M + N = {m+n : m € M and n € N}. Prove that for all n € N, the number sup(M + N) − n is an upper bound for M. -

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: show that the Cantor set is a closed subset of R and has measure zero.

A: We know the definition of Cantor set, The Cantor set is defined as the intersection C = ∩n=1∞Cn ,…

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: on as 3. Let A and B be bounded subsets of IR such that for each a₂ EA, there exists a b1 EB…

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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: let A be the subset of defined by A={1-((-1)/n) : n€N}. Find all cluster points of A and justify…

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Q: Determine which of the following subsets of R are well-ordered

A: A well-ordered set is a partially ordered set in which every non-empty subset has a least…

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

Q: or which compact sets A does there exist an m e N such that every open cover of A contains a bcover…

A: Please check the answer in next step

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

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Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

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Q: Let A be a denumer able set. Prove that A has a denumer able subset B such that A - B is…

A: Let A =set of multiples of 4, B=set of multiples of 3. Both are denumerable. Intersection of A and B…

Q: Consider the measurable space (N, F, P) and a random variable X defined on it. Show that the…

A: The objective of this question is to prove that the collection of subsets {X^-1(B): B ∈ B} forms a…

Q: Let A be the set of letters in the English alphabet with the usual order. Define ⟨A2, ≤⟩ such that…

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Q: Let A= {1, 2, ...10}. Give an example of a partition P of A such that |P|=3

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Q: Let D be the set of letters in the English alphabet with the usual order. Define ⟨D² , ≤⟩ such that…

A: Given D is a set of letters in English alphabet and the order in the set is considered as the usual…

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

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Q: Show that if any subset E of H is convex then x+E is also convex.

A: The objective of the question is to prove that if a subset E of a set H is convex, then the set…

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

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Q: Show that the countable collection B = {(a,b) × (c, d)| a <b and c< d,

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Q: S Property, B be a

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Q: Consider the following sets. A = {0, 1}5 A₁ = {20 : x € {0, 1}4} A₂ = {201 : x ≤ {0,1}³} A3 = {2011…

A: A1 = {x0: x∈ {0,1}4}A2 = {x01 : x ∈ {0,1}3}A3 = {x011 : x ∈ {0,1}2}For the above sets to form a…

Q: Suppose we take our universe to be the set U = { 1, 2, 3, 4, 5, . . . , 20 }, and set A = { x | x ∈…

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

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Q: 1 1 2 Let A = 3 1 0 -4 1 1.1 Find a basis B1 for im(A), and a basis B2 for null(A). 1.2 Explain…

A: The answer of given problem is given as follows :

Q: Let < be a partially order on a set A. Show that there exists a linear order 3 on A such that if a…

A: Given that ≤ is a partial order on a set A. The objective is to show that there exists a linear…

Q: Suppose that S is a nonempty subset of R and k is an upper bound of S. The number k = sup S if and…

A: Given: S is a non-empty subset of R and k is an upper bound of S. To prove : Number k=sup S if and…

Q: Let A, and Az be two subsets of R. Show that m. (A, x A2) S m*(A,) m*(A2).

A: To prove: m*(A1 * A2) ≤ m*(A1) m*(A2)

Q: Let S be a non-empty open subset of R that is bounded above. Show that sup S exists but that max S…

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Q: B) Show that every countable discrete family {F} of finite subsets of T3, then there exists a family…

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Q: Let S be an ordered set. Let A ⊂ S be a nonempty susbset that is bounded above. Suppose A exist and…

A: Let x1 ∈ A ( that's because A is nonempty, so at least one suchx1 exists)

Q: Let F be an event space. Show that F is closed under countable intersections.

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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: Consider the set A = {xy : x ∈ N, y ∈ N} = {1, 2, 22, 23, 24, ...., 3, 32, 33, 34,......, 4, 42, 43,…

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