# Thread: Is this an example of Well-definition?

1. ## 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.

2. Originally Posted by davismj

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?