klarfeld-sec-2-3-2-4 - Jennifer Klarfeld Homework 6 Sec 2.3...

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Jennifer Klarfeld, Homework 6, Sec 2.3 and Sec 2.4 1 Section 2.3 1(j). Find the union of the intersection of: ℳ = ࠵?ℤ ∶ ࠵? ∈ ℕ , ࠵?࠵?࠵?࠵?࠵? ࠵?ℤ = … − ࠵?࠵?, −࠵?, ࠵?, ࠵?, ࠵?࠵? … ࠵? = ℤ 3 ࠵?4࠵? ࠵? 3 ࠵?4࠵? = {࠵?}
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Jennifer Klarfeld, Homework 6, Sec 2.3 and Sec 2.4 2 3(a). Prove part (b) of Theorem 2.3.1: For every set B in the family ࠵?, ࠵? ⊆ ࠵?. ࠵?∈࠵? ࠵?࠵?࠵? ࠵? ࠵?࠵? ࠵? ࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵? ࠵?࠵?࠵? ࠵? ∈ ࠵?. ࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵? ∈ ࠵?. ࠵?࠵? ࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵? ࠵?࠵? ࠵? ࠵?࠵?࠵?࠵?࠵?࠵?, ࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵? ࠵? ࠵?࠵?࠵? ࠵? ∈ ࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵? ∈ ࠵? ࠵?࠵? ࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵? ࠵? ∈ ࠵?. ࠵?∈࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?, ࠵? ⊆ ࠵?. ࠵?∈࠵?
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Jennifer Klarfeld, Homework 6, Sec 2.3 and Sec 2.4 3 3(b). Prove part (b) of Theorem 2.3.2: If ࠵? ⊆ ࠵? ࠵?࠵?࠵? ࠵?࠵?࠵? ࠵? ∈ ࠵?, ࠵?࠵?࠵?࠵? ࠵? ࠵?∈࠵? ⊆ ࠵?. ࠵?࠵?࠵? ࠵? ⊆ ࠵? ࠵?࠵?࠵? ࠵?࠵?࠵? ࠵? ∈ ࠵?. ࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵? ∈ ࠵?. ࠵?∈࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵? ࠵? ࠵?࠵?࠵? ࠵? ∈ ࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵? ∈ ࠵?. ࠵?࠵?࠵?࠵?࠵? ࠵? ⊆ ࠵?, ࠵? ∈ ࠵?. ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?, ࠵? ࠵?∈࠵? ⊆ ࠵?.
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Jennifer Klarfeld, Homework 6, Sec 2.3 and Sec 2.4 4 5. Let the universe be , and let ࠵? be the empty family of subsets of ℝ. Show that ࠵? ⊆ ࠵? ࠵?∈࠵? ࠵?∈࠵? is false in this case by proving: (a) ࠵? ࠵?∈࠵? = ℝ ࠵?࠵?࠵? ࠵? ∈ ࠵? ࠵?∈࠵? . ࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?, ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵? ࠵? ∈ ࠵?, ࠵?࠵?࠵?࠵? ࠵? ∈ ࠵?. ࠵?࠵?࠵?࠵?࠵? ࠵? ࠵?࠵? ࠵?࠵?࠵?࠵?࠵?, ࠵?࠵?࠵?࠵? ࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?࠵?࠵?࠵?࠵?, ࠵?࠵?࠵?࠵?, ࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?. ࠵?࠵?, ࠵?࠵? ࠵? ∈ ࠵?, ࠵?࠵?࠵?࠵? ࠵? ∈ ℝ ࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵? ࠵? ࠵?∈࠵? ⊆ ℝ. ࠵?࠵?࠵? ࠵?࠵?࠵? ࠵? ∈ ℝ. ࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵? ࠵? ∈ ℝ, ࠵?࠵?࠵?࠵? ࠵? ∈ ࠵? ࠵?∈࠵? . ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?, ࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?, ࠵? ∈ ࠵? ࠵?∈࠵?
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