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problems-2-224-S08

# problems-2-224-S08 - MATH 224 PROBLEM SET 2 DUE 5 FEBRUARY...

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Unformatted text preview: MATH 224 PROBLEM SET 2 DUE: 5 FEBRUARY 2008 When you hand in this problem set, please indicate on the top of the front page how much time it took you to complete. Reading. § 4.3–4.5. Problems from the book: • 4.3.1, 4.3.4 • 4.4.3 • 4.5.2, 4.5.4 Additional problems: 1. Define f : [ 0, 1 ] → R by f ( x ) = braceleftbigg sin ( 1 x ) x > 0 x = . Is f integrable on [ 0, 1 ] ? 2. Let Q denote the set of rational numbers. a. For α, β ∈ R , we say α ∼ β if α- β ∈ Q . Show that ∼ is an equivalence relation. b. For any α ∈ R , let X α = { β ∈ R | α ∼ β } be the equivalence class of α . Show r ∈ Q implies that X α = X α + r . c. For each distinct X α , choose a representative in [ 0, 1 ) , and let Y denote the set of representatives (this requires the Axiom of Choice ...). For z ∈ R , define it’s reduction mod 1 to be the unique number x z ∈ [ 0, 1 ) such that z- x z ∈ Z . For X ⊆ [ 0, 1 ) , and y ∈ R , we define X ˜ + y to be the set of reductions mod 1 of all...
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problems-2-224-S08 - MATH 224 PROBLEM SET 2 DUE 5 FEBRUARY...

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