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    Mathematical Reasoning: The Sum of the Divisors Function

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    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.

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    https://brainmass.com/math/discrete-math/mathematical-reasoning-sum-divisors-function-448470

    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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