Answer
a)
Explaining a proof using strong induction that
E(n) is true for all integers n ≥ 4 runs into difficulties.
b)
-Showing that we can prove that E(n) is true for all integers
n ≥ 4 by proving by strong induction the stronger
statement T (n) for all integers n ≥ 4,
- which states that in every triangulation of a simple polygon, at least two of the triangles in the triangulation have two sides bordering
the exterior of the polygon.
Work Step by Step
a)
When we try to prove the inductive step
and
-find a triangle in each sub polygon with at least two sides bordering the exterior,
- it may happen in each case that the triangle we are guaranteed in fact borders the diagonal (which is part of the boundary of that polygon).
-This leaves us with no triangles guaranteed to touch the boundary of the original polygon.
b)
--Let P(n) be the statement that
- if a simple polygon with n sides is triangulated,
- then at least two of the triangles in the triangulation have two sides that border the exterior of the polygon.
-We will prove ∀n ≥ 4 P(n).
-The statement is clearly true for n = 4,
-because there is only one diagonal, leaving two triangles with the desired property.
-Fix k ≥ 4 and assume that P(j) is true for all j with 4 ≤ j ≤ k.
Consider a polygon with k + 1 sides, and some triangulation
of it.
-Pick one of the diagonals in this triangulation. First suppose
that this diagonal divides the polygon into one triangle
and one polygon with k sides.
-Then the triangle has two sides
that border the exterior.
- Furthermore, the k-gon has, by the inductive hypothesis,
-two triangles that have two sides that border the exterior of that k-gon, and
-only one of these triangles can fail to be a triangle that has two sides that border the exterior of the original polygon.
-The only other case is that this diagonal divides the polygon into two polygons with j sides and k + 3 − j sides for some j with 4 ≤ j ≤ k − 1.
--By the inductive hypothesis,
-each of these two polygons has
two triangles that have two sides that border their exterior, and
in each case only one of these triangles can fail to be a triangle
- that has two sides that border the exterior of the original
polygon
