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Then the following are equivalent (1) Bis finite. (2) There is a surjective function from a section of the positive integers onto B. (3) There is an injecdve function from B into a section of the positive integers. Proof f:{1 (2). Since B is nonernpty, there is, for some n, a bijective function (1) n)—÷B. (2) (3). 1ff : n) (1 —÷ B is surjective, defineg . B —÷ (1 n} by the equation g(b) = smallest element of ({b}). Because f is surjective, the set f1{(b)} is nonempty; then the well-ordenng property of Z÷ tells us that g(b) is uniquely defined.

4. (a) Prove by induction that given n E Z÷, every nonempty subset of (1 has a largest element. (b) Explain why you cannot conclude from (a) that every nonempty subset of has a largest element. n} The Integers and the Real Numbers §4 35 5. Prove the following properties of Z and Z÷: (a) a, b E X = {x a+bE I x E JR Z÷. ] and a + x E Z÷ (b) aE a Z÷ U (0). ] C + d E Z and c — d E Z. [Hint. ] (e) 6. Let a E JR. Define inductively a1 =a, a for n E Z÷. ) Show that for n, m E Z÷ and a, b E JR. ] 7. Let a E R and a 0.

Let B be a proper subset of A Then there exists no bijection 0) there does exist a bijection h : B —÷ n}; but (provided B g : B —÷ (1 (1 m}forsomem

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