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    One-To-One Functions

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    FUNCTIONS

    1. Find all functions from X = {a, b, c} to Y = {u, v}.
    2. Let F and G be functions from the set of all real numbers to itself. Define new functions F - G: R R and G - F: R R as follows:
    (F - G)(x) = F(x) - G(x) for all x  R,
    (G - F)(x) = G(x) - F(x) for all x  R.
    Does F - G = G - F? Explain.
    3. Let A = {2, 3, 5} and B = {x, y}. Let p1 and p2 be the projections of AB onto the first and second coordinates. That is, for each pair (a, b)  AB, p1(a, b) = a and p2(a, b) = b.
    Find p2(2, y) and p2(5, x). What is the range of p2?
    4. Let X = {1, 5, 9} and Y = { 3, 4, 7}. Define g : XY by specifying that g(1) = 7, g(5) = 3, g(9) = 4.
    Is g one-to-one? Is g onto? Explain your answers.
    5. Let X = {1, 2, 3}, Y = {1, 2, 3, 4}, and Z = {1, 2}.
    a) Define a function g : X Z that is onto but not one-to-one.
    b) Define a function k : X X that is one-to-one and onto but is not the identity function on X.
    6. Let X = {1, 2, 3, 4}, Y = {2, 3, 4, 5, 6}, Z = {1, 2, 3}.
    a) Define a function f: X  Y that is one-to-one but not onto.
    b) Define a function g: X  Z that is onto but not one-to-one.
    c) Define a function h: X  Y that is neither onto nor one-to-one.
    d) Define a function k: X  X that is onto and one-to-one but is not the identity function on X.
    7. List all the functions from the three element set {1, 2, 3} to the set {a, b}. Which functions, if any, are one-to-one? Which functions, if any, are onto?
    8. Define f: R  R by the rule f(x) = 2x2-3x+1
    a) Is f one-to-one? Prove or give a counterexample.
    b) Is f onto? Prove or give a counterexample.
    9. Define g : Z  Z by the rule g(n) = 3n - 2, for all integers n.
    a) Is g one-to-one? Prove or give a counterexample.
    b) Is g onto? Prove or give a counterexample.
    10. Let X = {a, b, c, d, e} and Y = {s, t, u, v, w}. A one-to-one correspondence F: XY is defined by: F(a) = t, F(b) = w, F(c) = s, F(d) = u, F(e) = v. Define F-1 (Please, specify each function value, i.e. F-1(s) = ...what-ever, or draw the diagram).
    11. Give a real-world example of a function which is both one to one and onto

    © BrainMass Inc. brainmass.com October 10, 2019, 8:22 am ad1c9bdddf
    https://brainmass.com/math/recurrence-relation/one-one-functions-622198

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    FUNCTIONS

    1. Find all functions from X = {a, b, c} to Y = {u, v}.

    Solution:

    The functions will be
    {(a,u),(b,u),(c,u)}, {(a,v),(b,v),(c,v)}, {(a,u),(b,u),(c,v)}, {(a,u),(b,v),(c,u)}, {(a,v),(b,u),(c,u)}, {(a,v),(b,v),(c,u)}, {(a,v),(b,u),(c,v)}, {(a,u),(b,v),(c,v)}

    2. Let F and G be functions from the set of all real numbers to itself. Define new functions F - G: R R and G - F: R R as follows:
    (F - G)(x) = F(x) - G(x) for all x  R,
    (G - F)(x) = G(x) - F(x) for all x  R.
    Does F - G = G - F? Explain.

    Solution:
    No,
    since
    F(x) - G(x) ≠ G(x) - F(x)
    (F-G)(x) ≠ (G- F)(x)

    Therefore, F - G ≠ G - F

    3. Let A = {2, 3, 5} and B = {x, y}. Let p1 and p2 be the projections of AB onto the first and second coordinates. That is, for each pair (a, b)  AB, p1(a, b) = a and p2(a, b) = b.
    Find p2(2, y) and p2(5, x). What is the range of p2?

    Solution:
    p2(2, y) = y
    p2(5, x) = x

    Range of p2 is B = {x, y}.

    4. Let X = {1, 5, 9} and Y = { 3, 4, 7}. Define g : XY by specifying that g(1) = 7, g(5) = 3, g(9) = 4.
    Is g one-to-one? Is g onto? Explain your answers.

    Solution:

    g is one-to-one because g(1) ≠ g(5), g(1) ≠ g(9), and ...

    Solution Summary

    This posting explains concepts of one-to-one and onto functions in detail.

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