So my answer is "A map is considered to be well-defined if the operation does not depend on choice of operation. i.e. $\phi: \mathbb{Z}_6 \rightarrow \mathbb{Z}$ where $\phi([n])=n.$ This is not well defined because $\phi([4])=4$ and $\phi([10])=10$ but [4]=[10]." Do you think this is a good way of describing well defined?
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$\begingroup$ Yeah it looks right to me. You showed that elements from the same equivalent class gives you something different. $\endgroup$– LemonCommented Dec 4, 2014 at 0:50
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$\begingroup$ It boils down to showing that the relation is acually a function. $\endgroup$– copper.hatCommented Dec 4, 2014 at 1:07
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You have the right idea, though your use of the word "operation" doesn't quite make sense. You could just as well simplify the statement to
A map is considered to be well-defined if it does not depend on a choice.
(Note also that you misused "i.e.". You should have instead used "e.g.".)
However, it's actually quite difficult to explain "well-defined" since it's kind of a bad choice of terminology. Timothy Gowers has an excellent blog post, Why aren’t all functions well-defined?, that goes into this further.