Solution to Exercise 2.20.
Complete the proof that the Ruler Placement Postulate is not independent, Theorem 2.5.
Geometry is a skill of the eyes and the hands as well as the mind.
—Jean Pedersen

 

Theorem 2.5. The Ruler Placement Postulate is not independent of the other axioms.

Proof. Let P and Q be two distinct points. Then by Postulate 1, there is a line l that contains P and Q. Assume  is a ruler, such a ruler exists by Postulate 3. Set

 

Define  by g(X) = k[f(X)  f(P)] for all X on line l. We now show g satisfies the conditions of Postulate 3.
      We first show g is a one-to-one function. Let A and B be points on l. Suppose g(A) = g(B). Thus k[f(A)  f(P)] = k[f(B)  f(P)]. Since k is nonzero, f(A)  f(P) = f(B)  f(P). Hence f(A) = f(B). Thus, since f is a one-to-one function, A = B. Therefore, g is a one-to-one function.
      Next show g is onto the real numbers. Let r be a real number. Since f is onto the real numbers, there is a point A on line l such that f(A) = r/k + f (P). Thus

 

Hence g is onto the real numbers.
      Show g satisfies the distance condition. Let A and B be points on l. Thus

 

      Finally, show the signs of the coordinates of the two points.

g(P) = k[f(P)  f(P)] = 0

and

 

Hence g(P) = 0 and g(Q) > 0.
      Therefore, the Ruler Placement Postulate follows from the other axioms and is not independent.//