Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.

Q: Let S = {1,2, .., 30} Let A = {25,26, ...,30} - Determine if A is bounded above and find the upper…

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Q: Suppose that A₁, ..., An are bounded subsets of R. Prove that A₁ U… U An is bounded and that sup(A₁…

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Q: Show that if A is such that m (A)> 0, then there exists a bounded subset BCA such that m'(B) > 0.

A: By properties of Measurable set,         If then  .

Q: 60. Let A be a bounded subset of R. Which of these statements is/are not true about A? II: Vx € A,…

A: Option (B) is correct.

Q: Let A and B be bounded NON empty sets of R, and let A + B := { a+b: at A, bt B3. Prove that supla…

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Q: Suppose that A and B are subsets of R which are nonempty and bounded above, and suppose that BCA.…

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Q: Use the Heine-Borel Property from (5.1), which states that a set of real numbers is compact if and…

A: See solution below. Here we used definition of bounded and closed sets.

Q: A relation < is defined on R? by (x1, x2) < (y1, y2) if and only if x1 < yi and x1 + x2 2 yı + y2.…

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Q: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

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Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

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Q: Consider the set T = {..., −34, −17, 0, 17, 34, ... } with the usual definition of less than (a0).…

A: Completeness property of real numbers:  Every non-empty set of real numbers that has an upper bound…

Q: Q1\ Define the compact set. Then, prove that: if A is a compact subset of F , there exists a best…

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Q: A and B are bounded subsets of R, then prove that UB and AnB are bounded.

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Q: 1.6.5) Given a nonempty set SC R, prove the following statements about suprema. Also formulate and…

A: Note: As per bartleby instruction when more then three questions for subpart is given only three…

Q: Let A= { n e Z :2<n<1 m2 } A relation R on A is defined by aRb if and only if either "a divides b"…

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Q: Let S be a nonempty subset of R that is bounded below. Prove that inf S= -sup{-s:s∊S}

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Q: (b) Let F be a non-empty subset of R. Prove that if OF C F, then F is closed.

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Q: (BOunded st) show that a subset M in a nor med space Xis bounded if and only If there is a positive…

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Q: inf Bs inf A s sup A s sup B.

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Q: (3) X is not the union of two disjoint non-empty separated sets. (4) X is not the union of two…

A: as given all the conditions are equivalent  (4) To show - X is not the union of two disjoint non…

Q: 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: Prove that if R: R -> R is continuous, theis b2n the image of a bounded set over f is bounded

A: As per the question we have to prove that if any function f : ℝ → ℝ is continuous, then the image of…

Q: Prove that if f : R → R is continuous, then the image of a bounded set over f is bounded.

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Q: (i) Show that a subset A of a topological space X is closed if and only A = A (ii) Prove that A…

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Q: b Let A be a continuous fuzzy quantity. If all the a-cuts of A are closed intervals then show that A…

A: In the question provided it is given that A is a continuous fuzzy quantity. The objective is to show…

Q: 3.6 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

Q: Let A = {AC R| A is countable} and A2 = {A C RI A° is countable} which is a measurable space? (IR,…

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Q: Let E₁ and E₂ be open dense subsets of a topological space X. (a) Show that E₁ E2 is dense in X. (b)…

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Q: Decide if the following matrices are reducible or irreducible:

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Q: Let A = {2, 3, 4, . . .} be ordered by “x divides y”. (i.) Determine the minimal elements of A.…

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Q: = (4) Suppose that A is nonempty set that is bounded from above and let B Prove that B is bounded…

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Q: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

Q: Prove that a set T1 is denumerable if and only if there is a bijection from T1 onto a denumerableset…

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Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

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Q: Let A, B be bounded subsets of R. Let A+B = {r+y:1 € A, y E B} and AB = {ry : 1 € A, y € B}. Which…

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Q: If a, <a, then prove that A2A, where denotes a crisp superset (i.e., suppositions) relation.

A: The concept of alpha-cuts (α-cuts) in fuzzy set theory represents the crisp set of all the elements…

Q: 4. Use the least upper bound property of R to prove that every nonempty subset of R that is bounded…

A: Solution

Q: Prove that if f: R → R is continuous, then the image of a bounded set over f is bounded.

A: We need to prove that, if  is continuous, then the image of a bounded set over  is bounded.We know…