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problems

# problems - Review problems for the final 1 Let f(x = 1(1(x...

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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)?

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