4. Let E be a measurable subset of [0, 1) and E+y for y E [0, 1) be a translation modulo 1 of E. Taking into account that under above-mentioned assumption the set E+y is measurable and (E + y = mE, construct nonmeasurable set by a suitable m choice of E.

Q: Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7. (i) Prove that…

A: Given that X,Y,Z are subsets of {1,2,3,...,10} |X|=|Y|=|Z|=7. X∪Y≤10 i)X∩Y≥4 X∪Y=X+Y-X∩YX∩Y=X+Y-X∪Y…

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: Prove that there is no set to which every se Show that for any nonempty set A, there xE Bex belongs…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

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Q: 4. Let E=]0,4[U]5, 6[ . (a) Prove that xo (b) Decide whether E is open, closed or compact, and…

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Q: Let (X, F) be a o – algebra | , and be two probability measures on (X, F) so that both are ergodic…

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Q: 6. Compute the norms for (a)||f()||, (b) ||F(x)||2, (c)||f(x)||, where f(x) = -V1-r", defined on [0,…

A: Given

Q: Theorem 9. The following statements regarding the set E are equivalent: (i) E is measurable. (ii)…

A:

Q: be a collection of sets in S'. Show that we can express U E, as the disjoint union U, Er of sets in…

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Q: 7. Suppose that X and Y are disjoint sets and k E Z>o. Prove that the function k so:U(P:(X) × x…

A: Let us solve step by step.... X and Y are disjoint sets...

Q: Theorem 9. The following statements regarding the set E are equivalent: (i) E is measurable. (ii)…

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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: Consider X = C[0, 1] with the norm = max |x(1)|, IE[0,1] and let T: X - X be the mapping given by…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Prove that ', 12 and lº norms satisfy the properties (N1)-(N3). (N1) (Positivity) ||x|| > 0, and ||x…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Show that the Grundy index Γ′(T) exists for every tree T and that ∆(T)≤Γ′(T)≤2∆(T)−1

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Q: Exercise 6.65. Prove that if E₁ CR and E₂ C Rm are convex, then E = E₁ × E2 is a convex subset of…

A: Given :  E1⊂ℝn and E2⊂ℝm be any two convex sets. To show :  E = E1×E2 is a convex subset of ℝn+m…

Q: Let g(x) and h(x) belong to Z[x] and let h(x) be monic. If h(x) divides g(x) in Q[x], show that h(x)…

A: Given: Let's take g(x) and h(x) belongs to Z(x) and take h(x) as monic.

Q: If vo and V, are respectively the t-conorms that are dual to the t-norms Ao and A5 respectively.…

A: In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary…

Q: of points of S. 6.30 Show that a set consisting of a finite number of points in E is bounded. Gnita…

A: Given, a set consisting of a finite number of points in E.

Q: Suppose (X, S) is a measurable space, E1, . . . , En are disjoint subsets of X, and c1, . . . , cn…

A: Let's suppose that ϕ:X→R defined by, ϕ=c1XE1+c2XE2+⋯+cnXEn So, this function is well defined as…

Q: Every nonempty class C has an E-minimal element.

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Q: Q) Make up a Kelly table for set G= [1,-1, i,-i} with respect to multiptication and then explain…

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Q: Show that X and Y are independent iff E(f o X – go Y)² E(f o X – E(f o X))² for all Borel-measurable…

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Q: Now let G = S_10, acting on the set {1,2,3,4,5,6,7,8,9,10}. What is the index in G of the stabilizer…

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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: Let n, k E Nand define the family of graphs Gnk= (Vn, Enk) on the vertex set %3D Vn = {1,2, ...,n}…

A: Given family of graphs is Gn,k=Vn, En,k on vertex set Vn=1, 2, 3,..., n with edge set is…

Q: 2.6.13 LetE be a subset of a metric space X. Prove that E has empty interior (E° = Ø) if and only if…

A: Dense set and interior set

Q: 4. Let r E R" be such that r = (E1,... , En). Show that the norms ...+ En and = V are equivalent so…

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Q: Let G=(V, E) be a finite

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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 E be a measurable set with m(F np with ISp <∞. Further if Lim (E) then lI f llo = %3D

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Q: 2 49) Given nonempty subsets of R², say Y....., let Y = nin enner-xer Fix P E R². For a nonempty…

A: Given: Non empty subset To find: Solution

Q: 18: Consider the two linear maps T, and T2 on V, defined as T(X, X, x) = (0, xz x) and T,(x , x, x)…

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Q: Let X be a nonempty set and fix x E X. Define (1, rE A (A) = 10, 4 A

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Q: 2.1 Let C C R" be a convex set, with x1, ..., xk E C, and let 01,..., Ok E R satisfy 0i > 0, 01 +…

A: The solution of 2.1  is in given below

Q: Let E and F be measurable sets each having finite measure. Prove that m(EAF) = 0 if and only if m(E)…

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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: Let x, y ∈ Z. Prove that x − y is even if and only x and y are of the same parity.

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Q: [2.13] Let S be a nonempty set in R". Show that S is convex if and only if for each integer k > 2,…

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Q: 2. Consider a measure space (X. A. u) and let s be a nonnegative measurable simple function on X.…

A: We have a measure space χ,Α,μ. Where A is a σ-Algebra and μ is a measure. We are given a function…

Q: Let (X, E) be a measurable space and E e E. If E1 e £ is such that E1 C E and ƒ is measurable on E…

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Q: 12. If E is a bounded above subset of Z, then A. sup E exists and sup E E E B. sup E exists and sup…

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Q: a) Suppose that E is a set. If there exists a function f from E onto N, then E is at most countable.

A: (a) False f:E → INLet E= IR  Thenf: IR → INdefened as f(x) =  1  , ∨ x ∈ IRi.e There exist a…

Q: Let X be an infinite set. Define u : E = P(X) → Ř by %3D if A is finite µ(A) = 1 if A is infinite.…

A: Let X be an infinite set and ∑ =P(X) Let μ=P(X)=∑ → ℝ is given by μ(A)=0if A is finite1if A is…

Q: Suppose that E is an alphabet, and that f :E* → E* has the property that f(ơ) = o for every o e E an…

A: Given information: Given that Σ is an alphabet. The given function is f.

Q: 2. Write a preference E on R as a subset of R', where a bundle a E Rª is at least as good as a…

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Q: Let X = {a, b, c, d, e} and let T ={X, Ø, {a}, {c,d}, {a,c,d), {b,c,d,e}), therm %3D O (X,T) is…

A:

Q: Let E ⊆ R. Show that E is measurable if and only if χE is measurable

A: