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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

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

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.4: Binary Operations
Problem 9E: 9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of...
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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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