Assignment:
(The Sum of the Divisors Function)
Let s be the function that associates with each natural number the sum of its distinct natural number factors. For example, s (6) = 1 + 2 + 3 + 6
= 12.
1. Calculate s (k) for each natural number k from 1 through 15.
2. Are the numbers 5, Ï?, and - 6 in the domain of the function s? What is the domain of the function s?
3. Does there exist a natural number n such that s (n) = 5? Justify your conclusion.
4. Is it possible to find two different natural numbers m and n such that s (m) = s(n)? Explain.
5. Are the following statements true or false?
(a) For each m N, there exists a natural number n such that s (n) = m.
(b) For all m, n N, if m n, then s (m) . s (n).
(Creating Functions with Finite Domains)
Let A = {a, b, c, d}, B = {a, b, c}, and C = {s, t, u, v}. In each of the following exercises, draw an arrow diagram to represent your function when it is appropriate.
1. Create a function f : A ' C whose range is the set C or explain why it is not possible to construct such a function.
2. Create a function f : A ' C whose range is the set {u, v} or explain why it is not possible to construct such a function.
3. Create a function f : B ' C whose range is the set C or explain why it is not possible to construct such a function.
4. Create a function f : A ' C whose range is the set {u} or explain why it is not possible to construct such a function.
Q1. (a) Draw an arrow diagram that represents a function that is an injection but is not a surjection.
(b) Draw an arrow diagram that represents a function that is an injection and is a surjection.
(c) Draw an arrow diagram that represents a function that is not an injection and is not a surjection.
(d) Draw an arrow diagram that represents a function that is not an injection but is a surjection.
(e) Draw an arrow diagram that represents a function that is not a bijection.
Q2. (a) Let g : R ' R be defined by g(x) = x cube. Is the function g an injection?
Is the function g a surjection? Justify your conclusions.
(b) Let f : Q ' Q be defined by f (x) = x cube. Is the function f an injection?
Is the function f a surjection? Justify your conclusions.
Provide complete and step by step solution for the question and show calculations and use formulas.