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Math Help - Set to the power of another Set?

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    Set to the power of another Set?

    How do you take a set to the the power to another set? For example, if A = {0, 1, 2} and B = {a, b}, then what is A^B?
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    Quote Originally Posted by TheMute View Post
    How do you take a set to the the power to another set? For example, if A = {0, 1, 2} and B = {a, b}, then what is A^B?
    This is very standard notation. It is surely in your textbook and/or lecture notes.
    The A^B notation is the set of all functions from the set B to the set A.
    That means that if f\in A^B then f:B\to A.
    If |X| stands for the cardinally of set X then \left| {A^B } \right| = \left| A \right|^{\left| B \right|} .
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    MHF Contributor undefined's Avatar
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    Quote Originally Posted by TheMute View Post
    How do you take a set to the the power to another set? For example, if A = {0, 1, 2} and B = {a, b}, then what is A^B?
    A^B denotes the set of all functions from B to A. These are

    {
    {(a,0), (b,0)}, {(a,0), (b,1)}, {(a,0), (b,2)},
    {(a,1), (b,0)}, {(a,1), (b,1)}, {(a,1), (b,2)},
    {(a,2), (b,0)}, {(a,2), (b,1)}, {(a,2), (b,2)}
    }

    Note that \left| A^B \right| = |A|^{|B|}
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by undefined View Post

    Note that \left| A^B \right| = |A|^{|B|}
    One needs to be careful with that. For finite sets this makes sense, but for infinite sets we actually define \alpha^\beta=\text{card }A^B where \text{card }A=\alpha,\text{card }B=\beta and so the logic is a little backwards.
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