# Well-defined

One of the most common places in mathematics in which the term well-defined is used is in dealing with cosets in group theory. It is as important that we check that we get the same result regardless of which representative of the coset we choose as it is that we always get the same result when we perform arithmetical operations (e.g., that it never happens that $\displaystyle 2+3\neq5$ ).
More generally, given a set X, an equivalence relation ~ on X, and a function f on X, one may be interested to know whether f can be viewed as a function on the quotient set X/~. That is to say, if [x] is an equivalence class in X/~, then one may attempt to define f([x]) = f(x). If the function satisfies f(x)=f(y) whenever x~y, then the definition makes sense, and f is well-defined on X/~. Although the distinction is often ignored, the function on X/~, having a different domain, should be viewed as a distinct map $\displaystyle \tilde{f}$ . In this view, one says that $\displaystyle \tilde{f}$ is well-defined if the diagram shown commutes. That is, that f factors through π, where π is the canonical projection map XX/~, so that $\displaystyle f=\tilde{f}\pi$ .