Q. Let (X, B(R), µ) be a Borel measure space. If AnC = ¢, BnC = ¢, µ(An B) = 0. Determine µ(AUBUC).

Q: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

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Q: If X =R and d(x.y)30, then X metric space not metric semi metric

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Q: (b) Is (R2,d) a metric space, where d((x.y1).(x2.y2)) 1x1-x2l; for all (x, , y1 ), (x2 2) E R2 ?…

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Q: Given: YX - XZ, WX bisects ZYXZ prove: AWYX = AWZX

A: Given: Prove:

Q: Suppose that l is a line. Let H be one of the two half-planes bounded by l. Give a proof that l U H…

A: Answer is mentioned below

Q: If U = span {x, y, z} in Rn, then show that U = span {x + tz, y, z} for all values t∈R.

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Q: Let X = R. and d is discrete metric, Find S (2,2) OR O (2) O {0} (2)

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Q: What is the image of the vertical strip  {zϵC:−1<Re(z)<1} under e^z

A: A  conformal map is a one-one and onto analytic map. So, Re(z)=1 to a point. Here f1 is defined by,…

Q: Let (X,T) be a discrete space, then Va e X, the set {a} is а. орen b. closed c. clopen d. is neither…

A: Given problem is :

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

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Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: suppose that S = [a, b] x {0} is a closed line segment %3D in R?. Show that H' (S) = b a.

A: Here given as S=a, b×0 is a closed line segment in ℝ2. We have to prove that H1S=b-a

Q: Let X and Y be two T, spaces, then X×Y. :is

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Q: Date. AC) Lat normed linear space X, Y= spomE and aex only if fla) = 0 wheneres f EX' omd f30…

A: To show the necessary part- To show that for each f∈X' whenever f vanishes on E, i.e.if f=0 then…

Q: Exercise 8.2. Let X, Y, Z be normed spaces and suppose that T E B(X, Y) and S e B(Y, Z). Show that…

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Q: Compute ff, x²ydA,where Dis the triangle bounded by x = 2, y = 1, y = 1 –

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Q: Let X = N and d be the usual metric, d (x, y) = |x – yl then d can induces a norm on X True O False…

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Q: For every x, y E R, let p(x, y) = |x² – y²\. Is p a metric on R?

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: I if é osy =xe, fiel y' by, use Implicite di fferen ional ? %3D

A: Authoring guidelines:…

Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: (b) Let (M, d) be a metric space. Define a function e: MXM-R by e(x, y) = min {1,d(x, y)}; vx,y E M.…

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Q: (f) The set B = {(x, y) E R² : x² + y² < 1} as a subset of R? with indiscrete metric is…

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Q: Consider the sets X = {(x, y) E R²: x² + y² ≤ 1) and Y = ((x, y) = R²: x ≥ 0]. Sketch on R²the sets…

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Q: Let, T:R' → R';T(x,y,z) = (x + y+ z,2y+z,2y+3z) 2. Find the dimension of KerT and ImT

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Q: 2. Let d1, d2 be two metrics on X + 6. Show that: d(r, y) = d1(x, y) + d2(x, y).

A: I have proved all four conditions of metric for d

Q: If (x, y) is any point on the unit circle, and t is the distance from (1, 0) to (x, y) along the…

A: x, y is any point on the unit circle and t is the distance from 1, 0 to x, y along the circumference…

Q: Let r, y EN such that y # 0, then y+r#0.

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Q: 2.1.12. Determine which of the following is a metric on R. (a) d(x, y) = x – y. (b) d(r, y) = (x –…

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Q: (f) The set B = {(z, y) E R² : ² + y < 1} ; is disconnected. as a subset of R? with indiscrete…

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Q: a) Assume that y is a continvoudly differentichle fone b) Celculote tontseetdt + 3.

A: We have to solve these

Q: Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

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Q: Is the set S = {1,, 3, ...} with metric d(x, y) = |æ – y| for all x, y e S complete ? prove your…

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Q: Consider the Simple Linear Model: 4 = Bo + Biz, + e, where i = 1,...,n, and e ~ IID(0, o?). Suppose…

A: Simple linear regression model: A simple linear regression model is given as y = b0 + bx, where y is…

Q: Let (S, d) be a metric space and suppose that ρ:S×S→R is defined by ρ(x, y) =d(x, y)/(1 +d(x, y))…

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Q: d(r, y) Let (X, d) be a metric space and let p(r, y) = for all r, y E X. Show that (X, d) 1+d(x, y)…

A: According to our company policy we are supposed to answer only first question i.e. (4) 4) in this…

Q: Let, T: R' → R';T(x, y,z) = (x+ y+ z,2y + z,2y +3z) Find the dimension of KerT and ImT

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Q: Consider a unit square with four vertices in a two-dimensional Euclidean space, i.e., {(x1,x2)|x¿ €…

A: As per the question we have a unit square and we have to choose two points P,Q inside it uniformly…

Q: If w = f(u, v), u = x + y, and v = x - y, show that w,wy = w - w?. %3D %3D

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Q: In a plane, if line f is perpendicular to line m and then fln. mln f || n n || f m || n

A: In the question we have to choose correct option.

Q: - Consider noncoplanar points A, B, C, and D. Using three points at a time (such as A, B, and C),…

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Q: _Show that the following are metric on R": d,(x,y) max, isn Ixi - vil : d,(x,y) E X-Yil.

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Q: Iw (X, D) discrete space A = {2,3,4s A A Bd(A), Euct CA)? EXercise %3D Fined

A: Given that A=2,3,4 The aim is to find A,A°, Bd(A), Eucd(A) A=2,3,4 And, A°=∅ Here A is closure of A…

Q: 12. Show that d((x1, Y1), (x2, Y2)) = |x1x2- Y1Y2| is not a metric on X = R %3D %3D

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Q: Let d(x, y) = be a metric on R* then the open ball of center 2 and radius is: O ]1/2,1[ O None of…

A: I have used definition of open ball in metric space.

Q: Suppose the matri x A is 27 X 36 and the rank of A is 14. Find the nullity of A a e dimension of…

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Q: 2. Does d(x, y)= (x-y)? define a metric on the set of all real numbers?

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Q: Let X = {a, b, c} Define d: X2[0, 0) by d(a, a) = 0, d(b, b) = 0, d(c, c) = 0, d(a, b) = 1, d(a, c)…

A: Metric Space: Let X be a set and a function d :X×X→ℝ Then X, d is a metric space if d satisfies the…

Q: y (b, f (b)) (a, f (a))

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Q: Show that a triangle with vertices (x,Y;), (x2,y2), and (x;,y;) has area + x,y3 + X3Y, – X,Y3 – X,Y,…

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