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    Set Theory

    Is every one-one function from N to itself onto? In other words, does the pigeon-hole principle hold for N itself?

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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by taypez View Post
    Is every one-one function from N to itself onto? In other words, does the pigeon-hole principle hold for N itself?

    Thanks for the help
    No. Consider the function f:N->N defined by f(n) = 2n, or the function g:N->N defined by g(n) = 2n + 1. Both these functions are one to one, but not unto. f only maps to the even integers in N, while g only maps to the odd integers in N
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    Quote Originally Posted by taypez View Post
    Is every one-one function from N to itself onto? In other words, does the pigeon-hole principle hold for N itself?
    No, Dirchlet's Pigenhole Principle only works for finite non-empty sets.
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