Math 99 (Notes)

Final Exam Review Sheet:

For the final exam, it would help you to do the following things:

1. 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 "("x)(xÎA Þ xÎB)" and this can be read as "everything in A is also in B".

2. 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).

3. Understand these challenging concepts (plus the ones from the midterm):

1. The definitions of one-to-one and onto.
2. The image of a set under a function and how xÎA relates to yÎf(A).
3. The inverse image of a set under a function and how yÎB relates to xÎf-1(B).
4. How f-1(f(A)) relates to A and f(f-1(B)) relates to B.
5. The difference between onto and f(A).
6. The difference between f-1(B) (the set) and f-1 (the function).
7. The difference between a|b and b/a.
8. The fact that Öx and x2 are not inverses.
9. What it means for a function to be well defined on a quotient space.

4. 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.

1. If f and g are bijections then so is gof (PS5#5)
2. If f and g are invertible then (gof)-1 = f-1og-1 (PS5#5)
3. f(AÇB) need not equal f(A) Ç f(B) (PS6#2)
4. if f: X -> Y is one-to-one, then f-1(f(A)) = A for all AÍX (PS6#4)
5. ºn on Z, defined by aºnb iff 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 Comments to: `dpvc@union.edu`