37. For a set A, let P(A) be the set of all subsets of A. Prove that A is not equivalent to P(A). (Hint: Suppose f: A→ P(A) and define C = (x:x EA and x # f(x)}, Show Cim fil

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set of algebraic numbers is a countable set.
37. For a set A, let P(A) be the set of all subsets of A. Prove that A is not equivalent to P(A).
(Hint: Suppose f: A → P(A) and define C =
38. Let a, b, c, and d be any real numbers such that a < b and c < d. Prove that [a, b] is
{(x:x EA and x # f(x)}. Show C# im f.)
equivalent to [c, d]. (Hint: Show that [a, b] is equivalent to [0, 1] first.)
0.5 REAL NUMBERS
Transcribed Image Text:set of algebraic numbers is a countable set. 37. For a set A, let P(A) be the set of all subsets of A. Prove that A is not equivalent to P(A). (Hint: Suppose f: A → P(A) and define C = 38. Let a, b, c, and d be any real numbers such that a < b and c < d. Prove that [a, b] is {(x:x EA and x # f(x)}. Show C# im f.) equivalent to [c, d]. (Hint: Show that [a, b] is equivalent to [0, 1] first.) 0.5 REAL NUMBERS
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