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Math 220–Homework #6 due at the beginning of class Wednesday, March 2, 2005 I. D’Angelo and West, p. 96, #4.21. (Hint: the example in the problem has to do with adding in or removing the integer n from the various subsets.) II. D’Angelo and West, p. 97, #4.25 III. Define the following set A, a subset of R2 : A = {(m, m) : m ∈ Z} ∪ {(n, −n) : n ∈ Z}. Prove that A is countable. IV. (a) Give an example of a set S and a function f : S → S such that f is injective but not surjective. (b) Give an example of a set T and a function g : T → T such that g is surjective but not injective. V. (a) Give an example of two functions f , g : R → R where g ◦ f is injective but g is not injective. (b) Show that if g ◦ f is injective, then f is also injective. √ VI. This problem concerns the two sets A = (0, 1) ∩ Q and B = (0, 2) ∩ Q. (a) What is wrong with the following “proof” that A and B have the same cardinality? “Define a function f : A → B by f ( x) = x + 31 . This function is injective because it is increasing, and it is surjective because given any y, we can input x = y − 13 into the function and obtain the value y. Therefore f is a bijection, so A and B have the same cardinality.” (b) What is wrong with the following “proof” that A√and B have the same cardinality? “Define a function f : A → B by f ( x) = x 2. This function is injective because√it is increasing, and it is surjective because given any y, we can input x = y/ 2 into the function and obtain the value y. Therefore f is a bijection, so A and B have the same cardinality.” (c) Give a correct proof that A and B have the same cardinality. (Hint: there is a way that doesn’t involve finding a specific bijection between the two sets.)