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Math Help - Definition of a function

  1. #1
    Member oldguynewstudent's Avatar
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    Definition of a function

    Q: Give an example of a function
    f:{1,2,3,4} \rightarrow {6,7,8,9}
    which has no inverse.

    A: x={1,2,3,4], y={6,7,8,9} f(x) = \emptyset

    Marked wrong because f is not a funciton.

    From Rosen:
    DEFINITION 1: Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A.

    Also from Rosen:

    THEOREM 1: For every set S, (i) \emptyset \subseteq S
    and (ii) S \subseteq S

    If null is a subset of every set, isn't it an element in that set? And therefore f(x) = \emptyset should be a function?

    Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by oldguynewstudent View Post
    Q: Give an example of a function
    f:{1,2,3,4} \rightarrow {6,7,8,9}
    which has no inverse.

    A: x={1,2,3,4], y={6,7,8,9} f(x) = \emptyset

    Marked wrong because f is not a funciton.

    From Rosen:
    DEFINITION 1: Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A.

    Also from Rosen:

    THEOREM 1: For every set S, (i) \emptyset \subseteq S
    and (ii) S \subseteq S

    If null is a subset of every set, isn't it an element in that set? And therefore f(x) = \emptyset should be a function?

    Thanks
    It's a 'function' depending on how you define function. It is not a well-defined function which is what was assumed you were suppoesd to find. A function has to map element of \text{Dom }f to exactly one element of \text{Im }f. Your 'function' satisfies the most commonly emphasized point that each element is mapped to at most one elment...but it fails to map each element to at least one element.


    EDIT: A function \phi:X\mapsto Y with x\mapsto\varnothing is a function iff X=\varnothing for the conditions of being a function are satisfied vaccuously since there is no x in \text{Dom }\phi
    Last edited by Drexel28; November 4th 2009 at 09:55 PM.
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