(A'n B) = A' U B'.

Q: j) A U A = A k) A U $ = A

A:

Q: If A and B are bounded subsets of R, then prove that AUB and AB are bounded.

A:

Q: Suppose that E1 and E2 are measurable sets. Prove that m(E1 U E2) + m(E n E2) = m(E1) + m(E2).

A: We have to prove that the given statement.

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F open?

A: A set is open if and only if it contains all of its limit points. Also recall that the union of open…

Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…

A:

Q: [3.12] Let S be a nonempty convex set in R", and let f. R" → R be defined as follows:

A: Let S be a nonempty convex set in Rn, and  Let f:Rn→R be defined as follows:…

Q: Prove that the set {a E R: a2 not equal to 6} is open.

A:

Q: Let S and T be sets. Prove that SUT = T iff S is a subset of T.

A: To Prove S∪T=T iff S is a subset of T

Q: Let X, Y be two sets. Prove that X × Y C P(P(X U Y )).

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

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Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on…

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Q: Let A = {AC R| A is countable} and Az = {A c R| A° is countable} which is a measurable space? (R,…

A:

Q: prove for every two sets A and B that A=(A-B)U(A n B)

A: To prove: We want to prove  for every two sets A and B that A=A-B∪A∩B

Q: Prove that if A is a nonempty finite set and B is denumerable, then A × B is denumerable.

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

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Q: 1. Prove that for all sets X, Y , if XU Y = Xn Y, then X = Y. %3D

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Q: be any nonempty set and let µ* be an outer measure on P(X) and let A, B e P(X). If at ne of A, B is…

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Q: Show that there exists a set N C (0, 1) that is not Lebesgue measurable.

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Q: 5. Let X be a set and p E X. b) If 72 = {A C X : p ¢ A}U{X}is a topology on X, then determine…

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Q: Suppose that |x| = n. Find a necessary and sufficient condition on r and s such that (x") C (x*).

A: Given,

Q: Theorem 4.9. A topological space X is normal if and only if for each closed set A in X and open set…

A: Consider X,I be a normal space and U⊂X be an open neighbourhood of a closed set A⊂X, then A⊂U To…

Q: 7. Let X be a set with |X| = n € Z4. Prove that for any set Y, |Y| = n if and only if there exists a…

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Q: Prove that P(Sn T) P(S) n P(T) for any sets S and T

A: Let, S and T be a any set. also, P(S) be any power set of S and P(T) be any power set of T.x∈P(S∩T)…

Q: Let E be a nonempty subset of an ordered set; suppose a is a lower bound of E and B is an upper…

A: Let X be a non empty set. Let a∈X. Then a is said to be a lower bound of the set X if a≤x for all…

Q: prove that the outer measure of Cantor Set C (x), 0<x< 1, is 1-x the generalized

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Q: Theorem 2.3. A set U is open in a topological space (X,J) if and only if for every point x€ U, there…

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Q: Show that E is a measurable set if and only if for every E > 0 there is a closed set F and an open…

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Q: Prove that if R, S, and T are nonempty sets, then R×T ⊆ S×T if and only if R ⊆ S.

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Q: Let A and B be sets. Let X = A n B, Y = A \ X and Z = B\ X. Prove that X n Y = Ø, X n Z = Ø, Y n Z =…

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Q: 2. Let X be any uncountable set, Prove that: 7. = (GC X: G is countable} U {0} is a topology on X.

A: Since you have posted multiple question... according to company rule we are supposed to answer first…

Q: Show that Let ECR" is measurable set. If µ(E) >0, then E have a non-measurable subset

A: Here, we introduce the notation Gr :=G∩[0, r) + 1-r ∪ G∩[r,1) - r for G⊂[0,1) and r∈R. Note that…

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

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Q: Identify all nonempty set/s from the following A = {x|x € N, 2x + 7 = 3} B = {x|x € Z, 2x + 7 = 4} C…

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Q: 1. Let T and S be topologies in X. Show that SnT is also a topology in X.

A: 1.

Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

A:

Q: For any set A and B prove that: a) А - В %3D А - (ВПА) b) BC A' iff A n B = ¢ с) (А — В) U В %3D…

A: The given question can be solved as shown in step 2 and 3. As per the guidelines of bartleby I'm…

Q: Let {An : n E N} be the indexed collection of sets defined by 1 2, 2+ 2n An Prove that: nMeN An = Ø

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Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: Prove using Venn diagram that An (B U C) = (A n B) U (A n C)

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Q: Let X and Y be two discrete spaces, then * X is homeomorphic to Y if and only if X and Y have the…

A: X and Y are two discrete spaces then X and Y   The topology is  discrete then for x∈X  is an open…

Q: 4. Prove that the set Z is a closed set in R.

A:

Q: For nonempty sets A, B, and C, let f: A -> B and g: B-> C. If g is onto, then gof is onto.

A:

Q: Show that there must be two distinct subsets A, BC S such that

A: Every value in S is between 0 and 100 , So the sum of any subset will be at least 0 and  at most…

Q: 3. Prove if x is an isolated point of a set S, then x E bd S.

A: Since you have posted a multiple question according to guildlines I will solve first question(Q3)…

Q: Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)}…

A: Given that A is non empty set which is bounded below by 3. Given that B=x∈R|∃a∈A,b=1a The objective…

Q: Prove that S UT = SN T for all sets S and T using the rules for complements of sets.

A: rules of complements x∈A⇔x∉A¯

Q: Let F and G be a non-empty families of sets with SG. Show that ng s UG.

A: Given: F and G are two non-empty family of sets with F⊆G, To show: ∩F⊆∪G