1. Let E be a measurable subset of [0, 1) and E+ y for y = [0, 1) be a translation modulo 1 of E. Prove that the set E+ y is measurable and m (E + y) = mE.

Q: 1. (a) Consider the set S {n(-1)" : n e N}. Verify the existence or otherwise of Min(S) and Max(S).…

A: given that (1) (a)                                    S=n(-1)n:n∈N=2k(-1)2k:n=2k,…

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…

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Q: 1. Let Ω = (0, ∞). Let A = (3, 8] and B = (5, 10]. Find the smallest collection of subsets of Ω that…

A: Definition: Let Ω be a non-empty set. Let Σ be a collection of subsets of Ω. The collection Σ is…

Q: 1. Let u, ER" prove that lũ ơi <ll||||||

A: we want to shoe that  u→·v→≤u→v→

Q: 1. (a) Consider the set S = {n(-1)" : n e N}. Verify the existence or otherwise of Min(S) and…

A: a) The given set is,S=n-1n:n∈ℕ

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: (2.1) Let h: Z → Z be a function given by h = |r|, Va € Z. Show that (a) h is not one-to-one (b) h…

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Q: Y= {1, 2, 3} and  Z={k, e}. How many onto function are there from Y to Z

A: Formula: If X has m elements and Y has 2 elements, the number of onto functions will be 2m-2 So in…

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: 3. Let (X, 7) be a Hausdorff space. If X is finite, then show that 7 = P(X), where P(X) is a power…

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Q: Exercise 6.41. Let E be a nonempty set in Rn. Then the distance of a point x € R¹ to E is defined by…

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Q: Show that the function f: [1, 2] → [1,8], f(x) = 7x - 6 is a bijection. What relationship exists…

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Q: Prove that the Frobenius norm is consistent. That is, for A, B € Rxn, show that ||AB|| ≤ ||A||||B||F

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Q: 1. Let f : X → R and Y = f-1(R). Then f is measurable iff ƒ-1({-∞}) E M, f-'({∞}) e M, and f is…

A: see the attachments

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: 3) Let R be the set of all real numbers with the euclidean topology and f: R → R be continuous…

A: Yes, the image of a continuous function from a path-connected space is always path-connected.…

Q: Let X = {o, p, s, w}, Y = {a,b, e, g, k}, and Z = {1, 2, 3, 5, 7, 9}. Let ƒ : X → Y where f = {(o,…

A: X=o,p,s,wY=a,b,e,g,kZ=1,2,3,5,7,9f:X→Yf=o,a,p,g,s,e,w,bg:Y→Zg=g,3,e,2,b,9,k,7,a,1

Q: Exercise 6.44. Let E be a subset of Rn. (i) Prove that E is closed if and only if it contains all…

A: This is a question of Topology.

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: 4. Let EC Rd be a measurable set such that |E| > 0, and assume that ƒ : E → [-∞, ∞] is a measurable…

A: Since |E| &gt; 0 and E is measurable, we can find a sequence of closed cubes  in Qn∈ℝd such that E…

Q: I. Let the universal set U = {1, 2, 3, ..., 10} . Let A = {1,4, 7, 10}, B = {1, 2, 3, 4, 5}, and C =…

A: According to our guidelines we can answer only first three sub-parts and rest can be reposted.

Q: (#4) Show that a set SCR" is closed if and only if bdry(S) CS.

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Q: 6. Let z E C. Let C be set of points z satisfying Im(z) 0 and Im Prove that C = {ZEC : |Z| = 1}.…

A: Given that  Im1+z+z21-z+z2=0 We know that if Imz=0 then Im1z=0 Therefore  Im1-z+z21+z+z2=0 Finding…

Q: Let A, BCR x R defined by A= {{n,m)|n, m € N, 0 < 2m(n + 1) – 3 < 5}, and B = {(1,1), (1,2), (2, 1),…

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Q: Exercise 3.3.20. Let A and B be sets. Suppose that BCA. Prove that AxA-BxB = [(A-B) XA]U[AX (A -…

A: We have B⊆A  first we prove that  AxA-BxB⊆  [Ax(A-B)]∪[(A-B)xA]   Then we prove that  AxA-BxB ⊇…

Q: dw a) Draw a dependence tree for this situation, and find an expression for dr b) Use your…

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Q: 5. Let the functions f: A→B, g:B→C, and h:C→D be defined by the figure below. A В C D f C4 y – 3 a)…

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Q: 4.4.4 List the relation on N defined bv h(x, y) e N x N : x > y } 9 Scanned with CS CamScanner as a…

A: The relation given is already a subset of the Cartesian product N x N. It can be written as follows.

Q: Prove that the Joukowski map 1 p(2) : (z + maps ONTO T - (-1, 1)

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Q: U is a neighborhood of x0 and U ⊂ V, The V is a neighborhood of x0. (2) If U1 ...., U2 are…

A: Solution ::

Q: Let the sets A, B and C be fuzzy sets defined on real numbers by the membership functions: μ₁(x)=…

A: Note: We are entitled to solve only one question at a time and up to 3 sub-parts only. As the…

Q: 3. Show that is a closed subset of R². {(x, y) = R²: y = x²}

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Q: If geRla,b] and if h(w)= gcu) encept %D finite number f points for [a, b],'then and in of prove that…

A: First we have to divide the set into finite subsets of x intervals required.

Q: In R4, let S = span{(1, 1, 1, 1)T ,(1, 2, 1, 3)T }. Find S⊥.

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Q: Let ECR. Prove that m*(E)= 0 implies that E is Lebesgue measurable.

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Q: Let E₁ and E2 be measurable sets (in R). (a) Use that the union of any two measurable sets is also a…

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Q: 2. Let A = {a, b, c}, B = {x, y, z}, C = {r, s,t}. Let f : A → B and g : B → C be defined by: f =…

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Q: 1. Let A={1,3,5,7}. Let f:A→Z be the function f={(1,2),(3,4),(5,6),(7,8)}. Let g:Z→Z be defined by…

A: Given, Set A=1,3,5,7 and a two functions f and g as follows, f: A → Z by f=1,2,3,4,5,6,7,8 g: Z →Z…

Q: Show that p(y]x, z)p(x]z) p(y|z) p(x]y,z) = Show that P(x|z) = > P(x]y, z)P(y|z) %3D Show that…

A: If A and B two events then by definition of conditional distribution, p(A|B) = p(A∩B)p(B) We have to…

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: Q5. Let (X, B(R), µ) be a Borel measure space. Suppose that {A, = ["+1_ 4n-1 Show that A1 C A2 C…

A: