The net effect of factors that are powers of two is

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divided in half three times, and so on. The net effect of factors that are powers of two is to defeat the simple argument that f grows “on average”.
40 CHAPTER 2. BASIC SET THEORY Problem 2.46 True or false (and explain): The function f ( x ) = x - 1 x +1 is a bijection from the real num- bers to the real numbers. Problem 2.47 Find a function that is an injection of the integers into the even integers that does not appear in any of the examples in this chapter. Problem 2.48 Suppose that B A and that there exists a bijection f : A B . What may be reasonably deduced about the set A ? Problem 2.49 Suppose that A and B are finite sets. Prove that | A × B | = | A | · | B | . Problem 2.50 Suppose that we define h : N N as follows. If n is even then h ( n ) = n/ 2 but if n is odd then h ( n ) = 3 n +1 . Determine if h is a (i) surjection or (ii) injection. Problem 2.51 Prove proposition 2.6. Problem 2.52 Prove or disprove: the composition of injections is an injection. Problem 2.53 Prove or disprove: the composition of surjections is a surjection. Problem 2.54 Prove proposition 2.7. Problem 2.55 List all permutations of X = { 1 , 2 , 3 , 4 } using one-line notation. Problem 2.56 Suppose that X is a set and that f , g , and h are permutations of X . Prove that the equa- tion f g = h has a solution g for any given permu- tations f and h . Problem 2.57 Examine the permutation f of Q = { a, b, c, d, e } which is bcaed in one line notation. If we create the series f, f f, f ( f f ) , . . . does the identity function, abcde , ever appear in the series? If so, what is its first appearance? If not, why not? Problem 2.58 If f is a permutation of a finite set, prove that the sequence f, f f, f ( f f ) , . . . must contain repeated elements. Problem 2.59 Suppose that X and Y are finite sets and that | X | = | Y | = n . Prove that there are n ! bijections of X with Y . Problem 2.60 Suppose that X and Y are sets with | X | = n , | Y | = m . Count the number of functions from X to Y . Problem 2.61 Suppose that X and Y are sets with | X | = n , | Y | = m for m > n . Count the number of injections of X into Y . Problem 2.62 For a finite set S with a subset T prove that the permutations of S that have all mem- bers of T as fixed points form a set that is closed under functional composition. Problem 2.63 Compute the number of permuta- tions of a set S with n members that fix at least m < n points. Problem 2.64 Using any technique at all, estimate the fraction of permutations of an n -element set that have no fixed points. This problem is intended as an exploration. Problem 2.65 Let X be a finite set with | X | = n . Let C = X × X . How many subsets of C have the property that every element of X appears once as a first coordinate of some ordered pair and once as a second coordinate of some ordered pair? Problem 2.66 An alternate version of Sigma ( ) and Pi ( producttext ) notation works by using a set as an index. So if S = { 1 , 3 , 5 , 7 } then summationdisplay s S s = 16 and productdisplay s S s = 105 Given all the material so far, give and defend rea- sonable values for the sum and product of an empty set.

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