James' Homework 3

1. I think you have made it a little more complicated: f(x) = 2x suffices as a bijection from N to evens and f(x) = 2x+1 from N to odds. 2. Good, on the reals, x^2 is the usual example of a non-injective function as well as a non-surjective one. 3. Good! 4. Close: the problem with this method is that you cannot discern permutations (e.g, ABC cannot be discerned from BAC or CAB) so injectivity does not work. But you can remedy this: instead of associating each letter to a prime, you can associate the i-th position with the i-th prime and have the letter correspond to the power of that prime. Then it will work. 5, 6. Good. 7.1, 7.3 Good! Sorry to make you do these: every math person must fill out a multiplication table at least once in their life. 7.5. You have the right idea, but you need to make this a little more rigorous. Check out Julian's solution of this problem. 7.8. GREAT! Good choice of homomorphism. We also already know this function is a bijection on (0, infinity)