6. Let's say the set X= {1, 2, 3, 4 is given as the whole space and M = {Ø,{3},{4},{1, 2},{3, 4}, {1, 2, 3},{1,2,4}, {1,2,3,4}} is given as the o-algebra. If there is a suitable measure u: M-[0, 1] and (X, M, u) forms a probability space and satisfies u({3})= u({4})= =, explain in detail that 3 u({1, 2, 4})= ³ is established. 4

Q: 6. How many mappings are there from A to B where A={1, 2, 3, 4} and B = {Low, Middle, High}. How…

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Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A: 3.3 is also known as Hausdorff Property. d(x1, x2) > 0 for every x1 ≠ x2   (metric space…

Q: 3. In each case suppose R" has the norm ||l inclusions: = max {ril,., r.}. Prove each of the set (a)…

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Q: 9. Let X = NU {v2, v3} be a set. Consider the topology 7 on X such that T = {AC X : ACN or if A…

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Q: Let A and B be sets. Prove that A ≤ B in each of the following. (i) A = {x²: x € Z} and B = Z (ii) A…

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Q: 9. If (X, M, ) is a measure space and E, FEM, then μ(E)+μ(F) = μ(EUF)+ μ(EnF).

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Q: a) range 1 0 0 11 00 10 b) range 1 1 01 1 c) span ((1,-1, 1), (−1, 1, 1), (1, 1, −1), (1, 1, 1)) d)…

A: Note : '' As per guidelines we will solve first three subparts. If you want any specific subpart to…

Q: 15. If t={X,Ø,{1,4},{4,3},{1,4,a},{b}} is a topology on X-{1,2,3,4}, then: * a=4, b=2 O a=3, b=1 O…

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Q: algebra (sigma algebra) genecated by Crif Eind the smallest on _a) 요.R. cofu20.55).c15.00)} and and…

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Q: 4) Define on the cartesian product of sets R:=R x R = {(^, z) | A, z E R } an addition and a…

A: We first show that R is a commutative ring. That is we show that R is commutative group under…

Q: 10. Let (X, Tx) and (Y, Ty) be two topological spaces, such that X = {a, b,c, d}, Tx = {0, X, {a},…

A: Definition: A function f is continuous from X to Y if inverse image of each open set in Y is open in…

Q: Q1. Let T={x,y,z}. L={Ø,x},{x,y},T} is a o-algebra over T. Prove that: (a) (b) (T,L) is a measurable…

A: (a) Given that T=x,y,z, L=∅,x,x,y,T is a σ-algebra over T. Since every sigma algebra is measurable.…

Q: 5. Let (X, B(R), µ) be a Borel measure space. Suppose that {An = ["+1, 4n-1]}, be a family of…

A: Let (X, B(R), µ) be a Borel measure space

Q: 2] -1 1] a. Find SVD from A 1. A = [4 | b. Find SVD from A invers 2. A = [; : : 8] 1 0 0 0 a. Find…

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Q: Let U, A, B, and C be defined as shown below. U= [ a, b, c, d, e, f, g, h, i, j } A = { d, e, g, h,…

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Q: Please solve 4,5,6,7 The Borel o-algebra on Ris generated by any of the following collections of…

A: As per the company policy, in a multipart question like this, we can solve only the first two…

Q: (A,7) be the topological space defined in question 2. Let A = {1,2,3}, then the set of limit points…

A: Solution: (4) Answer: The set is  A=1,2,3 Clearly, set A is finite.  We know that a finite set does…

Q: 9. If (X, M, u) is a measure space and E, FEM, then u(E)+μ(F) = μ(EUF)+ μ(EnF).

A: Introduction: A measurable function is a function between the underlying sets of two measurable…

Q: Fix a prime number p. For x EQ - {0}, write x = p" () where a, b € Z, a and b are not divisible by…

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Q: Is T ={∅, X, {1, a}, {1, b}, {2, c}, {1, a, b}, {1}} a topology on X = {1, 2, a, b, c}? Why?

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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. Let (X, d) be a metric space and xnx where each x, e X and x e X. Let A be the subset of X which…

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Q: Q1. Let A be a set contained in a given Hilbert space H. State the conditions for A to be…

A: According to the policy, we are solving only first question only. Please resubmit it and specify…

Q: 15. Let x = (x1, x2). Using the feature mapping show that *(2,3)·¤((4,4) –-(2,3)·(4,4)*

A: Given that for x=x1,x2, the feature mapping is…

Q: 3. Let X = {a,b,c ,d, e}, f ={{a,b}.{b.c}.fc.e}.fe}}cP(X) Find the topology z on X generated by B.

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Q: 13. If t={X,Ø,{2,4},{1,4},{1,2,4},{4}} is a topology on X={1,2,3,4}, let A={1,3,4}, then Int(A)= *…

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Q: 1. Decide if the following statement is True or False: %3D Let A1, A2, ..., Ak E Mnxn(R) be such…

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Q: Define complemented lattice. Let (L, *,Ð, ', 0,1) be a Boolean algebra. Then show the following:…

A: Given: (L,*,⊕,',0,1) is a Boolean algebra.We need to define complemented lattice and show that…

Q: c) Let u be a usual topology on a set of real numbers R, and E= {2,3,6 Show that whether E is…

A: Given that R is endowed with the usual topology and that E={2,3,6}. We shall state a definition and…

Q: the topology 7, = {X,ø{a},{b,c}} 1; = {Y.ø.{u}} B={X ×Y,6.{(a,u}…

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Q: 1. define For R", one can consider the following norm structures. For all x = (1,,n) R", we • ==…

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Q: 69. If S,, S, are two subsets of an inner product space V(F) then show that

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Q: Q#6. Given any sets A and B define the symmetric difference of A and B (ΑΘΒ) -(Α-Β ) υ (B - Α) prove…

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Q: 10. If t={X,Ø,{2,4),{1,4},(1,2,4},{4}} is a topology on X={1,2,3,4}, let A={2}, then the closure of…

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

A: Use the following definitions, to prove the result. If a set contains all its interior points, then…

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

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Q: 5. Show the span {(1,0, 1), (–1,2,3), (0, 1, – 1)} is all of R³. 6. Determine whether span {(1,…

A: (5) The given set is 1,0,1,-1,2,3,0,1,-1. The main objective is to show that the span of the given…

Q: 2. Consider the topology 7, = {X,ø,{a},{b,c}}_on X = {a,b,c} and 1, = {Y,ø.{u}} Y = {u,v} . the…

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