**Maintainer:**admin

This was the last lecture of the semester. In this lecture, Professor Darmon answered some questions about the final, and went over some of the questions in the practice final.

*1*Review session¶

Victoria de Quehen, one of the TAs for this course, is giving a review session tonight (Thursday, December 6) from 7pm-9pm in Burnside 1205.

*2*General questions¶

*2.1*Well-defined functions¶

When do we need to show that a function is well-defined?

We only really need to do this when the domain is a quotient, with an equivalence relation dividing it into equivalence classes. If we create a function that relies on the label used for an equivalence class, then we might run into trouble, because we could have cases where $[a] = [b]$ but our function $g$ assigns different values to $a$ and $b$ (with $g(a) \neq g(b)$).

Also, the TA who marked my assignment wrote this: "Please tell people you know in this class the only way a mapping can not be well-defined is if you map a class to an element. In this case you need to check $\phi$ doesn't depend on choice of representative."

For more information on this, take a look at this blog post: Why aren't all function well-defined?

*3*Practice final¶

Some of the questions were explained in class. See this page for student-provided solutions. Darmon has indicated that he will post official solutions at some point, perhaps Sunday night, by popular request.