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)?