Let S be a bounded set in R and. let a > 0, and let aS := {as : s E S}. Prove that: i. inf(as) = a inf S. a sup S. 11. sup(aS) %3D

Q: (6) Let (X,M.) be a measure space, and {Ej}cM Prove that : (I) If E,SE,S…. ,then j (II) If E,2E2...…

A: The given problem is related with measure theory. Given that X , M , μ is a measure space and Ej1∞ ⊆…

Q: 21. Let A and B be two on-empty bounded sets of positive embers and let С%3D {ху: х€ А and y € B}.…

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Q: Let S be a nonempty subset in R". Prove or disprove by a counterexample that Span S = (S+)+

A: We have to prove or disprove by counterexample that Span S=S⊥⊥

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Q: below. If y = inf S, show that, for each delta> 0, there is s ∈ S such that y ≤ s < y + delta .

A: Adding equation 1 and 2 we get our result.

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}. 1…

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Q: Let S be a nonempty bounded subset of R and let m = sup S. Prove that m is in S iff m = maxS

A: According to the given information, it is required to prove that:

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A: The Chebyshev's Inequality:                    Let X be a random variable. If X has finite mean μ,…

Q: * Let R be with the discrete topology. If A = {1, 3,5, 7, .}, then Aº A O N O RO

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Q: If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.

A: Since S ⊆ T it follows taht inf T ≤ t, ∀ t ∈ T, then  inf T ≤s , ∀ s ∈S:Therefore inf T is  a lower…

Q: Let S be a non-empty bounded subset of R and let a be an upper bound of S. Show hat sup S = a if and…

A: Follow the steps.

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A: Let W be a non-empty subset of ℝ is bounded above, and define the set B as follows, B=Ω∈ℝ: Ω is an…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/xe A}. 1…

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Q: . Let S be a bounded set in R and let T be a nonempty subset of S. Show that inf S <inf T< sup T <…

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Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

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Q: Consider the set S = {z E C : |Z| ≥ 2, Re(z) ≥ −2, Im(z) ≤ 2, Arg(z) ≥ 7 }. Z 109 0 (a) Sketch S on…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

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Q: Consider the particular point topology, PPX, on a set X. Determine Int(A) and Cl(A) for sets A…

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Q: Let A, B Be two subsets af R, such that B is bounded and YaE A ]bEB such that show that: a sb O A is…

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Q: let U and y be bounded sets. if u er i tre M+ (V) ≤M₂ (V) and M^ (U) SM³ (V) 1

A: Given :  U and V are bounded sets with  U⊆ V. To show :  m*(U) ≤ m*V  and  m*U ≤ m*V,  where m*U and…

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Q: . Let E be a measurable set with m(F np with ISp <∞. Further if Lim (E) then lI f llo = %3D

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Q: Let us assume the facts for all a, 6 E R, i.e. as < bi whenever a < b. Let SCR be bounded above and…

A: Given that S~=x∈ℝ|x3∈S And S⊂ℝ is bounded above. We say that S⊂ℝ is bounded above if  and if…

Q: If A and B are bounded subsets of R, then AuRand AOR are bounded: prove that

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Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}. 1…

A: Remark: Completeness property of R: Every bounded above non-empty subset of R has a least upper…

Q: Let E and F be measurable sets each having finite measure. Prove that m(EAF) = 0 if and only if m(E)…

A: To prove: The statement "Let E and F be measurable sets each having finite measure. Then, mE∆F=0 if…

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Q: Suppose the set N is a bounded, open region in R", and that for r > 0, rN := {(rx1,..., rxn) : (x1,…

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Q: 16. If S cT c R, where S # $ , then show that (i) If T is bounded above, then sup (ii) If T is…

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Q: Let A and B be nonempty subsets of R such that A is bounded above and B is bounded below and let A –…

A: Let A and B are two non-empty sets of ℝ.Given A is bounded above and B are bounded below sets.Claim:…

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Q: 23. Let A be a non-empty set of positive numbers, which is ounded above and let B= {xe R: 1/x e A}.…

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Q: X be a limit point of set ECR, then every o infinitely many points of terval containing

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Q: then: Suppose that f:[a,b]→R is bounded , P is parition of [a,b] and Q is refinement of P, L(f;Q) <…

A: It is a mcq type question and not asked about justification so i have provide direct the option.

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Q: A necessary and sufficient condition so that f € R[a, b] is Ve > 0, 3 a partition P. such that U (S,…

A: True

Q: 7. Assume that f:XY is a function from one metric space (X,d,) to another(Y,d,). Prove that f is…

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Q: let E be a set that baunded. Prove that Inf E = Sup -E) where R, XE E y

A: To prove for a buonded set E inf(E) = -sup(-E) , where -E = {-x, x belongs to E} , we first prove…

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

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Q: Assume that SN is a bounded C™ domain ofR° and that u E H™ (N). What is the smallest integer m such…

A: In this question, 5 will be the smallest integer m such that we can guarantee that u is continuous.…

Q: Consider the set R. Prove: (i) VxER, x 0 = 0;

A: To prove: For any x∈ℝ, x·0=0.

Q: Suppose that f:[a,b] R is bounded, P is parition of [a,b] and Q is refinement of P, then: L(f;P)…

A: Correct option :- B Detailed Explanation :

Q: Let A be a non-empty set of positive nunıbers, which is bounded above and let B = {x e R: 1/x e A}.…

A: