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Math Help - Simple Question

  1. #1
    Junior Member
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    Simple Question

    I'm trying to write the definition of a one-one and onto functions using logical quantifiers. Is the following correct:

    Let f be a function from S to W.

    One-one:
    If f(s) = w then for all s' in S such that s' != s, f(s') != t.

    Onto:
    For all w in W, there exists at least one s in S such that f(s) = w.

    I'm pretty sure about what I've written for onto, but not about one-one. Thanks for helping me clarify these simple concepts.
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  2. #2
    Super Member Matt Westwood's Avatar
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    One-one: How about:
    For all s, t in S: if s != t then f(s) != f(t).

    (not sure what yours means, what's the w and the t?)

    Alternatively:
    For all s, t in S: if f(s) = f(t) then s = t.

    Onto:
    Yep, that seems okay to me.
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  3. #3
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    Let:

    (x) mean for all x
    Ex mean there exists an x
    E!x mean there exists a unique x
    Let f:S------>W
    And :

    (f is one to one and onto) iff (w)[ wεW------>E!s( sεS & (s,w)εf)]........1

    Now go into another thread in Miscellaneous to see the meaning of unique

    Actually 1 straight away defines a function from W to S
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