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Math Help - equivalence relations and functions

  1. #1
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    equivalence relations and functions

    im having trouble on a problem

    A={1,2,3,4,5,6} f:A->A

    i need to find an example where f is bijective, but not 1A
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  2. #2
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    Quote Originally Posted by kdigga View Post
    im having trouble on a problem

    A={1,2,3,4,5,6} f:A->A

    i need to find an example where f is bijective, but not 1A
    If you map A into A, then f is both one to one and onto.
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  3. #3
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by dwsmith View Post
    If you map A into A, then f is both one to one and onto.
    I don't understand this statement. f: A -> A doesn't require that A be the image but only the codomain, so for example the function f(x) = 1 \;\forall x \in A would conform to f: A -> A.

    But I also don't see the difficulty in finding a bijective function. Just write down

    f(1) = \boxed{ }

    f(2) = \boxed{ }

    ...

    f(6) = \boxed{ }

    and fill in with values from A such that you use every single element, then it will be bijective. (Sorry the rectangles are a bit uncentered vertically, couldn't find the proper LaTeX.)
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    Quote Originally Posted by undefined View Post
    I don't understand this statement.
    1\to 1
    2\to 2
    3\to 3
    4\to 4
    5\to 5
    6\to 6

    This is what I meant. A mapped into A
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  5. #5
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by dwsmith View Post
    1\to 1
    2\to 2
    3\to 3
    4\to 4
    5\to 5
    6\to 6

    This is what I meant. A mapped into A
    Oh. I thought the original poster was referring to that as 1A and was asking for a different one.
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  6. #6
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    Quote Originally Posted by undefined View Post
    Oh. I thought the original poster was referring to that as 1A and was asking for a different one.
    I have no idea what 1A is supposed to mean.
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  7. #7
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by dwsmith View Post
    I have no idea what 1A is supposed to mean.
    Yeah, it was just a guess based on context. Well in any case it's the identity function. I'd never heard "f maps A into A" meaning f is the identity function, and seemed f: A -> A would naturally be read as "f maps A into A" hence my original response.
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  8. #8
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by undefined View Post
    Yeah, it was just a guess based on context. Well in any case it's the identity function. I'd never heard "f maps A into A" meaning f is the identity function, and seemed f: A -> A would naturally be read as "f maps A into A" hence my original response.
    The word `into' is sometimes used to mean `injection' (much like `onto'). When there exists an injection, f, from A to B then it means you can find a copy of A in B, and so f maps A into B.

    I would also presume, as did undefined, that 1A means the identity function ( 1_A or I_A) as this would be too easy an answer and noone would exclude any other bijection.

    Now, bijections between finite sets are just permutations. So, permute your elements. For example,

    1 \mapsto 2
    2 \mapsto 1
    3 \mapsto 3
    4 \mapsto 4
    5 \mapsto 5
    6 \mapsto 6

    will be a function which switches 1 and 2. It is thus a permutation, as it permutes these two elements and keeps everything else fixed.
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