible by (x - y). The sum of two terms both divisible by (x - y) is also divisible by (x - y). Therefore, what we started with, x^(k + 1) - y^(k + 1) is divisible by (x - y) Therefore, by the principle of mathematical induction, x^n - y^n is divisible by (x - y) for all natural numbers n.

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