Definition of "well defined" in mathematics I have encountered this term "well defined" in many places in maths like well-defined set, well-defined function, well-defined group, etc. What are the contexts in which we can talk about well definedness and what does it mean in each context?

The term well-defined (as oppsed to simply defined) is typically used when a definition seemingly depends on a choice, but in the end does not.
In most (but not all) cases, this applies to the definition of a function $f\colon A\to B$ in terms of two given functions $g\colon C\to A$ and $h\colon C\to B$: For $a\in A$ we want to define $f(a)$ by first picking an element $c\in C$ with $g(c)=a$ and then let $f(a)=h(c)$. Two problems arise with this: First of all, we must make sure that for each $a\in A$ there exists $c\in C$ with $g(c)=a$, in other words: $g$ must be surjective. But we also must make sure that the choice of $c$ is irrelevant, that is: Whenever $g(c)=g(c')$ it must also be true that $h(c)=h(c')$. Only if $g,h$ fulfil these conditions the above construction will actually define a function $f\colon A\to B$. To repeat: After this, $f$ is in fact defined. The definition itself does not become a "better" definition by saying that $f$ is well-defined. Instead, saying that $f$ is well-defined just states the (hopefully provable) fact that the conditions described above hold for $g,h$, and so we really have given a definition of $f$ this way. If the conditions don't hold, $f$ is not somehow "less well defined", it is not defined at all.
