Final Exam Review Sheet:

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

1. Learn the definitions you needed to know for the quiz 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". Also learn the definitions of:
   1. The official definition of a function
   2. Graph(f)
   3. f restricted to X
   4. a | b
   5. p is prime
   6. x is rational, x is irrational

   
   

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 image of a set under a function and how x being in X relates to f(x) being in f(X).
   2. The inverse image of a set under a function and how y being in Y relates to x being in f^-1(Y).
   3. How f^-1(f(X)) relates to X and f(f^-1(Y)) relates to Y.
   4. The definitions of one-to-one and onto.
   5. The difference between f^-1(Y) (the set) and f^-1 (the function).
   6. The difference between a|b and b/a.
   7. Mathematical induction.

   
   

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. f: A ® B is one-to-one iff f^-1(f(X)) = X for all X Í A (theorem 4.16)

   2. f: A ® B is onto iff f(f^-1(Y)) = Y for all Y Í B (PS7#4a)

   3. f: A ® B is one-to-one iff f(X) Ç f(Y) = f(XÇY) for all X,Y Í A (PS7#4b)

   4. If f and g are bijections then so is g o f (theorem 4.20)

   5. If f: A ® B and g: B ® C are bijections then (g o f)^-1 = f^-1 o g^-1 (theorem 4.24)

   6. There are an infinite number of primes (theorem 6.7)

   7. If n² is even then so is n (lemma 6.10)

   8. The square root of 2 is irrational (theorem 6.11)