Solution Found!
Show that when you substitute (3n + 1)- for each
Chapter 2, Problem 32E(choose chapter or problem)
Problem 32E
Show that when you substitute (3n + 1)- for each occurrence of n and (3m + 1 )2 for each occurrence of m in the right-hand side of the formula for the function f (m, n) in Exercise 31. you obtain a one-to-one polynomial function Z ×Z → Z. It is an open question whether there is a one-to-one polynomial function Q × Q → Q.
Questions & Answers
QUESTION:
Problem 32E
Show that when you substitute (3n + 1)- for each occurrence of n and (3m + 1 )2 for each occurrence of m in the right-hand side of the formula for the function f (m, n) in Exercise 31. you obtain a one-to-one polynomial function Z ×Z → Z. It is an open question whether there is a one-to-one polynomial function Q × Q → Q.
ANSWER:
Solution:
Step 1:
In this problem we need to show that the polynomial function with
(when you substitute
for each occurrence of n and
for each occurrence of m in the right hand side of the formula for the function f(m, n))is one -to-one.
One -to-one function: A function for which every element of the range of the function corresponds to exactly one element of the domain.
Test for one -to-one functions : If f(a) = f(b) implies that a = b , then f is one-to-one.