Inducing a well-defined function on a set What does it mean to say that $f$ induces a well-defined function on the set $X$?
I'm confused about what the term induce means here, and what role the set 
$X$ has.

It means that a function is such that we can define a(nother) well defined function on some set $\,X\,$ that'll depend, in some definite way, on the original function.
For example: if $\,f:G\to H\,$ is a group homomorphism and there's some group $\,N\leq \ker f\,$ , with $\,N\triangleleft G\,$ , then $\,f\,$ induces a well-defined group homomorphism 
$$\overline f:X:=G/N\to H\,\,,\,\,\text{defined by}\,\,\overline f(gN):=f(g)$$
Please do note that the original function's domain is $\,G\,$ whereas the induced function's domain is $\,G/N\,$ . These two domains are both different sets and different groups.
