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Thread: One-to-one and Onto Functions

  1. #1
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    One-to-one and Onto Functions

    Ok the question is:
    Give an example of a function from N to N that is
    (a) one-to-one but not onto
    (b) onto but not one-to-one
    (c) both onto and one-to-one
    (d) neither one-to-one nor onto


    (a) My answer is the function from {a,b,c} to {1,2,3,4} with f(a) = 2, f(b) = 3, f(c) = 1. Is this the correct example to this question?

    What does it mean from N to N?

    I kind of lost on how to show these examples. Thanks for any help!
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  2. #2
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    Quote Originally Posted by VGDude85 View Post
    Ok the question is:
    Give an example of a function from N to N that is
    (a) one-to-one but not onto
    (b) onto but not one-to-one
    (c) both onto and one-to-one
    (d) neither one-to-one nor onto

    What does it mean from N to N?
    Normally $\displaystyle \mathbb{N} = \left\{ {0,1,2,3, \cdots } \right\}$.
    Some authors/texts do not include 0 in $\displaystyle \mathbb{N}$ so check your textbook.

    Here is an example for (a) $\displaystyle f(n)=2n$.
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  3. #3
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    Quote Originally Posted by VGDude85 View Post
    Ok the question is:
    Give an example of a function from N to N that is
    (a) one-to-one but not onto
    (b) onto but not one-to-one
    (c) both onto and one-to-one
    (d) neither one-to-one nor onto


    (a) My answer is the function from {a,b,c} to {1,2,3,4} with f(a) = 2, f(b) = 3, f(c) = 1. Is this the correct example to this question?

    What does it mean from N to N?

    I kind of lost on how to show these examples. Thanks for any help!

    (c) $\displaystyle {\color{blue}n \mapsto n [\backepsilon n \in \mathbb{N}]}$
    (d) $\displaystyle {\color{blue}1 \mapsto 2, 0 \mapsto 2}$
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  4. #4
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    Thanks for the examples guys. So the N stands for natural numbers, I totally forgot what that meant. My old example I could tell was for Z.
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