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Mathematical Reasoning: The Sum of the Divisors Function

Prepare written answers to the following assignments:

Preview Activity 3 (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).

Activity 6.6 (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.

Exercises 6.3

1. (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.

4. (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.

Solution Preview

See the attached file.
1.

S(1) = 1
S(2) = 1+2
S(3) = 1+3
S(4) = 1+2+4
S(5) = 1+5
S(6) = 1+2+3+6
S(7) = 1+7
S(8) = 1+2+4+8
S(9) = 1+3+9
S(10) = 1+2+5+10
S(11) = 1+11
S(12) = 1+2+3+4+6+12
S(13) = 1+13
S(14) = 1+2+7+14
S(15) = 1+3+5+15

2.

no, they are not
The natural numbers {1,2,3,4...} (0 may be included depending on your class/text)

3.

No
S(1) = 1
S(2) = 1+2
S(3) = 1+3
S(4) = ...

Solution Summary

The solution discusses mathematical reasoning regarding the Sum of the Divisors Function.

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