3. Let A = {a} be a bounded subset of R. Which of these three statements about A is/are not true? I: max A does not exist. II: sup A = infA Ш: min A < max A A I and II only B. IlI only CI and III only D. I, II and III E None of the above. 4.

Q: 1.* Let U C R? be an open and bounded set and u e C²(U) n C(U) satisfy -1 Ди(х) : in U. 1 + |x|

A: Since I have solved similar ones, Let us show that u attains its minimum on the boundary

Q: 5. Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E) %3D

A: This is a problem of Real topology, metric topology.

Q: 2. Consider the set A = { n+m² n2+m m eN}. : п, тEN (a) Prove that sup(A) = ∞. (b) Prove that A is…

A: Set and infimum value

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Q: 1.4 Determine whether the following fuzzy sets are convex or not. a) A- Ju,(x)/x where ,(x) =1(1 +…

A: (a) A=∫μA(x)/x  where μA(x)=11+x2, -∞&lt;x&lt;∞ It is clear that z=λx+(1-λ)y, 0≤λ≤1 For x,y ∈ℝ…

Q: Show that if A is closed in Y and Y is closed in X, then A is closed in X.

A:

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:

Q: 2.28 let {Ei} i Ej be a Measurable pairwise dis joint sets. prove that for any bounded set A…

A:

Q: [} [2] » 21. Let u = and v = Show that is in k Span {u, v} for all h and k.

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Q: O Show that if Si and S2 are convex sets in R™X", then so is t mxn S = {(x, y1 + y2) | x € R", y1,…

A: Convex set: , If a and b are points in a vector space S. Then the vector space is called convex set…

Q: Prove that ifS is a bounded set, then inf(S) = -sup(-S) %3D

A:

Q: 3. Let A = {a} be a bounded subset of R. Which of these three statements about A is'are not true? I:…

A: Given set is A={a} which is bounded subset of R

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A: Follow the steps.

Q: 21. Let A and B be two non-empty bounded sets of positive numbers and let C = {xy : x e A and y e…

A:

Q: 2. Let S is a subset of R. Prove that S is closed if and only if S contains all of its accumula-…

A:

Q: 5. Show that if A and B are bounded subsets of R, then A U B is a bounded set. Show that sup(A U B)…

A:

Q: . Let A CQ satisfying the following conditions: (a) A + Ø and A # Q. (b) If p E A and q E Q with q <…

A:

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

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Q: Prove that if S is a bounded set then S is bounded.

A:

Q: Let M and N be nonempty bounded subsets of R and let M + N := {m + n : m ∈ M and n ∈ N} . Prove that…

A:

Q: 2) Let A be a finite set and a EA.Lf a EA is not a maximal element of A, then a has an îmmediate SUC…

A: Given A is a finite set. a∈A is not a maximal element of A. Now we have to prove 'a' has an…

Q: Q3. Let N = {1, 2, 3, ..} equipped with the cofinite topology. Prove that M = {7,14, 21} is a closed…

A:

Q: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

Q: 6. Let A and B be bounded nonempty subsets of R, and let A+ B:= {a +b: a e A, be B). Prove that…

A:

Q: Let A be a nonempty subset of R that is bounded below. Define the set B = {b: b is a lower bound of…

A:

Q: 7. Let A be a bounded subset of R and let sup A = B. Which of the following is/are not an attribute…

A:

Q: 2.6.4 Let A CR be a non – empty set. Suppose that A is bounded above. Let U = {x E R|x is an upper…

A:

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

A:

Q: 2.12 let A and B bounded sets For which there is an ε >o such that la-blze for all ac A, be B .…

A: Consider A and B are bounded sets.

Q: Q3. Let N = {1, 2, 3, ...} equipped with the cofinite topology. Prove that M = {7,14, 21} is a…

A: Given below the detailed solution

Q: a. Show that the null-space N(A) is a convex set. Is it true for every A?

A: As both questions are of different type,we shall solve first Que only. For other questions kindly…

Q: 19. If CI(A) denotes the closure of A, which of the following is not correct in general for a set A:…

A: Definition: Let X be a topological space and let A⊆X. The interior of A is the set IntA=∪V | V⊆A…

Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

A:

Q: 5. Suppose that A and B are bounded subsets of R. Prove that AUB is bounded and that sup(AUB) =…

A: Given that A and B are bounded subsets of R.   We have to Prove that AUB is bounded and that…

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A:

Q: 22. Let A be a bounded subset of R and let sup A = B. Which of the following is/are not an attribute…

A:

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A:

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A:

Q: Prove or disprove: All uncountable measurable subsets of ℝ have positive measure.

A: All uncountable measurable subsets of ℝ have positive measure.

Q: Let x= sup (A) and if a E A, with x# a then which of the following statements is true? There is rEA…

A:

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Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤…

A:

Q: *7. Let S be a nonempty bounded subset of R and let ke R. Define kS = {ks : se S}. Prove the…

A: Note: We'll answer the first question since the exact one wasn't specified. Please submit a new…

Q: Prove that if A and B are disjoint denumerable sets, then A UB is denumerable.

A:

Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup up A exist? Show that (a) inf…

A:

Q: 3. If is a collection of dosed sets in R such that every finite subcollection has a nonempty…

A: Given : θ is a collection of closed sets in ℝ such that every finite subcollection has a non empty…

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A: Explanation of the answer is as follows

Q: Q Let X be an Infinite sel with topdogy ?. Prove or disprove : (a) If every mfinite Suhsel of x i…

A: