Review problems for the final

1. Let f(x) = 1/(1 - (x + x^2 + x^3)). Show that the coefficient of x^n is positive for each n, when we expand f(x) as a power series in x. What's a recurrence relation for the coefficient of x^n?
2. Let X = {1,2,3}, Y = {1,2,3,4,5,6}, and Doub : X -> Y be the doubling function. (1->2, 2->4, 3->6)

Can you find two functions f,g from Y -> X, not equal, such that f o Doub = g o Doub? (These are functions X->X.)

Can you find two functions f,g from Y -> X, not equal, such that Doub o f = Doub o g? (These are functions Y->Y.)

If you can, give an explicit example of an f and a g.

3. Let h(x) = f(1/x), where f(x) is the function from question #1.
  • Is h(x) = 1 + O(1)?
  • Is h(x) = 1 + O(1/x)?
  • Is h(x) = 1 + O(1/x^2)?
    • 4. Let X be the numbers 1...m, Y the numbers 1...n. How many functions f are there from X to Y such that
  • 1+1=2? (I.e. how many functions, period?)
  • f is "strongly order-preserving", so i less than j implies f(i) less than f(j)?
  • f is "weakly order-preserving", so i less than j implies f(i) less than or equal to f(j)?
  • f is 1:1 and strongly order-preserving?
  • f is 1:1 and not strongly order-preserving?
  • f is strongly order-preserving and not 1:1?
  • f is weakly order-preserving and not 1:1?
    • 5. Same X and Y as in #4, but now also have Z, with p elements, where p is prime. Given 1:1 functions f:X->Y and g:Y->Z, we get a function h = g o f from X to Z.

    If you're only told h, how many possibilities are there for f? How many for g? How many for the pair (f,g)?

    6. Let P have n elements, Q have n-2. How many functions are there from P onto Q?
    7. Prove that every even prime is the sum of two odd numbers.
    8. Let p,q be prime, p less than or equal to q.
  • If q+p is prime, what are all the possibilities for p?
  • If q+p,q-p are both prime, what are all the possibilities for q?
    • 9. Let p,q be prime, m and n be nonnegative integers. If p^m = q^n + 1, what can you say about p and q? (Not a very well-defined question -- nothing this vague will be on the final -- but you should still think about it.)

    For example, 3^2 = 2^3 + 1 since 9 = 8+1.