Final Exam Review Sheet:
For the final exam, it would help you to do the following things:
- Learn the definitions you needed to know for the quizzes and
for the midterm exam. Know the definitions precisely. You should
know the statements in formal language, as well as interpretations in
words. E.g., "A is a subset of B" means
and this can be read as "everything in A is
also in B".
- Do "blank-paper practice" for the problems on the
problem sets and the midterm. Note: You should be able to
do all the problems, including the hard ones. Avoid repeating a
mistake you made on the problem set (this is important).
- Understand these challenging concepts (plus the ones from the midterm):
- The definitions of one-to-one and onto.
- The image of a set under a function and how xÎA relates
- The inverse image of a set under a function and how yÎB
relates to xÎf-1(B).
- How f-1(f(A)) relates to A and
f(f-1(B)) relates to B.
- The difference between onto and f(A).
- The difference between f-1(B) (the set) and f-1 (the
- The difference between a|b and b/a.
- The fact that Öx and x2 are not inverses.
- What it means for a function to be well defined on a quotient space.
- In addition to the proofs listed for the midterm, know the proofs of
these key examples. You should not memorize them, but should remember the
central idea(s) and reconstruct the proof from that memorized core.
- If f and g are bijections then so is gof (PS5#5)
- If f and g are invertible then
(gof)-1 = f-1og-1
- f(AÇB) need not equal
f(A) Ç f(B) (PS6#2)
- if f: X -> Y is one-to-one, then
f-1(f(A)) = A for all AÍX (PS6#4)
- ºn on Z, defined by
n|b-a, is an equivalence relation (PS6#5)
||Math 99 (Fall 2001) web pages
Created: 08 Nov 2001
Last modified: 08 Nov 2001 08:44:51