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Math Help - Is this an example of Well-definition?

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    Is this an example of Well-definition?



    I'm just not really sure what well-definition implies. I think I get it, but I could use a little feedback.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by davismj View Post

    I'm just not really sure what well-definition implies. I think I get it, but I could use a little feedback.
    Yes, that is correct. Well-defined means if you pick two things from the same class and multiply (or do any other operation) you will always get the same answer - it doesn't matter which element from the class you pick. So, whenever you see something whose elements are of the form [a] or a+I or aN where I or N are some sort of algebraic structure (for instance, I an ideal or a ring, N a normal subgroup of a group) you would need to take two elements and see if they are always mapped to the same thing.

    For instance, let S=\{0, 1, 2, 3, \ldots, n\} and let T = \{[x]:[a] = [b] \Leftrightarrow n|(a-b)\} be the set of equivalence classes of the integers modulo n, and again let the operation be addition.

    Can you show that \varphi: [x] \mapsto x \text{ mod } n is well-defined?
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