Does there exist a non-empty open set A and two elements in A that are farthest apart (among all pairs of

Q: Prove that a subset of a set of measure zero has measure zero.

A:

Q: Let A be a set and let B be a proper subset of A. Is it possible for B to have the same cardinality…

A: Given: Let A be a set.And B be a proper subset of A.

Q: Show for every nonempty, finite set E that supE = maxE.

A: Given, E is a finite set. E is a non-empty set. To prove, supE = maxE We know that, sup is supremum…

Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not exactly?

A: No, there is no set A that satisfies A={A}. Let A is an empty set, i.e.  A=  or  A=ϕ, in which…

Q: Prove that the smallest member of a set, if exits, is the greatest lower bound of the set.

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Q: (2) Show that if f has support in a compact set A CR and g has support in a compact set B C R then f…

A:

Q: Prove that the set of all finite subsets of N is countable infinite.

A: We have to prove that the set of all finite subsets of N is countable infinite. Since,…

Q: 5. Suppose A, B are countably infinite sets, meaning that there exists one-to-one corre- spondences…

A:

Q: Suppose a set A is uncountably infinite, that U is countably infinite, and that UC A. Show that A -…

A:

Q: 2. Show that any subset of a countable set is countable. 3. Show that a nonempty set S is countable…

A: Hence any subset of a countable set is countable.

Q: 6. Every finite set in R is compact. Why?

A: The given question is related to real analysis. We have to show that every finite set in ℝ is…

Q: Show that every open set with respect to 7 is a union of open ball(s).

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Q: Prove that: m*(A) 2 0, for all sets A

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Q: Choose any that is a subset of the set (Ø} {{Ø),0} {{1}}

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Q: Exhibit a countable collection of open sets U; such that each open set OCR can be written as a union…

A: Given :- Uj is countable collections of open sets. To prove :Every open set O⊆ℝ can be written as a…

Q: Prove that empty set is subset of all sets: φ ⊆ A

A: Given: An empty set and set A is given. To Find: To prove that empty set is subset of all sets: ϕ⊆A

Q: Show that the o-algebra B of Borel sets contains all F,-sets.

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Q: 4. Let A be an open set in å nörmed space subset B of X .

A: Que  Ans :  Given ( X, ||-||) be a norm linear space,  A be any open subset of X    ( def- set A is…

Q: Is the Cantor set closed? Is it compact? Explain!

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Q: Let X be an n-element set. Prove that there are 3n ordered pairs (A, B) of subsets of X such that A…

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Q: Prove that A is an infinite set if and only if ∀n ∈ ω, A has a subset B such that B ≈ n.

A: given that A is an infinite set let B be any set let prove this contradiction suppose assume that B…

Q: uppose that ℱ is a non-empty family of sets, B is a set, and ∀A ∈ ℱ (A ⊆ B). Is ∪ℱ ⊆ B? Either…

A: Given: To explain the given statement as follows,

Q: Prove that the greatest member of a set, if exits, is the least upper bound of the set.

A:

Q: Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not ?

A:

Q: Prove that every infinite set S contains a proper subset similar to S.

A: We have to prove that every infinite set S contains a proper subset similar to S. Let X=S Let X be…

Q: List all the possible subsets of set G, where G = {-7,m,z}- %3D

A:

Q: prove that empty set is a subset of every set

A: We have to prove that empty set is a subset of every set = ?

Q: 5.3.8 Prove that if A CR is a finite set, then A is well-ordered.

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Q: Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not exactly

A: Set equality

Q: Give an example of sets {An}n≥1 such that for each n ≥ 1, the set An ⊂ [0, 1] is both compact and…

A: Construct sets Ann≥1 such that for each n≥1, the set An⊂0, 1 is both compact and connected, but the…

Q: Prove that every nonempty subset of a well-ordered set is well-ordered.

A: Given problem is :

Q: (4) Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf…

A:

Q: Show that for any nonempty set A, there exists unique set B such that for any x. xEB ex belongs to…

A: From your question, it is clear that "A" is a set of sets, i.e., members of the set "A" are sets.

Q: Give an example of 4 non-empty sets, A, B, C and D such that all the following are true: A ∩ B ̸ =…

A: We have to give an example of 4 non-empty sets A, B, C, D such that all the following are…

Q: Let A be a well-ordered set, then every subset of A is not well-ordered

A:

Q: Prove directly that the union of two disjoint sets, where one is countably infinite and the other is…

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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: sets are subsets of a un

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Q: (4) How many distinct pairs of disjoint non-empty subsets of A are there, the union of which is all…

A:

Q: Prove that every set with positive Lebesgue measure contain a non-measurable set

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Q: The cardinality of a finite set A, denoted by Al, can provide the of A as equal to 21A

A:

Q: Determine whether each of the following statements is true or false, and explain why.A set is a…

A: A set A is subset of another set B if all elements of the set A are elements of the set B.

Q: Prove that if A is a transitive set then UA is a transitive set.

A: If A is a transitive set then, It is proved that ∪A is a transitive set as shown below.

Q: Show that every closed subset F of R' is the intersection of a countable col- lection of open sets.

A: detailed explanation mentioned below.

Q: 2. Prove that the union of any finite collection of closed sets in RP is a closed set in RP.

A: A set A is closed if and only if every limit point of A belongs to A. Now we can prove the result.…

Q: Let A1, · · · , An be countably infinite sets. Prove that A1 × A2 × · · · × An is countably…

A:

Q: For any collection a of subsets of X, there is ntaining a.

A:

Q: Consider the set X={a,b,c,d} construct all the possible topologies of X that have only three open…

A: Given: The set X=a,b,c,d To find:  all the possible topologies of X that has only three open…

Q: - A non-empty set E in R is connected if and only if it is an interval

A: