Theorem: (i) If Fa is a closed set for each a er then Fa is closed. aer (ii) If Fk is a closed set for k = 1,2,..., n then U Fx is closed. k=1 Proof:Exercise

Q: 1. Let y be a positive real number. Prove that for every n e N there exists a unique positive real…

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Q: Consider R×R to be a universal set with subsets A, B, and C defined as follows. = {(x,y)x* + y° s1}…

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Q: Sketch the set M defined by M = {z € C | Re(z²) 1}.

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Q: Consider RxR to be a universal set with subsets A, B, and C defined as follows. A = {(x,y)\x² +y°…

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Q: Given A = {3, 6, 9, 10, 12} and B = {1, 6, 9, 12}, find each of the following set cardinalities:…

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Q: Show that if A, B E M with m(A), m(B) <∞ then m(AUB) = m(A) +m(B) - m(AB). Show that if A, B, C M…

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Q: F and Hare sets of real numbers defined as follows. F = {ww≤ 4} H = {w|w>5} Write FUH and FnH using…

A: We can find the union and intersection of sets in the below steps.

Q: 3) #36) Hint: {[a,b): a <b} then the Borel o-algebra this complete the pr

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Q: *12. Let D be a nonempty set and suppose that f: D→R and g: D → R. Define the function f +g:D→R by…

A: Solution 12: a: Since f and g are bounded above on set D. ⇒sup f and sup gexist (completeness axiom)…

Q: 9. (a) Prove: If y>0, then there exists ne N such that n-1s y<n. ☆ (b) Prove that the n in part (a)…

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Q: 2. (Problem 4.6) Let A, B be sets. (a) Prove that (b) Prove that P(ANB) P(A) P(B) P(AUB) CP(A) UP(B)…

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